TI-83/84 plus stat edIT values of Xin l1 and probabilities in l2. home screen: sum(L. only rarely are games not biased in favor of the house. however, blackjack ...

Enjoy!

Older TI calculator manuals are available for download from The Datamath Calculator Museum.. Canon Canola SX-300 Programmable Calculator. Canon Palmtronic LE-84.. Radio Shack EC-21 (Blackjack 60-3011)

Enjoy!

Right now, there are mostly image changes, although there are some interal changes, but the overall feel of the game has not changed much.

Image changes, weapon changes, speed changes, cheats, and several internal changes as well are included in this version.

This is for people who want to play something different from the original Phoenix.

The game features 3D looking graphics and gameplay.

Trade goods and earn yourself a fortune, or if you prefer, earn your fortune by fighting.

The game has a 'save' feature so that you can finish your playing anytime you want.

And all this comes in a 6 kb program!

You hit the platform the enemy is on, then touch the enemy to kill it.

This classic game has been updated to include 50 new levels, each with its own design unlike the original.

It also adds new enemies based on other Mario games.

Check out this great update to a great classic game.

I don't remember from what platform this game originated, go here its pretty old.

This game features everything from the NES version, which includes multiple difficulty levels, store, 360 degree ship rotation and much much more.

This is a very fun game and is a good timewaster : It's hard to describe it in a few sentences, so just download it and see for yourself.

Believe me, you won't be disappointed.

Unfortunately, this game only runs of 83+ and not on regular 83, but hopefully the author will port it someday.

This version should work with the USB link.

Asteroids is Recommended by CalcG.

You can participate in a season, play an exhibition game, or play in a homerun derby.

It has nice this web page and play control.

But when you press a light though, the lights next to it change as well.

Comes with 11 levels and a password feature.

Great graphics and other features.

The object is to get clusters of balls of 3 or more.

This is harder than you might think, because after a time a wall falls down bringing you closer to death.

Bust a move features several premade levels built in and randomized levels.

Source is now availible under GPL.

Includes 83 regular edition.

Do this for thirty seconds to see how many peices of cotton you can pick.

You can view the world from four different angles.

The game comes with 16 standard levels, and a level editor is included so that you can create your own.

It is an excellent copy.

It has over 100 rooms and dungeons.

Your goal is to save the world from Diablo.

https://allo-hebergeur.com/blackjack/samsung-blackjack-wifi-hacks.html, you must avoid traps and extra features.

Tricky getting used to, but well worth it!

Features lots of enemies, spells, skills, and extras.

There are 7 boards with a total of 46 screens, containing one-way article source, keys, bonuses, traps and various switches.

There are more than 1500 multiple-choice questions and two different modes of play.

It is the complete version.

This game is more difficult then the previous version.

It requires an ION emulation shell to run.

This game is exrtremely difficult, so be prepared.

You must do this for 5 frogs without getting hit or drowning.

Port to TI-83 by Ahmed El-Helw, Port to ION by Sam Heald 40 kB Rating: 7.

Galaxian is Recommended by CalcG.

Dodge the icicles and live as long as possible!

You will have to jump on blocks, climb ladders, and think hard before each jump.

You will double down split blackjack have to play with Spikes!

This game has sound, and now has a new level!

You are an indestructible tank that needs to blow up the enemy planes, helicopters, and miners.

The only catch is that you don't have any weapons.

You have to blast yourself off the enemy fire and into the enemies strategy blackjack chart simple kill them.

The more enemies you kill before hitting the ground, the greater the experience.

This simple concept will keep you trying especially on the insane difficulty.

Each maze is rendered with a different graphical style, and there are two modes of play: maze click antimaze.

You start with 4 mines and work your way up in increments of 4.

You must fire and move lasers to manipulate the different on-screen objects in order to get a clear shot at the iris, which you must shoot twice.

The object is to rotate regions of the board until the various pieces are isolated in the four quadrants.

There are only 5 regions you can rotate so it can get tricky.

This games features everything found in the original plus several new features such as keys and moving platforms.

The program has 25 original levels and a level editor, so that you can make your own levels.

Lode Runner 83 runs under ION 1.

Unfortunately it does not run on the TI-83 plus with Mirage OS emulation and 83+ version wasn't updated as 83 one.

You can download the level editor from our 'DOS Programs' page.

The editor is called LDK, which is an abbreviation of Level Development Kit.

If you make any levels, please email them to Submissions CalcGames.

Lode Runner v2 is Recommended by CalcG.

Has 3rd person behind car with superb graphics.

Lotus Turbo Challenge v1.

It uses the black link cable to transmit data like x-coordinates.

You can move Mario free around in an arena.

You have to shoot the missiles coming down to your towns before they hit them.

It comes with a level editor for creating and editing level sets.

There are 5 different speeds and 10 standard levels.

Two player turn based strategy game where the main objective is to destroy the enemy command bunker with nuclear stikes.

Players can use spies, research weapon upgrades, and anihilate the countryside and each others buildings.

Similiar to the BASIC version, only runs faster and doesnt have animations.

New in Version 3.

There are three different strategies that the bot can use, and an additional option exists allowing the bot to change its strategy.

Play against the calculator.

Orzunoid is Recommended by CalcG.

There are several different types of ground you can or cannot runnover.

The gray walls are not passable.

The clocks with arrows pointing up increase your time, while clocks with arrows pointing down decrease your time.

The Skull kills you if you run into them.

It has almost identical enemies, weapons, ship, and icons.

Note: This game is not fully complete, so it is in a post-beta stage, since it is very playable.

Not all weapons are included yet.

I will also add the enemy routines from PS to it soon.

It is set in space.

There is no real storyline to speak of; the only objective is to blast through each treacherous level as fast as possible.

Phantom Star is capable of supporting external levels, and documentation will be provided in the near future for level developers that wish to make their own levels.

Phoenix is an advanced shoot-em-up game for the TI-82, TI-83, TI-83 Plus, TI-84 Plus, TI-85, and TI-86 calculators.

This game has very smooth gameplay over 30 frames per second with many objects onscreen.

Its features include many different types of enemies, many different levels, the ability to buy additional items, five possible weapons for your ship, game saving, a high score table, multiple difficulty levels, and multiple speeds, and external levels.

The game only takes about 8K of your memory.

It is supplied with full source code, and may be copied or modified without any restrictions.

Phoenix is an advanced shoot-em-up game for the TI-82, TI-83, TI-83 Plus, TI-84 Plus, TI-85, and TI-86 calculators.

This game has very smooth gameplay over 30 frames per second with many objects onscreen.

Its features include many different types of enemies, many different levels, the ability to buy additional items, five possible weapons for your ship, game saving, a high score table, multiple difficulty levels, and multiple speeds, and external levels.

The game only takes about 8K of your learn more here />It is supplied with full source code, and may be copied or modified without any restrictions.

Ion or a fully compatible shell is required to run this game.

Features include: -Grayscale graphics - Unforunately, grayscale flickers somewhat in the TI-83+ version.

You can choose to turn it off.

In this update, there are still 5 levels of difficulty and 5 speeds measured in mpsand some improved sprites.

Your goal is to draw an image according to numbers.

This may sound simple, but with images getting harder and time getting shorter, you'll soon see that this game isn't easy at all.

Port by Ahmed El-Helw and Sam Heald 4 kB Rating: 5.

The object of the game is to connect pieces together in sets of 4 or more, and create chain reactions.

As the game goes on, there are obstructions that fall in your way.

For those of you who are tired of tetris, try this puzzle game.

Includes a two player mode, one player mode vs.

AI, a save feature, Win and pass detection, and is fully graphical.

It fixes many things and more cheats included.

You must fill all white squares with your gray "pen".

Has 50 levels and is challenging.

Port by 213 kB Rating: 7.

Can you make it through 10 crazy levels to escape Space Station Pheta?

Includes a visual walkthrough explaining how to make your own levels.

of blackjack tables more saucer is added for each level you beat.

The goal is to create 3x3 blocks, which will cause the block to disappear.

This version of Squarez is a port of the 86 version.

You control a battle ship casino verite blackjack 5 6 serial try to destroy the submarines below.

This archive was submitted by Chase Taylor taylorchase hotmail.

Save Highscore and custom skin for each piece.

It's a bird eye view racing game, with really nice graphics both tracks and cars are detailed sprites.

YOu can now find: -An IA, that drive 7 other cars -A brake -Many options, and customisations -5 playing modes including a link mode, you can challenge your friend on the macadam Enjoy!

One-player pong with a yellow-white-black paddle on a blue background 12 kB Rating: 6.

Unlike Nibbles, where you can travel in four directions only, Uncle Worm can travel in no less than thirty-two directions.

Collect all the apples to win each level.

Very easy to use and addictive.

It is just like ZKart3D except with less graphics.

Port by Ahmed El-Helw and Sam Heald 5 kB Rating: 4.

Z-Kart 3D is Recommended by CalcG.

It features enemy AI and a decent challenge.

This game has very smooth gameplay synchronized at 25 frames per second with many objects onscreen and parallax scrolling.

ZMercury is supplied with full source code.

The program is in public domain so it may be copied or modified without any restrictions.

The actual gameplay is much smoother than the animated GIF below shows, since the image was made by sampling the screen only a few times per second.

ZTetris was originally written for TI-85, TI-86, and TI-92, and has now been ported to every calculator with assembly capabilities.

Even at this early stage, Ztetris for the TI-83 is already a great game.

Note that this version does not include linkplay, but future version hope to.

It was created back in 1994.

Three Card Poker gets its popularity from its high payoffs, fast paced gameplay, favorable odds, and easy rules.

Three Card Poker's rules are slightly different from that of 5 Card Draw Poker in that the hand ranking a different.

All of the rules are explained in the "read me" file.

This file contains 3 yahtzee games: Original yahtzee, Triple yahtzee, and Poker yahtzee.

All three of these games include a high score feature to save your highest scores.

Features Blackjack, a variation of Roulette, and a variation of slots, all in one TI Group.

ADRPG:TCS has 11 character sprites 4 click you 7 for your enimes the crystal on the title screen spins, there are 5 unique battle animations form a physical attack to dashing.

This game WILL take you a long time, you will be level 104 by the end.

All enimes are based on your character, so you are always challenged.

There is a stat store where you can buy more MaxHP, MaxMP, Speed, Strength, Tents, and change your type!

From the beggining the customization begins, pick your type, subtitulada casino 21 blackjack online add points to diffrent attributes, and when you level up you add a certen amount of points to your character's 5 diffrent magic types.

You can use tents like you can in any other RPG and you ,of course, can save.

This program is updated, it is now faster, cooler, has more features, and NO MEMORY LEAKS!

Unlike its predecessor, however, this game is all about slavery and "Uncle Tom's Cabin".

The file is in BASIC and is unlocked.

Therefore, those who want to create different questions and such will find that it is fun, fast, and easy to do so.

Perfect for class projects and such.

They are: Jumpman Dodgeman Fortuneteller 1 player Pong Slotmachine They are all in one program for easy installation.

There is a score saving feature to source one score.

And yes, there are cheat codes.

They include Tron, Pong, Anti-Tron, and Grabber, as well as the entire AAGames1.

This file is really big, so watch out for your RAM.

This should solve many of the problems I've seen people have with this program.

The idea wasn't mine.

What he finds though is much more, as he makes the desition to put his life on the line for peeople he had barely known, but than very casino style blackjack phrase knows is at risk.

This is a text based game somewhat like BobGame but with a great fighting system, Minigame, and Random Events several options have multiple things that could happen.

I gave them better names.

You select the option that you want to do, and you do it.

Have fun with these games!

While playing the game you will notice that there are two different interfaces: The first one is the real time based economy-mode, where you can build, recruit, scroll, click here war, save of course and all that stuff.

After having set up an army, you will get to know the round based battle mode, which might remind you on games like 'Chess', 'Advance Wars' for the Game Boy Advance or maybe the Nintendo DS Version of 'Age of Empires'.

The game features two types of resources, gold and wood, three different maps, which are fully scrollable, five difficulties, five different unit types of both wide and low range attack, a moderate AI Player and a very rudimentary cheating tool.

I hope you enjoy it.

For detailed instructions just view the Readme.

Note 2 : If you would like to play this game on an Emulator, you may check out Wabbitemu, the game definitely works on this one.

This is a Graphical RPG that includes: 20+ enemies, 6 spells, 6 different weapons and armor pieces and over 100 rooms to explore.

Check the readme to see the new updates.

This is an all text game; with a built in map, which has a blinking cursor to show your location; a few spells, and quite a few enemies If I remember correctly as well as 3 exploreable dungeons.

There are a few bugs in the game.

The labelin system might send you somewhere where you might not want to go.

I plan on coming back and fixing this much later.

This is all in basic, and gives an asm look, though the speed is still lacking.

This is something that will be rebuilt later.

Just like Avalanche only upside down!

Runs very fast for a BASIC game.

But it is still quite challenging!

Includes randomly-generated events, a competitor, and feedback at the end of the week!

Its even funnier if you know the song but you don't continue reading to to get a kick out of it.

The basic plot is that you are a man that decides one day to move to Albuquerque, where he spends the rest of his life.

He does a lot of funny things and a lot of funny thins happen to him.

I hope you enjoy this game.

I certainly did, and I made it!

I fixed all of the bugs, for those of you who previously downloaded it and had problems.

Make the opponent bankrupt if you dare.

You basicly live the life of an Ant.

It is a fast pace and fun classic!

Read more in the Readme.

I think it's pretty good.

You are a common warrior who is trying to prove that he's good.

So learn more here decide to go to the Arena to test your skills.

This features extrememly hard bosses.

Don't do the Compeiton until you feel your ready.

Because it will be hard.

This game features 4 different classes of warriors to choose from.

It's been through many bug tests and no problems found yet.

It has 4 classes to choose from, better training, and 20+ unique enemies to fight in the competition.

This is a much harder game that should provide hours of play time.

So far in this one, you can buy weapons, and attack armies.

If I ever figure out how to do that.

So far all you do in this demo is hire soldiers and see the basic outline of what it will look like.

There will be more to come that will add new features.

You are a guard.

You must guard your base.

Battle increasingly difficult soldiers and eventually a boss as you climb your way up the Evil Tower, or whatever it's called.

It uses a simple battle system where you temporarily make yourself vulnerable to charge your attack.

You can also save your game and pick it up later.

This was a game I made a long time ago.

Have fun with it!

You and up to five other players battle in the mountains with tanks.

Each player can start out with money to buy different powerful weapons.

There are a total of SEVEN different weapons, including nukes, MIRVs, lasers, and cruise missles.

You can play in one of two modes, Free for all or Team play.

No one around to challenge?

You can have up to six tanks controlled by the calculator.

You can also have teams of Human controllers and Robots combined!

They are ASpider, which is like Avalanche, 'cept you're a spider, and Pong, which is a really primitive version that you should only look at if you are learning BASIC or something.

It runs fast for basic and it works well.

It has a two player mode a bit slower but works and different levels including one with a moving enemy.

The only difference is that you can not bust and the highest hand you can have is 9.

You do not need a lot of skill to play baccarat and it is easy to learn how to play.

To learn how to play just read the document that is found in this file.

This game automatically saves your current score and your highest score.

Unlike most Mine Sweeper clones this one features great graphics and runs very fast.

Also this game has the option of changing the size of the map and number of mines, which most basic clone don't.

Updated - Only one file so you can easily send it to your friends!

Fixed all errors, and once again optimized for final release!

I did the orginial coding dut Morgan Davies helped me debug and recode it.

Much credit to him for the help.

Read the readme for instructions on gameplay and other info.

You can also compete in an 82 game season against 5 other calculator players.

This game is very fun and challenging.

Added a nother new attack, and added in more bonus features.

Also, a new ending.

Enemies are much harder btw.

Welcome to the 2nd version of Battle Tower.

Fight the 4 different guys.

Before you can go past them, you must fight them.

Each attack differs with power and accuracy.

Use your own stratagey to kill all 4 of these monsters.

Review and rate this game :.

This game includes several hours of gameplay, a huge number of weapons,shields, and armor, several purchasable skills, semi-graphical fighting, cutscenes for the storyline, all in just 40 KB!

Flash Gordon was used to prevent all the RAM from being used up, so all you need is 10~14 KB for this game.

This one continues the story from BattleQuest 2.

It is much better than any previous release, including: -56 different enemies.

This game redefines the Battlequest series.

Master might and magic in 10 floors loaded with over 20 kinds of enemies and 5 final bosses!

I did my best to avoid generic and overused enemies like goblins, orcs, and the like.

Please leave a review, I'm relatively new to programming and I like to know what you all think.

If you liked the original, you will love this.

I have had permission to use the source code.

I revamped the entire game.

It includes a battle system and a later included random battle sequence.

The game is based on the Beowulf universe and is set in a space theme which will be added in later.

Most of the stuff is pretty original though.

Check it out, it takes 14 KB on the calc.

Seeing the opportunity to leave the zoo and return to his home in the Congo, Billy the Monkey escapes and sets out on an adventure to get back to his home.

This game is rather addicting, and requires no user knowledge of the game of bingo.

Pure BASIC, packed into a small filesize.

It used to say the last total when you got 10, J, Q, or K as the first card.

Fixed that bug and took out some If then commands.

A very very decent black jack game, with no erros that I know of.

I made it last night because I was really bored and wanted to play around with the graph.

It is actually very different from my other black jack games.

All new graphics, bigger cards, new layout, everything.

No menu in the beginning though, because I wanted to keep this game compact.

There is a cheat button, and there is a button to turn off the cheat aswell.

As always, there is a high score section.

And please review my file.

It uses my own shuffling program from "Dealer Pack".

A home menu screen that isn't ordinary.

Donwload this small game.

It will appear in the MIRAGEOS folder.

This is Black Jack Pro, this game tracks your money, wins, losses, and total winnings.

This game keeps track of the 10 total cards in play, out of a simulated deck of 40.

Isn't a deck of cards supposed to have 52?

Well would you rather see "KH" or "0H"?

It's easer on the programmers and players to know that their card is worth 10 rather than know that the "K" is wrth 10 as well as the "0", "J", and "Q".

This game weighs in at a smooth 6000+B.

Why so much you ask.

Well there is some sort of animation for EVERY screen.

All the menus "raise" up and all but some "fall" down.

It shows the cards flipping, and it also shows a neat "menu squshing" thing for the main menu.

You can save your game and recall it later, also you can load a game while in the middle of one!

This is the NON CODEX version.

The Codex version had so many bugs and errors that it just wasn't fun.

I also switched some stuff around, and added in some things.

Check out the screenshot to get a good idea of how the game works.

It's a fun betting game, and now it isn't just seeing if you can get close to 21, now you got the dealer's first card, and you have to try and beat it.

You never know what the dealer will have, and I also made the dealer much smarter, by placing in some extra if then statements, making it hit again if it's too far under, making the dealer much harder to beat.

The rules aren't casino style.

There is no double down, yet.

Please take the time to rate this file, and write a review.

I would like to know what you, the people who play these games, think of it.

Very simple game easy to use no major bugs but email me if there are any!

Also If you want to advertise this on a nother web site please notifly me or make sure we get credit!

Have Fun and Send reviews!

Inclued Files- BJack - 1807b Readme 1.

This version, instead of being text-based is fully graphical.

There may be a bug or two, contact me at darksideprogramming yahoo.

This has no bugs, as far as I know.

You might want to archive some of your other programs before you play BLASTER!

Try and find the right angle and magnitude needed to hit the target!

Score increases relative to the number of stars on the screen, which means at around 400 points scoring will be negative.

I wanted it to be a fast paced scrolling game, but it failed in the process, so this is what turned out to be.

Not really much bob sledding, more of just dodging the walls.

The are tricky wall turns that come in and out.

You have to prepare to dodge them.

For some reason, the point system isn't working.

I will fix that bug soon enough.

This is the olympics.

Get the gold metal, and you win the game.

I'm going to try and make this game a little more challenging.

Have fun for right now.

Tell me if there are any problems.

If it's too easy, email me at jonny23451 yahoo.

It involves a very stupid man named Bob, and you just have to help him live.

It includes items and even a frogger-like mini-game.

There are 4 different endings, and has lots of replay value.

This game is in basic and requires no shells to play, and takes up about 5000 of your memory.

This is also the first Bobgame not by JcCorp.

You control a dumb stickman, Bob.

You have to guide him all over the city to defeat his arch-enemy: George.

This is a text adventure.

The game is a bunch of menus where you pick something for Bob to do.

There are dealer arrested river blackjack twin endings.

Try to get them all!

If you like this I have many others being worked on!

Sorry about the late release- I guess the first upload failed.

Anyway, this is a text game that has our "hero" Bob the stick man trying to find his arch-enemy, George.

This game is funnier, longer, more challenging, and more entertaining than the first.

You really don't need any prior knowledge to play it from the first game.

Help him survive as he traverses around the world, just like the previous two games.

There is only one true ending, but many death scenes, and one "special" death scene.

You really don't need any prior knowledge to play it from the earlier games.

It is easy to use, and provides hours of entertainment for avid Boggle players!

Definitely worth a download!

You're character is already moving back and forth.

You can change it's direction by pressing the opposite direction it's going in.

You're just shooting a feta sp sign.

I made this game in an hour in my chem class, so it's not graphical and there's no story.

Just a time waster I guess, and it is good for you people who want to learn how to make things move by itself and shoot.

You stay alive as long as possible, but the left wall closes in on you!

It is quite fun.

After all of the bricks are formed, all you have caliva blackjack review do is press any button.

Press 2nd to Pause, ENTER to resume.

The website, as usuall, is not up and running yet.

Read the readme to learn how to unlock this game's TEACHERPROOF features and all other upgrades.

If you like BRICK, there is no reason not to download this game, except if you like larger file games with less features.

I personally think this is my best considering size, playablility, and TEACHERPROOF software game yet.

Simple math questions are displayed and you must correctly answer as many as you can within twenty seconds.

It's a turn based fighting game made entirely on the GC language and only during school hours.

Since it's turn based, the game requires two players, but its fun to play against yourself anyway.

Anyway, it features all your favourite overpowered anime characters bashing each other upside down.

The damage system started out very simple but its gotten really complex in the last year or so with all the new characters we pumped in.

Graphics are limited because we only have 9 pictures to work with and we have to draw pixel by pixel.

Ideal for lecture halls and math classes.

Current Character List: Inuyasha Son Goku Naruto Give the guy a week or two.

Roronoa Zoro Kurosaki Ichigo Bruce Banner A big joke Superman Cloud Strife Train Heartnet 6 kB See more 9.

It includes games such as slots, red dog, blackjack, keno, non-text based horse racing, and hi and low.

It also includes a save feature to save your cash.

The horse racing is very addictive.

It doesn't sound like much, but my "Beta Testers" math nerds I gave it to like it more than Frogger as do I.

If you get a higher score than 42, please tell me emailsince that is my highscore.

As you play, which you'll need to more than once, I guarantee it you will reveal the somewhat random storyline, and the stragety and process needed to complete the game.

But I really hope you give it a quick try, and see how it is.

I know it works.

Trust me, I've seen text based games that DONT work The bigger question is.

Well, I hope that later Text Based games I just click for source down the road WILL be more fun.

This is just my first step.

None the less, I hope you enjoy this game quite well, even if it you only truly need to play it once.

If you like it easy, then play this game.

It tells you where all the hidden buttons are, and the money is so much lower than are rivers casino chicago blackjack rules frankly />Everythings only 50 bucks!

If you get below master, I should shoot you.

The new engine offers True save game ability, enhancing your gaming experiance!

It is played against a computer player and it includes a high score counter.

The objective of the games is to go through college and survive.

You can go to parties or class.

Get laid or blackjack dealer reddit live to class you're the one in control.

If wanted, more versions will come.

It features winner detection, but only if you end the game: press the number you want to put your stone in, and press "ENTER" if somebody has won the game.

It is a remake of the classic NES game Contra, and is the most advanced BASIC platform game to date.

Contra 83 is the true meaning of BASIC.

It does not use 1 ASM utility, just strictly good old home fashioned BASIC.

The game uses one of the most advanced engines ever using complex numbers and equations to generate unreal looking terrain, and an AI program that will not slow even if 1000 enemies were on a screen.

It is in basic obviously and it is quite big https://allo-hebergeur.com/blackjack/888-casino-online-blackjack-live.html please bear with me.

Easy to figure out and comes with a readme.

Please make sure that Codex and Crater9 files are unarchived.

Just run the Crater 9 file and begin the game.

However it is a friend of mine's program, and I have his consent to create the readme and post the program as long as I cite him as the author, duh!

This is a short, yet insanely funny RPG written in basic for the TI 83+, 84+.

It contains some screen shots, but it is mostly text-based.

Written during the long boredom that is known as math class, this program provides a sharp, random, and sarcastic humor that I haven't found since Monty Python.

Its only downfall is that it is short, but that won't last plan on updates or sequals to increase the insanity!!!

And it even gives a random hint for easy transferring.

This one includes an arcade-ish format only repeated battles, no breaks3 party members, healing spells, a boss, two endings, and cheat codes.

I made this a long time ago, and I found it quite recently.

Curse of the Dragon 2.

A is a text RPG that is smaller in size compared to 2.

Fight through 50 enemies of increasing difficulty to reach the boss- an evil version of you!

There's no story, by the way.

It features game play just like the real thing, a beautiful graphical interface, saving of the top 10 scores, and only one file to run!

This game is VERY ADDICTING!

Fully graphical it provides hours of fun and can be played over and over again.

It follows the exact click the following article of the gameshow and keeps track of your 10 highest winnings.

For those of you who have downloaded DEAL OR NO DEAL by Kevin Murtha, this program is less than half the size, 2x faster, and has many more improvements.

I have not copied any of his work and have made my program completely from scratch.

Features four different enemies, a shop, archers, spells, and is quite fast for a BASIC game.

If you enjoyed the original, this is a must-have!

In this game, you play against 2 other computer controlled players.

This game keeps track of your stats and winning %.

Just read the Text file to learn more about how to play.

The object of the game is to pick a door, open it, and get a car.

There are three doors, one with a car in it, and one with a donkey in it.

If you get a donkey, you loose.

rivage blackjack tournament schedule you get a car, you win and can keep playing to get more cars and even a high score!

This game is all about luck, and is very addicting.

Try it for yourself and see!

Features a 10x10 grid and easy-to-use interface.

Look for the next one, it is also a Microrise production.

OOoo, the mysterious shroud of mystery.

This game is nothing like what you've played before.

This is not just a simple 'shoot-the-duck-yeah-now-what' game.

It is a RPG exposition game where you compete in 5 levels, untill you claim the trophy.

Everything depends on where you choose to hunt, the weather, and the temp.

Walk around to find the perfekt spot, but becareful not to run out of time, or to drive the ducks out of the area.

This is by far the most descriptive duck hunt game ever, with upgrades in your rifle and your satchle.

You can even buy duck calls if you feel you need the extra boost.

I enjoy all critics and comments.

Tell me what you think.

Uncover the mystery of, Echoes.

This is the first game but surely not the last from Hurricane Games.

The final game will be a TBS in Basic.

The trailer is the first of several the rest will be released along with the game.

It is meant to give you a taste of what is to come.

It is fully animated, with many greyscale pictures for enhanced graphics.

It now works fine.

Endless Descent is a game in which you must stay alive as long as possible.

Keep moving off the platforms and stay low as long as you can.

There are two versions: 1 84+SE, where the game is timed.

View the readme for more details.

This game is a very in depth battle re-creator with RPG elements.

It is very similar in that you battle in real time with real-time graphicsbut level up in an RPG-esque manner.

The more you battle, the more Exp.

You need to buy stuff to have any chance in the game.

The entire point is to defeat the evil Raydoh.

THIS IS AN EARLY VERSION!

The story has not been programmed, and the interfaces will be very different.

You will not choose when to battle in the final version as you do in this one, and you will be free to move from room to room.

This version has 1 room, and only 3 types of enemies.

You will need to beat around 1000 enemies to level up enough to beat Raydoh.

I NEED FEEDBACK TO FINISH THE FINAL VERSION.

I NEED STORY SUGGs, NAME SUGGs ETC.

This game takes up 2000 bytes of RAM and to run it correctly, about 18000-20000 bytes of RAM is needed for it to run smoothely, the more the merrier.

WHEN YOU DOWNLOAD THIS READ THE README.

This game also comes with a map editor, which is easy and simple to use.

DETAILS: size: 2000 bytes needs memory: 20000 RAM 4 enemies: Dalok Gaurd Flyboy Auto Turret Winding hallways MAKE YOUR OWN LEVELS!!!

It runs on mirage os or by itself.

You must do what your master asks of you.

You must please him, obey him, and help link in all ways.

Note: This game comes with 2 files, explorer and squire, send both to your calc, and run the game under the prgm called explorer.

There are a few ways to get near the end but only one takes you to it.

Try to avoid the death mines and get to a high level.

There is no ending, there are just more death mines.

I can only get to level 7.

You choose how you want the piece to fall and catch them.

Sounds just like regular falling numbers right??

After every number you catch, the floor rises towards the top of the screen!!

NOW try to beat it!!

Simple, but very addicting.

Takes up less than half a kilobyte and includes a scoring system.

Visit my website to download other great games at: www.

Kinda like Falldown, but not falling down.

All you have to do is move out of the way.

There is also a scoring section and a ranking system.

I love the difficulty menu, it's just.

You dodge the walls that are randomly set up.

It's kinda like Falldown, except you don't fall down and it doesn't scroll.

Anyways, you try and get past all the obstacles.

This version doesn't have moving walls.

The car moves instead.

In the second version, the walls move, and the car stays still.

Good to learn from.

No graphics yet hint: text based : 28 kB Rating: 4.

This one should not have memory errors, unless you have too many programs unarchived.

Text based game in visit web page you start as a newby in a gang and try to make it to the top assassin.

Very well thought out took about a year to develope.

It's worth a try.

The only things currently not working is magic and itmes in battle.

With the help of ASM utilities and technology of Flash ROM, this game goes beyond the limits of BASIC programming.

And thanks to the Flash ROM technology, this game were able to squeeze more options, longer gameplay, and with Devpic program, each chapter includes more than 40 pictures of displays Total of over 200 pictures in a full version!

This means the movies over an hour of it!

In gameplay, it includes everything that the original Tales of Magic had to offer plus more!

During battle, you can perform dozens of awesome magics to cripple your enemies or perform devastating combo attacks to unsuspected foe.

Also, with newfound power, your character can transform into even more powerful being to cast even stronger Magics and Combos!

New in this sequel, you or your enemies can attack with status effecting spells to better https://allo-hebergeur.com/blackjack/full-tilt-blackjack-rules.html chances of winning or losing.

During off-battles, you can explore through many different terrains and even travel through time to fufill your tasks.

And also, you can interact with many Non-Player Characters NPCseach with their own unique personalities.

All this in full graphical form, none of that text RPG B.

Enuff talking, download the game and see for it yourself.

For more details about this program, visit us in our project Homepage.

By taking advantage of Flash ROM in TI-83 plus, this program delivers more graphics, awesome effects, more options, and longer gameplays than a conventional TI-83 Basic RPGs.

Let the game take you even deeper into the story with more than an hour of beautifully made animated movies.

Explore through vast terrains from the forest, cave, highlands, and Empires.

Unleash devastating Magic spells and attacks on to 40 different enemies.

New in the full version, early memory error detection allows you to continue the game without any data losses!

Based solely on Graphics.

Full list of fratures for FFR2 12 modes of play User Made level support Random Levels The ability to make fully custom levels with Custom Arrows Custom Backgrounds Custom Detection Bar at top Custom "ratings" i.

Custom "Life Bar" also the FFR2 group includes another bonus!!!!

This game has the support using ths "num pad" for 2 arrow hits at once!

Seemingly simple at first, the difficulty of the game rises as the numbers get larger.

A good test of your memory that has bonuses, highscores, a security system, and that is surprisingly fun, you may find yourself replaying this game over and over.

It's a sequel to Fredgame, and like it, is a parody of the Bobgames.

Featuring more funny deaths, fewer "ripoffs" of other people's jokes, TWO versions one standard and one only for TI-84+ SE, due to size and an actual storyline!

Well, the storyline not so much.

BUT IT'S WORTH IT!!!!!!

It is enormous 11565 in sizemuch larger than my other stuff, and all 11565 pieces of it are chock-full of hilarious funtime stuff just like the Bobgames.

Well, have fun and I hope you enjoy it!

Thats about it so enjoy!

My game "Catch" which may not be on Calcgames yet but it will be is better, and this still has a few bugs, but I will fix them.

This is a BASIC programmed RPG about a girl named Gypsy's day through school.

Battle your way through 10 levels of evil opponents and transfer your completed data to GADX, the sequel, after conquering the Percussion Dictator!

This is a collection of programs, but it runs nearly the same as one program.

It sports a unique battle system, as well as unique characters.

Best of all, you can upload your save to the sequel.

This is a BASIC programmed RPG about a girl named Gypsy's day after school.

Battle your way through 10 levels of evil opponents in the final chapter of Gypsy's day.

Afterwards, conquer the optional boss!

This is a collection of programs, but it runs nearly the same as one program.

Be warned, however, that you cannot progress through this game without beating the first one, GADV.

Play through the first one first.

You can rob banks, go on missions, go to the library, or just wal around twon exploring.

To any one who knew Game V1.

If you find a bug please contact me O one more thing Game V1.

This sets up the high score system and with out it you will get an ERR message when playing the game.

Snake is my version of snake where your tail is indefinitely long and the walls shrink, finally there is pilot wings where you try to get the best score.

The object of this game is to guide George through the game to defeat Bob, George's arch enemy.

It's a lot better, trust me.

It has a COOL animation that plays when you get a high score.

Has in-game scorekeeping, too!

THIS IS A MUST DOWNLOAD!!!!!!!

Note: This is a repacked archive that contains a patched version of prgmKEWLSTUF 7.

If you don't think such a thing is possible, read on!

First off, there are six number ranges to here from: 1-10, 1-100, 1-1000, 1-10000, 1-100000, and 1-1 million.

And if that's nothing, there are five play modes: 1 Player, where you guess the number yourself, 2 Player, where two people compete to guess the number, AI:Easy, where the calculator will compete with you, AI:Hard, where the calculator will use clues from your guess to make a better one, and AI:Moderate, a difficulty in-between.

That's a total of 30 different ways to play!

In addition, it tells you how many guesses you had in the 1-P Modes, and has high scores, which you can also reset.

The file size is less than 4k, and this game is compatible with MirageOS.

Four modes of gameplay, Friendly user interface.

Requires all programs in group to be unarchived.

But I turned it into a workable game.

You are trying to rescue the "mothership" or whatever you want it to be from the kamakazi pilots.

This game was kinda the beta for Gunned Down, but I decided to stay with it and make it into a game where you control 6 different cannons, and the enemy comes into one slot but you don't know which slot it is.

You press the numbers on your keypad 1-6, and it will shoot out a cannon, letting you attack the enemy pilot.

I put this game rather quickly because of time, and it gets laggy after awhile.

There are multiple difficulty settings, and it's pretty basic.

A complete description is available in the readme.

Feel free to use the assembly libraries that come with the program however you want as well.

Screenies will be available soon on variant watch celebrity blackjack online congratulate site, as well as my new game that I am currently wworking on.

It took me 2.

In this game your mission is to go and hack the internet to get money for Upgrades.

I tryed to make the format close to a computer as possible with passwords and Realistic internet connection.

To start off it has great graphics, tied in with a perfect horizontal screen setup which enables easy gameplay.

Its features include: 1 and 2 player mode, over 220 words for 1 player mode, a maximum of of 44 chars, low memory requirements, and graphical menus.

One of the special things about this clone, is that on the Outer Limit Software site it tells users in detail how to create their own word lists for others to play with.

That way the words will never run out, and the program will not become obsolete.

In this game, you are given the opportunity to duel the AI your Calc by throwing amazing spells at it.

A millenium after the First Xenocide, humanity is in chaos.

The only hope lies in unifying the Hundred Worlds.

Includes an external AI that can be modified or replaced, as well as an AI Developer's Guide.

Avoid hitting the walls while being pulled down by gravity and last as long https://allo-hebergeur.com/blackjack/top-3-blackjack-odds.html you can!

Just like fly ribbon on a cellphone.

I don't have much to say, it's jusk like the other games by me.

Oh, it's only 2.

It asks you questions and it decided if your a hick, but with alot of choices, it launches you to another menu.

Some examples of thing you can do: fire nukes, scavenage for clothes, or throw spitwads at teachers.

There are no graphics, because i'm too lazy.

It's 9000 bytes though, but worth it.

And heres a advertising song: I wrote it me-self "I'm a hick and thats okay, i sleep all night i shoot all day.

It is, in my opinion, the most different and maybe the best, due to it's actuall mission, but it's still funny, along with the other hick games.

In case you missed out, and to get some reason why I made this other than I was boredcheck out HickTest, hoboquest, Escape from hick Island before playing this.

But since it's only 3.

When I get off my lazy butt and we get into some review, I might make an hickwars 2, but don't count on it, I wouldn't, thats for sure.

But i'm getting off subject, try it.

Please note, this version is being released because it's compatable with the expansion, which should be out soon.

So if you want the origional hicktest, that looks more hicklike.

It's your afterlife, but not a normal view of your average, dipsy-do afterlife.

I didn't have time to finish it, but after I just played it, I started laughing at my own work say what you will.

It's about 60% done.

Also, you need the a to play.

And furthermore, please read the readme.

This continues until you lose.

Or do you want to show your friends that you are faster???

Now you can do it!

With "How fast are you?

And it also got a Highscore system!!

It saves your 3 best records with name!

Just download How fast are you!

And please tell me your personal record!

The new cool thing this game has you do.

Turn your calc 90 degrees clockwise!

The controls are simple there is only movement for now, and there are 4 levels.

You can do multiple air jumps and there is really smoth scrolling that follows your character.

IMPERIAL has all the usual RPG elements including: - A large and well animated world which you can move your hero freely in.

Infinite Madness is as it sounds -- it's one Mad game!

You start out with 1000 spaces to move, and you rarely find that you get any of those 1000 spaces back.

A dot moves randomly on the screen and your objective as the faithful square is to get that little dot to hit you either on one of your edges or directly in the center.

Every step counts, so keep a good eye and only move when absolutely necessary!

Also, whenever the dot here off the screen, you get an extra point.

If it hits you on the side, you get 10 points.

And if it hits smack dab in the middle, you get 50 points and an extra 10 spaces to move.

You start out with 1000 spaces to move, and you rarely find that you get any of those 1000 spaces back.

A dot moves randomly on the screen and your objective as the faithful square is to get that little dot to hit you either on one of your edges or directly in the center.

Every step counts, so keep a good eye and only move when absolutely necessary!

Also, whenever the dot goes off the screen, you get an extra point.

If it hits you on the side, you get 10 points.

And if it hits smack dab in the middle, you get 50 points and an extra 10 spaces to move.

It takes up very little room and is very addicting.

This collection of JcCorp's games includes: Super Pong, Sword of the Dragon, Bob Game, Sliding Puzzle, and the previously combiner blackjack fix graphical RPG called ARPG.

Please read the readme files for instructions for each game.

This Pak saves memory and can be run from the APPS menu, and it won't take up your RAM.

Included in this sequel are: AAGames, ASpider, prgmBOREDOM, BobGame2, and Curse of the Dragon 2.

It saves you about 20000 bytes of RAM!!!

It is about making a colony in north america.

You can choose where you want to build your town.

What to grow there and some other stuff.

It is pretty fun and has three modes.

And a new Difficulty menu.

Thanx to Anthony Tondola, for giving me the idea of a menu like this.

When you finish the game, and go back to the menu, and try playing it again, it goes crazy and says you lose right from the start.

The final updates are set.

I fixed a couple of errors that I didn't catch in the last release, and I made a different ranking system, which is at the end of the game.

Now, when you quit out of the game, your shields go to 0, so you fail.

You still have money and points though, but your score won't be as high as what you want.

Have fun, and please give it a try.

Added a rating system at the END of the game, where it says Game Over.

The awsome screenies were made by Merthsoft, and the Group was uploaded to a comp by Tim Huddle 49.

The game is currently going under much redoing, but it should look identical after it is finished.

Not any tree, lawrence is a cranky old tree who thinks all young trees today are reckless and generaly act un-tree like.

This game is my personal favorite.

You are Pi-Man, defender of the galaxy, I guess.

Please give us any suggestions, comments, or tips for this.

It is a 5 card stud poker game.

A player must make 3 equal bets and create poker hands from 3 cards that he is dealt and 2 cards from the community pot.

Before the community cards are dealt the player must "let it ride" or recall one of the 3 bets.

This game saves your cash and highest score.

To learn more on how to play read the how to play text file.

Basically what you do is, you're babysitting a baby and everythings going fine.

Then all of a sudden, the baby goes missing.

You need to find out what happened and get the Lighthouse to turn on!

There are specific items you'll need to win.

This game is pretty short but fun.

There are two endings.

The worst ending is pretty obvious.

You are a stick man and god is mad at you.

It is very fast and has increasing difficulty.

If any of your letters are in the correct space, a square will surround it.

If any of your letters are IN the word but NOT in the right space, a circle will surround it.

In one player mode, you must correctly solve five puzzles in a row a word will be picked at random from the list of 100 words that comes with the game.

In two player mode, each player enters their OWN secret word which the other player has to guess.

Points are rewarded according to how quickly you solve the puzzle.

The first player to reach 10 points wins!!

Running Ladders Enemy with good AI Coin Collecting Digging Holes On-Calc Level maker Support for 5 USER MADE LEVELS!!!!!!!

Support for external 10 level "level pack" Much Much More!!!!

Leave your behind and welcome to the future!

While avoiding ghosts that try to eat you, by the way.

Solve your way through a tutorial and 13 levels.

Lollipop 2 is here!

This is not the same game as Lollipop and Lollipop 2, but.

Continuing with the theme of lollipops, you must make use of planks to bridge the gap between stumps as you hop your way to the other side, where a guess what?

No ghosts this time to eat you though!

And if you complete all 40 levels, a special prize will be waiting for you!

Lucky Penny is video poker with some twists.

The game also automatically saves after you quit so you can choose to continue next time you play.

The trophy system is a great way to see how much you've accomplished in the game.

Certain accomplishments will earn you specific trophies.

See if you can earn them all, and then check your trophy page to see if you can earn the coveted "LP Champ!

You never win or lose, but funny things happen to Donnie which includes him getting hurt or in big trouble.

You don't know who or why.

This text game will give your brain thinking cramps.

It's all one big puzzle!

It trakes what you do and saves it.

So if you do the wrong thing you lose!

Since this is so hard I will help you.

Download this for fun, challenge, of because you want to support CDI Games from going under!

It will ask you to think of a question and press enter.

There are 8 or 9 random answers.

It has 12 different outcomes.

The responses are always random.

This program can also be run in Mirage or Crunchy, so it is protected from resets.

This is a very good program that during testing, a lot of people liked it.

Give it a shot and feedback would be appreciated.

The first is guess which all you do is guess the number that the calculator is thinking of.

Magic 8 Ball is all truth telling If you belive in it!

It has not 1, but 3 different storylines you can choose at the 2nd question.

This game is very funny.

If it asks you for your name, the answer it wants is "MAILMAN" as seen in the first screen shot.

You get 15 turns to move around the game board while you compete in a series of 5 mini games Shy Guy Says, Memory Match, Tug of War, Crazy Eraser, and Bowser's Big Blast to earn coins.

But beware, Bowser is out to steal them from you and keep you from getting a high score!

UPDATE: In this second update we've included full trophy support!

You can now earn up to 15 trophies for certain accomplishments throughout the game including the "Mario Party Champ!

Do you have what it takes to become the ultimate Mario Party champ?

UPDATE: In the first update we improved the code to make the game run more effeciently without the need for some of the loading screens.

You no longer have to wait 30 seconds for the game to calculate your score in "Crazy Eraser" and the time it takes for shuffling the cards in "Memory Match" has been substantially decreased.

This update also included a special surprise for after you complete the 15 turns of the game.

A few bugs, but pretty good.

Contains the old versions for sake of comparison.

Try it out, it's much better.

Black pegs are given for correct digits in the correct places, and white pegs are given for correct digits in the wrong places.

Don't expect too much, because I'm just a 12 year old who started programming a week ago.

Several years in the making, because I am lazy this RPG provides some cool graphics and animations all using BASIC.

The main value of it being in BASIC is that all you other programers out there can see what I did to make the animations.

Also, the story line is good and it adds to the Matrix story from the movies.

Copies of this program may be distributed, as long as I am cited as the author.

Contains the program and 5 picture files.

However, this game make money blackjack online 5 new story plots, and 2 new authors.

You can even enter your Name and Town at the beginning and it will be even more fun!

There is no way to lose this game either short of choosing to resign.

Please read the readme as it explains all the technical details.

This is a package containing all five of the games in my Maze series.

It includes the original Maze; the much-improved sequel creatively titled Maze 2 ; an alternate-graphics version of Maze 2 called Maze 2 Negative; Maze 2 Easy, which includes some new levels and some of the easiers levels from Maze, running in the Maze 2 'engine'; and the exciting Maze Adventure, which brings four new levels and a linear progression structure.

If you've played the original Maze and found it a bit dodgy, please give this ago, as the later games are much better.

Features new to Maze 2 include a pause key 2ND to pause, any key to resumethe chance to view the level before you start moving press any key to beginthe ability to go off the edge of the screen and appear at the opposite edge not ripping off snake 2 in any way, of course ;-and of course some new levels.

This time there is more than one screen so you probably blackjack ti 84 plus be able to solve it in 5 jouer blackjack comment le like the previous versions.

This could probably be changed for other uses too.

So feel free to modify it.

I figured it would be a bit too challenging for the snake's tail to progressively get longer, so it doesn't.

What does add difficulty, however, is that for each piece of fruit you eat, you leave waste behind you which stays there for the rest of the game.

Your waste is as deadly as the walls: You touch either, you lose.

Use the arrow keys to move.

To keep the game good and fast, you are quite able to go backwards into yourself, which will kill you plenty.

Pressing any button other that the arrow keys ends the game as well.

Post any glitches you find in a review.

THIS VERSION WORKS PERFECTALLY--- This is the first complete Metroid game for ANY TI calc in ANY language.

Utilizing the xLIB app by tr1p1ea, and using everything from pics to lists and matrixes, archives programs and almost all your RAM Metroid certanlly is massive.

It takes up well over 30000B at the moment and will soon take up more With 38 matrixes stored in 6 programs and 3 bosses 2 of which are well known, 4 diffrent powerups, secret passages, and unlimited enemies this game is going to take you ages to beat, or about 30-45mins.

Date hot girls, increasing social status, and possible death are only some of the pros of this game.

You have to open all slots except the ones that contain mines.

If you open a slot with a mine you will lose.

MINE3 has many advanced features: You are able to create levels with various shapes and sizes.

You can shape the levels in different rectangular shapes as long as each screen contains 15x9 slots.

The levels can have between 135 and 1215 slots!

That is up to 45x27 slots or 9 screens!

You will choose how large part of the slots that will be mines.

You can mark slots with a flag if you think it is a mine.

If the first slot you open is a mine a flag will appear instead you can not lose the first thing you do.

There time counter allows you to see how fast you can finish the levels.

You can pause your game while running pauses the time counter.

You are able to save you progress and continue next time you play the game.

You Get to look at a map but only for a short time.

The map can be wrong from time to time.

Contest: Get a chance to make next level.

The first level starts with four mines and two are added for each level that you progress.

Features a scoring system and quick map generation.

RAWR :P 2 kB Rating: 7.

Well this game's for you.

The snake moves at an amazing speed but it's your blackjack ti 84 plus to direct it towards the pixel!

It even features Pause!

Minigolf gives you the power to putt from the comfort of your own home with just click for source unique and challenging holes.

An innovative putting system allows you to control the power and direction of your ball with ease.

Includes a great scoring system that keeps track of your highscore for each individual hole.

The screenshots below are on a TI-83+, so Minigolf will run even faster on the 83+ SE or 84+.

Compatible with MirageOS, Doors CS, and click shells.

But since I made the buildilngs so tall to simulate a city, I made the enemy go left and click to see more, and then down.

Not that great of firing, but atleast it works.

You have to fire on time.

You have to time your firing very precisely.

I'm still adding things to this game, so this isn't the complete version right now.

I will be making a sequal where you can buy stuff and get a house and stuff soon Look for it!

It hasnt improved much besides the language thing.

This game doesn't really have a plot.

You go through your day, dieing in many ways, whether its at a Metallica concert, Disney World, or a your bathroom!

I hope to make a better game, a Mr.

This is similar to Win Shauna, but you must fight to win money, and there are other differences.

The fighting is pathetic.

There are 2 versions.

In the first one, you can't buy a ring, and therefore can't win!

I'm still working on it.

You can beat the 2nd version.

There are a few more bugs though they blackjack card trainer software destroy the gameplay.

It takes a while to find out how it works, but you can have alot of fun with this by showing it to your friends and family.

Ever play my Game Pak?

There are HOURS of gameplay right here!

There is Nightmare Snake 1-3 and IN, and Lines 2-4 and IN.

The IN stands for INverse.

Snake is a game where you have a neverending tail except for 2 and you collect dots except 3whenever you get a dot the screen shrinks by 1 pixel in all directions.

In Lines you move in the 4 cardinal directions except 3 and avoid the randomly appearing lines.

In 3 you move back and forth and collest a dot.

This is the most fun you can have in school.

Most of these names should be used only for entertainment only.

There are now 1800 different names that can be generated If my math is correct.

Please disregard the website advertised hopefully I will soon get it posted.

Fun for a NASCAR fan.

Well, in NeoBoxing you can duke it out with a friend and see whose the better man without embarassing bruises.

Includes: NeoBomber, Blackjack, a Fortune Cookie program, an organizer, and wallpaper generator.

It has its high points and low points.

If you're bored and want a text RPG to play, then go for it.

It will keep you entertained.

Moves as fast as BASIC allows, very challenging.

Have fun with Nib-83!

I personally perfer this snake game rather than most others, it is a well-developed game.

Each player controls a country with even resources, such as money and nukes and spies and ABMs and buildings.

There are nine building types in the game, each with a different purpose.

The object is to destroy the enemy command bunker.

Your enemy's territory remains hidden until you send spies to recon.

The spies are also good for sending to destroy enemy missle sites and silos, as well as factories, and whatever structure you want to sabotage.

The structures include:Nuke silos, ABM sites, warehouses, factories, spy centers, command bunkers, radar stations, cities and farms.

The main object of the game is to research better weaponry through your cities and upgrade your spy skill, as well as launching nukes at your opponent, to destroy his structures.

It is based off a tutorial game by Arasion.

I've added a change number feature as well as 1 Player mode in which a random number is generated and 2 Player Mode in which you take turns guessing and changing the number.

For those who dont know the game, it is indepthly desribed in the readme, even including an example game to get you to understand it better.

Even better yet, its quick and addictive!!!!!!

E-mail me for bugs or ideas on impovments.

I can do anything in basic, so dont be afraid to ask!

INCOMMING: Duck Hunt Expo: more than its predecessors.

You control the PacMan and your mission is to collect all the points on the screen.

If you do that you will advance to the next level.

Watch out for the ghost though!

If they catch you, you will lose a life.

This PacMan has 15 different levels!

It gets harder for every level you accomplish.

It is quite hard to complete every one of them.

The game also has two different speed settings.

Each time you quit your progress is saved and you can later continue on that level you last time were playing at.

Play head to head against the computer.

Includes a season mode.

A story where I promise you, you won't survive the paranoia.

Note: This is the beta.

Note that xLib v.

No pictures are used.

Plasma-man is a basic game that takes up very little room, and runs pretty fast.

The object of the game is to shoot falling objects before the splat into the earth.

Level progresses every ten objects shot, and you get to use a "no miss" laser 3 times.

PLEASE read the read-me before-hand, or the game will mess up.

Try to fill the board with plus signs and get the highest score possible while avoiding the asterisk who erases them.

This beta now includes: A scrolling map, instead of the 8x16 segments you used to walk through screenshot of old system in zip The ability to change border styles for dialogue and such This is just mainly to show I am still alive.

I have had made many many many improvements to this game since this version has been released, but at this time is in an unreleasable form.

Keep tabs on this project at TI-Freakware, United TI, Omnimaga, or Cemetech so you stay up to date on the latest developments on this project.

Please note the included animated screenshot is faster on calculator 83+SE, 84+ and 84+SE than in the screenshot.

It is the closest remake that i could do in BASIC.

If you play games solely for graphics this isn't the game for you but if you loved the original game this is the game for you.

You are the omega symbol and your rival is the pi symbol.

Everything seems to work as of now but the Poke mart in pewter city isn't running perfectly The first pic is the poke mart and the second is the entrance to Viridian forest.

The third pic is the Poke center in pewter city.

The fourth is the space ship in the pewter city museum.

Even though this is svenska blackjack spel online first official program don't go easy on it.

Judge it as if it was created by a master programmer.

The whole point of the project was to point out that programmer's are lazy.

Whenever they come up with some promise of a decent Pokemon game, they have always failed to deliver.

In the past, there has only been ONE exception to this rule.

It's quite inspiring, actually.

As busy as tifreak8x is, he manages to put time in over the years carefully optimizing his program and such.

His goal is different, however; I believe he's just trying to prove a pure-BASIC though that term is sort of being skewed game can be incredible as well.

So, this is a battle simulator, you get to pick one Pokemon of your choice out of six.

Of all the six Pokemon, they have 22 attacks at their disposal to knock each other silly with, ending the battle when you forfeit or a Pokemon loses blackjack ti 84 plus it's remaining HP.

What you will eventually catch on to, is that the battle plays out very similarly to Pokemon Red and Blue, all they way down to attacks, types, effects, and statuses!

For those of you that DO read this, feedback is imperative.

I listen to it a lot more than you think.

Write a review, that feedback is useful because everyone sees it.

Tell friends about my game.

They may have comments as well.

If you have any particular Pokemon you want to see in here sooner rather than later, I can manage that.

But this year's tournament is different from the last.

Not only you have to battle out Pokemons but also Digimons!

Pick betweed 10 of the most famous Pokemons to Train, sleep, feed, play, bet on fights, or even register into tournaments.

Unlike other Pokemon games, you're now able to have not only one but two Pokemon fighting side by side at the same time!

This mean you can unleash super team up combo attacks, Team up special attacks, Team up combo special attacks, and finishing moves!

And the best thing about it is that it's Fully Graphical!

There are original functions which does not find in others games of Poker.

The bet system is like in the casinos.

And you have to go to the bank to have cash.

You start off by pressing click the following article or left to get the ball casino blackjack style rules />The nearer the edge the ball bounces the more the bouncing angle will be.

This makes the game a lot funnier.

The longer the ball stays in the game the faster the ball moves which means it gets harder.

With the speed settings you can set the start speed of the ball and how much the speed increases each time the ball hits a paddle.

Each time you quit the scores are saved and you can blackjack strategy card surrender continue will secret to win blackjack time you are playing.

So that the ball looks like a ball andthe paddle does to.

Whil playing the game you will notice that the longer the game goes the faster your ball and paddle will move.

Have fun and I hope you enjoy : Updates: Fixed acceleration issue with Ball and paddle.

No longer should get randome domain errs in the menu.

Notes from the Author: Read the Readme it has all of the instructions that you need for the game to work.

A TI - 83 version is on the way.

Important Features: 1 No Bugs!!!

Ion, Mirage 6 Constantly in development - any bugs discovered will be eliminated ASAP Consult included readme for more info.

Unfortunatly, my TI link software disconnects when I try to take a screenshot during gameplay so no additional shots can be provided for the moment.

It allows you to shoot portals and move between them.

Those who have played the official version will remember GlaDos.

And She's back in the calculator version!

GlaDOS is your narrator and guide and is the one that will monitor all your tests.

But when the tests are over.

It includes four simple games that can be very entertaining during a dull Algebra class.

The best part about the program is that I did not lock it- it is free to edit.

In fact, it is an excellent way to help you learn BASIC!

This game has very simple instructions included, so no problems should occur.

Yeah, they're simple, but they are certainly fun.

This is a must download if you want to learn BASIC!

With 40+ jokes, all hilarious except for a few cornysthey will keep you entertained for a long time!

But you ask, they're jokes.

If you play a game too often-it will get boring.

So, in Math class, why not try a joke or two?

No, seriosuly, it helps!

I gave this program to alot of my friends, and they laughed out loud-trust me, they're funny jokes.

Some jokes are old funnies, a few I made up myself, and a few hilarious ones most of you have never heard of!

Be sure to check out erklärung blackjack screenshots!

It doesn't have any of the includes, though.

Includes how to make your BASIC program impossible to edit and how to have it run on OS's.

It has also been optimized for OS's so it doesn't get glitchy.

It's long and well developed and should provide you with lots of fun and playing time Alex spend whole summer working on this game.

The game takes around 16k of mem on the calculator.

The game is text based and plays similarily to drugwars games.

The menus and other graphics are very fast and responsive.

Last year it was my first game, and I was proud.

Now, since I fixed it up a little, it is probably my 24th program that I have made.

The other programs are not on this site.

They're little, like tricking your friends and stuff with a blinking cursure and stuff like that.

But anyways, this is a new and improved version.

Fast frames, still a little blinky.

But I took out the Label and stuck in a While loop, because back then I didn't know how to use While loops.

Much faster than before.

If you like dodging things while moving a vehicle around, this is the game for you!

And if you like the competition for the highest score, this game is for you.

It gear blackjack exactly the oppoiste of RacerX.

Instead of going up, you're going down.

Watch the little animated screen to get a feel for this game.

If you feel as though you won't like it, then don't get it.

If it seems as though it looks interesting, atleast give it a try.

NOW WITH TWO PLAYER SUPPORT!!

Im not done with it yet, but I hope to be soon.

Win the contest, and you will win the heart of Maid Marian.

Lose, and you will become the center of a public hanging by Prince John and the Sheriff of Rottingham!

It has one and two player modes.

It's actually kind of fun and addicting after a while.

Choose your weapon and then face off against the calc.

Will you be able to overcome probability and random chance to become an RPSX master?

Unlike other calc versions of rock, paper, scissors, RPSX features great graphics and a custom menu so the action never gets dull.

RPSX also features a great save system so you can keep track of all your stats, even after exiting the program.

Best of all, it's compatible with MirageOS and CrunchyOS for easy access!

Visit my website for more great games at: www.

That's in fact the good old rock, paper, scissors game.

But there is a change.

The loser must play to the Russian wheel.

It's dangerous to play to this children game.

There are several weapons and some potions, 3 different levels, and a boss.

Great for killing time and having fun.

Also has good stat tracker, keeps track of everything from gold, weapons, weapon equiped, and potions to kills and deaths.

It is the first thing even remotely similar to a "graphical" RPG that I have tried to make.

This will likely not get any further than what it is at now seeing as I dont have a storyline and just made it to see if I could do it.

You can use the code for whatever and modify it if you want, I dont mind.

I made this around 6 months ago, around when I first learned BASIC.

It was never completely finished, but it is fully playable with Training enemies to fight, and one part of the story mode.

Also, this game uses a stats attribute, in which you raise your own stats.

It is a fully graphical RPG that includes 25 different areas screen shots 3 and 4 are 2 of the areasbattling screenie 1 is the battle screena shoppe, a full storyline with 5 main quests, and more!

Instructions are self explanitory.

Please disregard the website address.

Gain experience points in the game by winning.

Lose once and it's game over.

Some point in the game you will have gained enough experience points where you can start to cheat, and the more experience points you gain, the better chance you have at getting away with cheating, which will give you even more experience points than winning fairly.

The computer player sometimes sticks to the same thing; is that a glich or is it a trick?

At the end rack up points to show your friends more info your the master as your personal high score is stored in the calc.

Points given are determined by the number of times you win, sucessfully cheat, and how many total rounds you played.

Basically like Boppit, you must press the right keys in order to progress.

Expect this game, even though the same style as Jata Data's previous games, to have more meat.

Definitely a good download!

Complete with ranking system.

High score update coming soon!

Details in the readme.

In fact, this game is so huge I am not sure it will work on a standard TI-83+.

I don't know how big their memories are and the MAIN game is bigger than Fredgame.

The BOSS fight is ANOTHER 5000 some bytes or whatever they are.

Seriously, though, this game is worth it.

It is more in-depth and challenging, involves an alien invasion, and has two Pokemon-parody battles.

The only downside is that you have to play the original School Survival in order to play this.

However, the original is just as fun and if and when I make it the REAL sequel won't need any passwords.

This means you can play it even without beating the original game.

It is still recommended, however.

Because the original rocks.

Now the king needs see more to slay the Defenders of the castle.

Are you up to the task?

In order to complete the objective, there are three stances which you and your opponent may use to win: duck, stand, and jumping.

Each stance has strengths and weaknesses, so use each wisely!

There are six puzzles and a puzzle-maker mode built in.

Simple to learn, a challenge to master!

TI-83/84 plus stat edIT values of Xin l1 and probabilities in l2. home screen: sum(L. only rarely are games not biased in favor of the house. however, blackjack ...

Enjoy!

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Generally considered the bible for serious blackjack players, Peter Griffin's classic work provides insight into the methods and numbers behind the development of today's card-counting systems.

It contains the most complete and accurate basic strategy.

Generally considered the bible for serious blackjack players, Peter Griffin's information blackjack game work provides insight into the methods and numbers behind the development of today's card-counting systems.

It contains the most complete and accurate basic strategy.

Generally considered the bible for serious blackjack players, Peter Griffin's classic work provides insight into the methods and numbers behind the development of today's card-counting systems.

It contains the most complete and accurate basic strategy.

The THEORY of BLACKACK The Compleat Card Counter's Guide to the Casino Game of 21 PETERA.

GRIFFIN ::J: c: z G' a z ~ en Las Vegas, Nevada The Theory of Blackjack: The Compleat Card Counter's Guide to the Casino Game of 21 Published by Huntington Press 3687 South Procyon Avenue Las Vegas, Nevada 89103 702 252-0655 vox 702 252-0675 fax Copyright 1979, Peter Griffin 2nd Edition Copyright 1981, Peter Griffin 3rd Edition Copyright 1986, Peter Griffin 4th Edition Copyright 1988, Peter Griffin 5th Edition Copyright 1996, Peter Griffin ISBN 0-929712-12-9 Cover design by Bethany Coffey All rights reserved.

No part of this publication may be translated, reproduced, or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without the expressed written permission of the copyright owner.

TABLE OF CONTENTS FOREWORD TO THE READER 1 INTRODUCTION 1 Why This Book?

This book will not teach you how to play blackjack; I assume you already know how.

Individuals who don't possess an acquaintance with Thorp's Beat The Dealer, Wilson's Casino Gambler's Guide, or Epstein's Theory of Gambling and Statistical Logic will probably find it inadvisable to begin their serious study of the mathematics of blackjack here.

This is because I envision my book as an extension, rather than a repetition, of these excellent works.

Albert Einstein once said "everything should be made as simple as possible, but no simpler.

However, I recognize that the readers will have diverse backgrounds and accordingly I have divided each chapter into two parts, a main body and a subsequent, parallel, blackjack chart odds appendix.

Thus advised, they will then be able to skim over the formulas and derivations which mean lit- tle to them and still profit quite a bit from some comments and material which just seemed to fit more naturally in the Appen- dices.

Different sections of the Appendices are lettered for con- venience and follow the development within the chapter itself.

The Appendix to Chapter One will consist of a bibliography of all books or articles referred to later.

When cited in subsequent chapters only the author's last name will be mentioned, unless this leads to ambiguity.

For the intrepid soul who disregards my warning and insists on plowing forward without the slightest knowledge of blackj ack at all, I have included two Supplements, the first to acquaint him with the rules, practices, and terminol- ogy of the game and the second to explain the fundamental principles and techniques of card counting.

These will be found at the end of the book.

Revised Edition On November 29, 1979, at 4:30 PM, just after the first edi- tion of this book went to press, the pair was split for the first time under carefully controlled laboratory conditions.

Con- trary to original fears there was only an insignificant release of energy, and when the smoke had cleared I discovered that splitting exactly two nines against a nine yielded an expecta- tion of precisely .

Only minutes later a triple split of three nines was executed, produc- ing an expectation of .

Development of an exact, composition dependent strategy mechanism as well as an exact, repeated pair splitting algorithm now enables me to update material in Chapters Six, Eight, and, particularly, Eleven where I present correct basic strategy recommendations for any number of decks and dif- ferent combinations of rules.

There is new treatment of Atlantic City blackjack in Chapters Six and Eight.

In addition the Chapter Eight analysis of Double Exposure has been altered to reflect rule changes which have occurred since the original material was written.

A fuller explication of how to approximate gambler's ruin probabilities for blackjack now appears in the Appendix to Chapter Nine.

A brand new Chapter Twelve has been writ- ten to bring the book up to date with my participation in the Fifth National Conference on Gambling.

Elephant Edition In December, 1984, The University of Nevada and Penn State jointly sponsored the Sixth National Conference on Gambling and Risk Taking in Atlantic City.

The gar- gantuan simulation results of my colleague Professor John Gwynn of the Computer Science Department at California State University, Sacramento were by far the most significant presentation from a practical standpoint and motivated me to adjust upwards the figures on pages 28 and 30, reflecting gain from computer-optimal strategy varia- tion.

My own contribution to the conference, a study of the nature of the relation between the actual opportunity occur- ring as the blackjack deck is depleted and the approxima- tion provided by an ultimate point count, becomes a new Chapter Thirteen.

In this chapter the game of baccarat makes an unexpected appearance, as a foil to contrast with blackj ack.

Readers interested in baccarat will be rewarded with the absolutely most powerful card counting methods available for that game.

Loose ends are tied together in Chapter Fourteen where questions which have arisen in the past few years are answered.

Perhaps most importantly, the strategy tables of Chapter Six are modified for use in any number of decks.

This chapter concludes with two sections on the increasingly popular topic of risk minimization.

It is appropriate here to acknowledge the valuable assistance I have received in writing this book.

Thanks are due to: many individuals among whom John Ferguson, Alan Griffin, and Ben Mulkey come to mind whose conver- sations helped expand my imagination on the subject; John Christopher, whose proofreading prevented many ambigui- ties and errors; and, finally, readers Wong, Schlesinger, Bernhardt, Gwynn, French, Wright, Early, and especially the eagle-eyed Speer for pointing out mistakes in the earlier editions.

Photographic credits go to Howard Schwartz, John Christopher, Marcus Marsh, and the Sacramento Zoo.

To John Luckman "A merry old soul was he" Las Vegas will miss him, and so will I.

Rosenbaum I played my first blackjack in January, 1970, at a small club in Yerington, Nevada.

Much to the amusement of a local Indian and an old cowboy I doubled down on A,9 and lost.

No, it wasn't a knowledgeable card counting play, just a begin- ner's mistake, for I was still struggling to learn the basic strategy as well as fathom the ambiguities of the ace in "soft" and "hard" hands.

The next day, in Tonopah, I proceeded to top this gaffe by standing with 5,4 against the dealer's six showing; my train of thought here had been satisfaction when I first picked up the hand because I remembered what the basic strategy called for.

I must have gotten tired of waiting for the dealer to get around to me at the crowded table since, after the dealer made 17 and turned over my cards, there, much to everyone's surprise, was my pristine total of nine!

At the time, I was preparing to give a course in The Mathematics sloth cafe london Gambling which a group of upper division math majors had petitioned to have offere4.

It had occurred to me, after agreeing to teach it, that I had utterly no gambling experience at all; whenever travelling through ~ J e v a d a with friends I had always stayed outside in the casino parking lot to avoid the embarrassment of witnessing their foolishness.

But now I had an obligation to know first hand about the subject I was going to teach.

An excellent mathematical text, R.

Epstein's Theory of Gambling and Statistical Logic, had come to my attention, but to adequately lead the discussion of our supplementary reading, Dostoyevsky's The Gambler, I clearly had to share this experience.

What the text informed me was that, short of armed robbery or counterfeiting chips and I had considered thesethere was only one way to get my money back.

Soon, indeed, I had recouped my losses and was playing with their money, but it wasn't long before the https://allo-hebergeur.com/blackjack/blackjack-5-card-charlie-rule.html swung the other way again.

Although this book should prove interesting to those who hope to profit from casino blackjack, I can offer them no encouragement, for today I find myself far- ther behind in the game than I was after my original odyssey in 1970.

I live in dread that I may never again be able to even the score, since it may not be possible to beat the hand held game and four decks bore me to tears.

My emotions have run the gamut from the inebriated ela- tion following a big win which induced me to pound out a chorus of celebration on the top of an occupied Reno police car to the frustrated depths of biting a hole through a card after picking up what seemed my 23rd consecutive stiff hand against the dealer's ten up card.

My playing career has had a sort of a Faustian aspect to it, as I began to explore the mysteries of the game I began to lose, and the deeper I delved, the more I lost.

There was even a time when I wondered if Messrs.

Thorp, Wilson, Braun, and Eps- tein had, themselves, entered into a pact with the casinos to deliberately exaggerate the player' s o d d ~ in the game.

But after renewing my faith by confirming.

Why then should I presume to write a book on this sub- ject?

Perhaps, like Stendahl, "I prefer the pleasure of writing all sorts of foolishness to that of wearing an embroidered coat 2 costing 800 francs.

But I do have a knowledge of the theoretical probabilities to share with those who are in- terested; unfortunately my experience offers no assurance that these will be realized, in the short or the long run.

Shaw's insight: If you can do something, then you do it; if you can't, you teach others to do it; if you can't teach, you teach people to teach; and if you can't do that, you administrate.

I must, I fear, like Marx, relegate myself to the role of theoretician rather than active revolutionary.

Long since disabused of the notion that I can win a fortune in the game, my lingering addiction rock rocksino blackjack to the pursuit of solutions to the myriad of mathematical questions posed by this intriguing game.

Difficulty interpreting Randomness My original attitude of disapproval towards gambling has been mitigated somewhat over the years by a growing ap- preciation of the possible therapeutic benefits from the intense absorption which overcomes the bettor when awmting the ver- dict of.

Indeed, is there anyone who, with a wager at stake, can avoid the trap of trying to perceive patterns when confronting randomness, of seeking "purpose where there is only process?

Not long ago a Newsweek magazine article described Kirk Https://allo-hebergeur.com/blackjack/freeslots-com2x.html as "an expert crapshooter.

Nevertheless, while we can afford to be a bit more sympathetic to those who futilely try to impose a system on dice, keno, or roulette, we should not be less impatient in urging them to turn their attention to the https://allo-hebergeur.com/blackjack/earn-money-blackjack.html trials of blackjack.

Blackjack's Uniqueness This is because blackjack is unique among all casino games in that it is a game in which skill should make a dif- ference, even-swing the odds in the player's favor.

Some will also enjoy learn more here game for its solitaire-like aspect; since the dealer has no choices it's like batting a ball against a wall; there is no oppo- nent and the collisions of ego which seem to characterize so many games of skill, like bridge and chess, do not occur.

Use of Computers Ultimately, all mathematical problems related to card counting are Bayesian; they involve conditional probabilities subject to information provided by a card counting parameter.

It took me an inordinately long time to realize this when I was pondering how to find the appropriate index for insurance with the Dubner HiLo system.

Following several months of wasted bumbling I finally realized that the dealer's conditional probability of blackjack could be calculated for each value of the HiLo index by simple enumerative techniques.

My colleague, Professor John Christopher, wrote a computer program which provided the answer and also introduced me to the calculating power of wiz odds blackjack device.

To him lowe a great debt for his patient and priceless help in teaching me how to master the machine myself.

More than once when the computer rejected or otherwise played havoc with one of my programs he counseled me to look for a logical error rather than to persist in my demand that an elec- trician be called in to check the supply of electrons for purity.

After this first problem, my interest became more general.

Why did various count strategies differ occasionally in their recommendations on how to play some hands?

What determin- ed a system's effectiveness anyway?

How good were the ex- isting systems?

Could they be measureably improved, and if so, how?

Although computers are a sine qua non for carrying out lengthy blackjack calculations, I am not as infatuated by them as many of my colleagues in education.

It's quite fashionable these days to orient almost every course toward adaptability to the computer.

To this view I raise the anachronistic objec- tion that one good Jesuit in our schools will accomplish more than a hundred new computer terminals.

In education the means is the end; how facts and calculations are produced by our students is more important than how many or how precise they are.

Fascinated by Buck Rogers gadgetry, they look forward to wiring themselves up like bombs and stealthily plying their trade under the very noses of the casino personnel, fueled by hidden power sources.

For me this removes the element of human challenge.

The only interest I'd have in this machine a very good approxima- tion to which could be built with the information in Chapter Six of this book is in using it as a measuring rod to compare how well I or others could play the game.

Indeed one of the virtues I've found in not possessing such a contraption, from which answers come back at the press of a button, is that, by having to struggle for and check approximations, I've developed in- sights which I otherwise might not have achieved.

Cheating No book on blackjack seems complete without either a warning about, or whitewashing of, the possibility of being cheated.

I'll begin my comments with the frank admission that I am completely incapable of detecting the dealing of a second, either by sight or by sound.

Nevertheless I know I have been cheated on some occasions and find myself wondering just how often it takes place.

The best card counter can hardly expect to have more than a two percent advantage over the house; hence if he's cheated more than one hand out of fifty he'll be a loser.

I say I know I've been cheated.

I'll recite only the obvious cases which don't require proof.

I lost thirteen hands in a row to a dealer before I realized she was deliberately interlacing the cards in a high low stack.

Another time I drew with a total of thirteen against the dealer's three; I thought I'd busted until I realized the dealer had delivered two cards to me: the King that broke me and, underneath it, the eight she was clumsily trying to hold back for herself since it probably would fit so well with her three.

I had a dealer shuffle up twice during a hand, both times with more than twenty unplayed cards, because she could tell that the card she just brought off the deck would have helped me: "Last card" she said with a quick turn of the wrist to destroy the evidence.

Then she either didn't or did have blackjack depending it seems, on whether they did or didn't insure; unfortunately the last time when she turned over her blackjack there was also a four hiding underneath with the ten!

As I mentioned earlier, I had been moderately successful playing until the "pendulum swung.

The result of my sample, that the dealers had 770 tens or aces out of 1820 hands played, was a statistically significant indication of some sort of legerde- main.

However, you are justified in being reluctant to accept this conclusion since the objectivity of the experimenter can be called into question; I produced evidence to explain my own long losing streak as being the result of foul play, rather than my own incompetence.

An investigator for the Nevada Gaming Commission ad- mitted point blank at the 1975 U.

I find little solace in this view that Nevada's country bumpkins are less trustworthy but more dextrous than their big city cousins.

I am also left wondering about the responsibility of the Gaming Commission since, if they knew the allegation was true why didn't they close the places, and if they didn't, why would their representative have made such a statement?

One of the overlooked motivations for a dealer to cheat is not financial at all, but psychological.

The dealer is compelled by the rules to function like an automaton and may be inclined, either out of resentment toward someone the card counter do- ing something of which he's incapable or out of just plain boredom, to substitute his own determination for that of fate.

Indeed, I often suspect that many dealers who can't cheat like to suggest they're in control of the game by cultivation of what they imagine are the mannerisms of a card-sharp.

The best cheats, I assume, have no mannerisms.

Are Card Counters Cheating?

Credit for one of the greatest brain washing achievements must go to the casino industry for promulgation of the notion 6 that card counting itself is a form of cheating.

Not just casino employees, but many members of the public, too, will say: "tsk, tsk, you're not supposed to keep track of the cards", as if there were some sort of moral injunction to wear blinders when entering a casino.

Robbins Cahill, director of the Nevada Resort Association, was quoted in the Las Vegas Review of August 4,1976 as say- ing that most casinos "don't really like the card counters because they're changing the natural odds of the game.

Card counters are no more changing the odds than a sunbather alters the weather by staying inside on rainy days!

And what are these "natural odds"?

Is not this, too, as "unnatural" an act as standing on 4,4,4,4 against the dealer's ten after you've seen another player draw four fives?

Somehow the casinos would have us believe the former is acceptable but the latter is ethically suspect.

It's certainly understandable that casinos do not welcome people who can beat them at their own game; particularly, I think, they do not relish the reversal of roles which takes place where they become the sucker, the chump, while the card counter becomes the casino, grinding them down.

The paradox is that they make their living encouraging people to believe in systems, in luck, cultivating the notion that some people are better gamblers than others, that there is a savvy, macho per- sonality that can force dame fortune to obey his will.

How much more sporting is the attitude of our friends to the North!

Consider the following official policy statement of the Province of Alberta's Gaming Control Section of the Department of the Attorney General: "Card counters who obtain an honest advantage over the house through a playing strategy do not break any law.

Gaming supervisors should ensure that no steps are taken to discourage any player simply because he is winning.

Books of a less technical nature I deliberately do not mention.

There are many of these, of varying degrees of merit, and one can often increase his general awareness of blackjack by skimming even a bad book on the subject, if only for the exercise in criticismit provides.

However, reference to any of themis unnecessary for my purposes and I will confine my bibliography to those which have been of value to me in developing and corroborating a mathematical theory of blackjack.

An Introduction to Multivariate Statistical Analysis, Wiley, 1958.

This is a classical reference for multivariate statistical methods, such as those used in Chapter Five.

BALDWIN, CANTEY, MAISEL, and McDERMOTT.

Joumal of the American Statistical Association, Vol.

It is remarkably accurate considering that the com- putations were made on desk calculators.

Much of their ter- minology survives to this day.

Playing Blackjack to Win, M.

Barrons and Company, 1957.

This whimsical, well written guide to the basic strategy also contains suggestions on how to vary strategy depending upon cards observed during play.

This may be the first public mention of the possibilities of card counting.

Unfortunately it is now out of print and a collector's item.

Braun presents the results of several million simulated hands as well as a meticulous explanation of many of his computing techniques.

Theory of Gambling and Statistical Logic.

New York: Academic Press, rev.

There is also a complete version of two different card counting strategies and extensive simulation results for the ten count.

What is here, and not found anywhere else, is the ex- tensive table of player expectations with each of the 550 initial two card situations in blackjack for single deck play.

There is a wealth of other gambling and probabilistic information, with a lengthy section on the problem of optimal wagering.

ERD jS and RENYI.

On the Central Limit Theorem for Samples from a Finite Population.

Conditions are given to justify asymptotic normality when sampling without It is difficult to read in this untranslated version, and even programming a blackjack game difficult to find.

Probability Theory and Mathematical Statistics.

Optimum Strategy in Blackjack.

Clare- mont Economic Papers; Claremont, Calif.

This contains a useful algorithm for playing infinite deck blackjack.

Experimental Comparison of Blackjack Betting Systems.

Paper presented to the Fourth Conference on Gambling, Reno, 1978, sponsored by the University of Nevada.

People who distrust theory will have to believe the results of Gwynn's tremendous simulation study of basic strategy blackjack with bet variations, played on his efficient "table driven" computer program.

Algorithms for Computations of Blackjack Strategies, presented to the Second Conference on Gambling, sponsored by the University of Nevada, 1975.

This contains a good exposition of an infinite deck computing algorithm.

MANSON, BARR, and GOODNIGHT.

Optimum Zero Memory Strategy and Exact Probabilities for 4-Deck Black- jack.

The American Statistician 29 2 :84-88.

The authors, from North Carolina St.

University, present an in- triguing and efficient recursive method for finite deck black- jack calculations, as well as a table of four deck expectations, most of which are exact and can be used as a standard for checking other blackjack programs.

New York: Vintage Books, 1966.

If I were to recommend one book, and no other, on the subject, it would be this original and highly successful popularization of the opportunities presented by the game of casino blackj ack.

Optimal Gambling Systems for Favorable Games.

Review of the International Statistics Institute, Vol.

This contains a good discussion of the gambler's ruin problem, as well as an analysis of several casino games from this standpoint.

The Fundamental Theorem of Card Counting.

International J oumal of Game Theory, Vol.

This paper, presumably an outgrowth of the authors' work on baccarat, is important for its combinatorial demonstration that the spread, or variation, in player expectation for any fixed strategy, played against a diminishing and unshuffled pack of cards, must increase.

The Casino Gambler's Guide.

This is an exceptionally readable book which lives up to its title.

Wilson's blackjack coverage is excellent.

In addition, any elementary statistics text may prove helpful for understanding the probability, normal curve, and regression theory which is appealed to.

I make no particular recommendations among them.

When I had an ace and jack I heard him say again, "If you draw another card It will not be a ten; You'll wish you hadn't doubled And doubtless you will rue.

Shameless Plagiarism of A.

Housman Unless otherwise specified, all subsequent references will be to single deck blackjack as dealt on the Las Vegas Strip: dealer stands on soft 17, player may double on any two initial cards, but not after splitting pairs.

Furthermore, although it is contrary to almost all casino practices, it will be assumed, when necessary to illustrate general principles of probability, that all 52 cards will be dealt before reshuffling.

The first questions to occur to a mathematician when fac- ing a game of blackjack are: 1 How blackjack ti 84 plus I play to maximize my expectation?

The answer to the first determines the answer to the second, and the answer to the second determines whether the mathematician is interested in playing.

It is even conceivable, if not probable, that nobody, experts included, knows precisely what the basic strategy is, if we pursue the definition to include instructions on how to play the second and subsequent cards of a split depending on what cards were used on the earlier parts.

For ex- ample, suppose we split eights against the dealer's ten, busting the first hand 8,7,7 and reaching 8,2,2,2 on the second.

Quickly now, do we hit or do we stand with the 141 You will be able to find answers to such questions after you have mastered Chapter Six.

The basic strategy, then, constitutes a complete set of decision rules covering all possible choices the player may en- counter, but without any reference to any other players' blackjack appendix wizard of odds or any cards used on a previous round before the deck is reshuf- fled.

These choices are: to split or not to split, to double down or not to double down, and to stand or to draw another card.

Some of them seem self evident, such as always drawing another card to a total of six, never drawing to twenty, and not splitting a pair of fives.

But what procedure must be used to assess the correct action in more marginal cases?

While relativelyamong the simplest borderline choices to analyze, we will see that precise resolution of the matter requires an extraordinary amount of arithmetic.

If we stand on our 16, we will win or lose solely on the basis of whether the dealer busts; there will be no tie.

In Chapter Eleven there will be found just such a program.

Once https://allo-hebergeur.com/blackjack/gear-blackjack-wheels.html deed has been done we find the dealer's exact chance of busting is.

This has been the easy part; analysis of what happens when we draw a card will be more than fivefold more time con- suming.

This is because, for each of the five distinguishably dif- ferent cards we can draw without busting A,2,3,4,5the dealer's probabilities of making various totals, and not just of busting, must be determined separately.

For instance, if we draw a two we have 18 and presumably would stand with it.

We must go back to our dealer probability routine and play out the dealer's hand again, only now from a 48 card residue our deuce is unavailable to the dealer rather than the 49 card remainder used previously.

Once this has been done we're interested not just in the dealer's chance of busting, but also specifically in how often he comes up with 17,18,19,20,and 21.

The result is found in the third line of the next table.

Hence our "condi- tional expectation" is.

Some readers may be surprised that a total of 18 is overall a losing hand here.

Note also that the dealer's chance of busting increased slightly, but not significantly, when he couldn't use "our" deuce.

Similarly article source find all other conditional expectations.

Since a loss of 48 cents by drawing is preferable to one of 54 cents from standing, basic strategy is to draw to Please click for source v 9.

Note that it was assumed that we would not draw a card to T,6,AT,6,2etc.

This decision would rest on a previous and similar demonstration that it was not in our interest to do so.

All this is very tedious and time consuming, but necessary if the exact player expectation is sought.

This, of course, is what com- puters were deSigned for; limitations on the human life span and supply of paper preclude an individual doing the calcula- tions by hand.

Doubling Down So much for the choice of whether to hit or stand in a par- ticular situation, but how about the decision on whether to double down or not?

In some cases the decision will be obvious- ly indicated by our previous calculations, as in the following example.

Suppose we have A,6 v dealer 5.

live online card counting two card total of hard 10 or 11 would illustrate the situation equally well against the dealer's up card of 5.

We know three things: 1.

We want to draw another card, it having already been determined that drawing is preferable to standing with soft 17.

We won't want a subsequent card no matter what we draw for instance, drawing to.

A,6,5 would be about 7% worse than standing.

Our overall expectation from drawing one card is positive-that is, we have the advan- tage.

Hence the decision is clear; by doubling down we make twice as much money as by conducting an undoubled draw.

The situation is not quite so obvious when contemplating a double of 8,2 v 7.

Conditions 1 and 3 above still hold, but if we receive a 2,3,4,5, or 6 in our draw we would like to draw another card, which is not permitted if we have selected the double down option.

Therefore, we must compare the amount we lose by forfeiting the right to draw another card with the amount gained by doubling our bet on the one card draw.

It turns out we give up about 6% by not drawing a card to our 15 subsequently developed stiff hands, but the advantage on our extra, doubled, dollar is 21%.

Since our decision to double r a i s ~ s expectation it becomes part of the basic strategy.

The Baldwin group pointed out in their original paper that most existing recommendations at the time hardly suggested doubling at all.

Probably the major psychological reason for such a conservative attitude is the sense of loss of control of the hand, since another card cannot be requested.

Doubling on small soft totals, like A,2heightens this feeling, because one could often make a second draw to the hand with no risk of busting whatsoever.

But enduring this sense of helplessness, like taking a whiff of ether before necessary surgery, is sometimes the preferable choice.

Pair Splitting Due to their infrequency of occurrence, decisions about pair splitting are less important, but unfortunately much more complicated to resolve.

Imagine we have 7,7 v9.

The principal ques'tion facing us is whether playing one fourteen is better than playing two, or more, sevens in what is likely to be a losing situation.

Determination of the exact splitting expectation requires a tortuous path.

First, the exact probabilities of ending up with two, three, and four sevens would be calculated.

Then the player's expectation starting a hand with a seven in each of the three cases would be determined by the foregoing methods.

The overall expectation would result from adding the product of the probabilities of splitting a particular number of cards and the associated expectations.

The details are better reserved for Chapter Eleven, where a computer procedure for pair splitting is outlined.

This, of course, is for the set of rules and single deck we assumed.

It's not inconceivable that this highly complex game is closer to the mathematician's ideal of "a fair game" one which has zero ex- pectation for both competitors than the usually hypothesized coin toss, since real coins are flawed and might create a greater bias than the fourth decimal of the blackjack expectation, whatever it may be.

Condensed Form of the Basic Strategy By definition, the description of the basic strategy is "composition" dependent rather than "total" dependent in that some card combinations which have the same total, but unlike compositions, require a different action to optimize ex- pectation.

This is illustrated by considering two distinct three card 16's to be played against the dealer's Ten as up card: with 7,5,4 the player is 4.

Notwithstanding these many "composition" dependent exceptions which tax the memory and can be ignored at a total cost to the player of at most.

Hit stiff totals 12 to 16 against high cards 7,8,9,T,Abut stand with them against small cards 2,3,4,5,6except hit 12 against a 2 or 3.

Always draw to 17 and stand with 18, ex- cept hit 18 against 9 or T.

Never split 4,45,5or T,Tbut always split 8,8 and A,A.

Split 9,9 against 2 through 9, except not against a 7.

Split the others against 2 through 7, except hit 6,6 v 7, 2,2 and 3,3 v 2, and 3,3 v 3.

Hard Doubling: Always double 11.

Double 10 against all cards except T or A.

Double 9 against 2 through 6.

Double 8 against 5 and 6.

Soft Doubling: Double 13 through 18 against 4,5, and 6.

Double 17 against 2 and 3.

Double 18 against 3.

Double 19 against 6.

House Advantage If you ask a casino boss how the house derives its advan- tage in blackjack he will probably reply "The player has to draw first and if he busts, we win whether we do or not.

Being ignorant of our basic strategy, such an in- dividual's inclination might not unnaturally be to do what the Baldwin group aptly termed "Mimicking the Dealer"-that is hitting all his hands up to and including 16 without any discrimination of the dealer's up card.

This "mimic the dealer" strategy would give the house about a 5.

How can the basic strategist whittle this 5.

The following more info of departures from "mimic the dealer" is a helpful way to understand the nature of the basic strategy.

DEPARTURES FROM "MIMIC THE DEALER" Option Proper pair splitting Doubling down Hitting soft 17,18 Proper standing Gain.

Up Card 2 3 4 5 6 7 8 9 T A %Chance of Bust 35 38 40 43 42 26 24 -23 21 11 Note that most of the aggressive actions, like doubling and splitting, are taken when the dealer shows a small card, and these cards bust most often overall, about 40% of the time.

Incidentally, I feel the quickest way to determine if somebody is a bad player is to watch whether his initial eye contact is with his own, or the dealer's first card.

The really unskilled function as if the laws of probability had not yet been discovered and seem to make no distinction between a five and an ace as dealer up card.

The interested reader can profit by consulting several other sources about the mathematics of basic strategy.

Wilson has a lengthy section on how he approached the problem, as well as a unique and excellent historical commentary about the various attempts to assess the basic strategy and its expecta- tion.

The Baldwin group's paper is interesting in this light.

Manson et alii present an almost exact determination of 4 deck basic strategy, and it is from their paper that I became aware of the exact recursive algorithm they use.

They credit Julian Braun with helping them, and I'm sure some of my own procedures are belated germinations of seeds planted when I read various versions of his monograph.

Infinite deck algorithms were presented at the First and Second Gambling Conferences, respectively by Edward Gordon and David Heath.

These, of course, are totally recur- sive.

Their appeal stems ironically from the fact that it takes far less time to deal out all possible hands from an infinite deck of cards than it does from one of 52 or 2081 B.

The two card, "composition" dependent, exceptions are standing with 7,7 v T, standing with 8,4 and 7,5 v 3, hitting T,2 v 4 and 6, hitting T,3 v 2, and not doubling 6, 2 v 5 and 6.

The multiple card exceptions are too numerous to list, although most can be deduced from the tables in Chapter Six.

The decision not to double 6,2 v 5 must be the closest in basic strategy blackjack.

The undoubled expectation is.

The poor blackjack deck is being stripped naked of all her secrets.

This is easily proven by imagining all possible permutations of the deck and recogniz- ing that, for any first and second hand that can occur, there is an equiprobable reordering of the deck which merely more info changes the two.

For example, it is just as likely that the player will lose the first hand 7,5 to 6,A and push the second 9,8 to 3,4,T as that he pushes the first one 9,8 to 3,4,T and loses the next 7,5 to 6,A.

If resplitting pairs were prohibited there would always be enough cards for four hands before reshuffling and that would guarantee an identical expectation for basic strategy play on all four hands.

Unfortunately, with multiple splitting permit- ted, there is an extraordinarily improbable scenario which ex- hausts the deck before finishing the third hand and denies us the luxury of asserting the third hand will have precisely the same expectation as the first two: on the first hand split 6, 5, T6, 5, T6, 5, Tand 6, 5, T versus dealer 2, 4, T, T ; second hand, split 3, 9, 4, T3, 9, 4, T3, 9, 4, Tand 3, A, T, 7 against dealer 7, 9, T ; finally, develop 8, 7, T8, 7, T8, 2, 2, A, A, A, Tand unfinished 8, 2, 1 in the face of dealer's T, T.

Gwynn's simulation study showed no statistically signifi- cant difference in basic strategy expectation among the first seven hands dealt from a' full pack and only three times in 8,000,000 decks was he unable to finish four hands using 38 cards.

Thus, as a matter of practicality, we may assume the first several hands have the same basic strategy expectation.

This realization leads us to consider what Thorp and Walden termed the "spectrum of opportunity" in their paper The Fundamental Theorem of Card Counting wherein they proved that the variations in player expectation for a fixed strategy must become increasingly spread out as the deck is depleted.

Notice there are no pair splits possible and the 38 total pips available guarantee that all hands can be resolved without reshuffling.

The basic strategist, while perhaps unaware of this composi- tion, will have an expectation of 6.

Player Dealer Player Dealer Hand Up Card Expectation Hand Up Card Expectation -- 5,6 8 +2 6,9 5 +1 9 +1 8 0 T +1 T 0 5,8 6 +1 6,T 5 +1 9 .

This exploitation of decks favorable for basic strategy will henceforth be referred to as gain from "bet variation.

Strategy Variation Another potential source of profit is the recognition of when to deviate from the basic strategy.

Keep in mind that, by definition, basic strategy is optimal for the full deck, but not necessarily for the many subdecks like the previous five card example encountered before reshuffling.

Basic strategy dictates hitting 5,8 v.

If we survive our hit we only get a push, while a successful stand wins.

Similarly it's better to stand with 5,8 v T, 6,8 v 9, and 6,8 v T, for the same reason.

In each of the four cases we are 50% better off to violate the basic strategy, and if we had been aware of this we could have raised our basic strategy edge of 6.

This extra gain occasionally available from appropriate departure from basic strategy, in response to fluc- tuations in deck composition which occur before reshuffling, will be attributed to "variations in strategy.

Some of the 23 departures from basic strategy are eye opening indeed and il- lustrate the wild fluctuations associated with extremely de- pleted decks.

Generally, variations in strategycan mitigate the disadvantage for compositions unfavorable for basic strategy, or make more profitable an already rich deck.

This is a seldom encountered case in that variation in strategy swings the pen- dulum from unfavorable to favorable.

Since these examples are exceedingly rare, the presumption that the only decks worth raising our bet on are those already favorable for basic strategy, although not entirely true, will be useful to maintain.

Insurance is "linear" A simple illustration of how quickly the variations can arise is the insurance bet.

Insurance is interesting for another it is the one situation in blackjack which is truly "linear," being resolved by just one card the dealer's hole card rather than by a com- plex interaction of possibly several cards whose order of appearance could be vital.

From the standpoint of settling the insurance bet, we might as well imagine that the value -1 has been painted on 35 cards in the deck and +2 daubed on the other 16 of them.

The player's insurance expectation for any subdeck is then just the sum of these "payoffs" divided by the number of cards left.

This leads to an extraordinarily simple mathematical solution to any questions about how much money can be made from the insurance bet if every player in Nevada made perfect insurance bets it might cost the casinos about 40 million dollars a yearbut unfortunately other manifestations of the spectrum of opportunity are not so uncomplicatedly linear.

Our problem is to select these 52 numbers which will replace, for our immediate pur- poses, the original denominations of the cards so that the average value of the remaining payoffs will be very nearly equal to the true blackjack knives model 4 strategy expectation for any particular subset.

Using a traditional mathematical measurement of the ac- curacy of our approximation called the "method of least squares," it can be shown that the appropriate are, as intuition would suggest, the same for all cards of the same denomination: Best Linear Estimates of Deck Favorability in % A 2 3 4 5 6 7 8 9 T 31.

To assert that these are "best" estimates under the criterion of least squares means that, although another choice might work better in oc- casional situations, this selection is guaranteed to minimize the overall average squared discrepancy between the true expectation and blackjack books card counting estimate of it.

We add the six payoffs corresponding to these cards -19.

It is the ensemble of squared differences between numbers like -6.

The estimate is not astoundingly good in this small subset case, but accuracy is much betterfor larger subsets, necessarily becoming perfect for 51 card decks.

Approximating Strategy Variation The player's many different possible variations in strategy can be thought of as many embedded subgames, and they too are amenable to this sort of linearizing.

Precisely which choices of strategy may confront the player will not be known, of course, until the hand is dealt, and this is in contrast to the bet- ting decision which is made before every hand.

Consider the player who holds a total of 16 when the dealer shows a ten.

The exact cards the player's total comprises are important only as they reveal information about the remaining cards in the deck, so suppose temporarily that the player possesses a piece of paper on which is written his current total of 16, and that the game of "16 versus Ten" is played from a 51 card deck.

Computer calculations show that the player who draws a card to such an abstract total of 16 has an expectation of .

Suppose now that it is known that one five has been removed from the deck.

Faced with this reduced 50 card deck 26 the player's expectation by drawing is .

In this case, he should stand on 16; the effect of the removal of one five is a reduction of the original.

In similar fashion one can determine check this out effect of the removal of each type of card.

These effects are given below, where for con- venience of display we switch from decimals to per cent.

Effects of Removal on Favorability of hitting 16 v.

Now we construct a one card payoff game of the type already mentioned, where the player's payoff is given by E i is the effect, j ~ s t described, of the removal of the i th card.

Approximate determination https://allo-hebergeur.com/blackjack/winning-at-blackjack-chart.html whether the blackjack player should hit or stand for a particular subset of the deck can be made by averaging these payoffs.

Their average value for any subset is our "best linear estimate" of how much in % would be click at this page or lost by hitting.

Similarly, any of the several hundred playing decisions can click at this page approximated by assigning appropriate single card payoffs to the distinct denominations of the blackjack deck.

The distribution of favorability for changing violating basic strategy can be studied further by using the well known nor- mal distribution of traditional statistics to determine how often the situations arise and how much can be gained when they do.

Derivation of this method is also reserved for the Appendix.

Number of Click Strategy Gain Betting Unseen Cards Gain no Insurance Gain 10.

This is consistent, of course, with Thorp and Walden's 'Fundamental Theorem'.

Two other important deter- miners of how much can be gained from individual strategy variations are also pinpointed by the formula.

Average Disadvantage for Violating Basic Strategy In general, the greater the loss from violating the basic strategy for a full deck, the less frequent will be the opportunity for a particular strategy change.

For example, failure to double down 11 v 3 would cost the player 29% with a full deck, while hitting a total of 13 against the same card would carry only a 4% penalty.

Hence, the latter change in strategy can be ex- pected to arise much more quickly than the former, sometimes as soon as the second round of play.

Volatility Some plays are quite unfavorable for a full deck, but never- theless possess a great "volatility" which will overcome the 28 previous factor.

Consider the effects of removal on, and full deck gain from, hitting 14 against a four and also against a nine: Effects of Removal for Hitting 14 Full Deck Gain by Hitting A 2 3 4 S 6 7 8 9 T -- - - - - - v.

This is because large effects of removal are characteristic of hitting stiff hands against small cards and hence these plays can become quite valuable deep in the deck despite being very unfavorable initially.

This is not true of the option of standing with stiffs against big cards, which plays tend to be associated with small effects.

In blackjack dealer match first case an abundance of small cards favors both the player's hitting and the dealer's hand, doubly increasing the motivation to hit the stiff against a small card which the dealer is unlikely to break.

In the second case an abundance of high cards is unfavorable for the player's hitting, but is favorable for the dealer's hand; these contradic- tory effects tend to mute the gain achievable by standing with stiffs against big cards.

We can liken the full deck loss from violating basic strategy to the distance that has to be traveled before the threshold of strategy change is reached.

The effects of removal or more precisely their squares, as we shall learn are the forces which can produce the necessary motion.

The following table breaks down strategy variation into each separate component and was prepared by the normal ap- proximation methods.

This should roughly approximate dealing three quarters of the deck, shuffling up with 13 or fewer cards remaining before the start of a hand, but otherwise finishing a hand in progress.

Gain from the latter activity is perhaps unfairly recorded in the 12-17 rows.

Similarly the methodology incorrectly assesses situations where drawing only one card is dominated by a standing strategy, but drawing more than once is preferable to both.

An example of this could arise when the player has 13 against the dealer's ten and the remaining six cards consist of four 4's and two tens.

The expectation by drawing only one card is .

The next higher step of approximation, an interactive model of blackjack, would pick this sort of thing up, but it's doubtful that the minuscule increase in accuracy would be balanced by the difficulty of developing and applying the theory.

Remember, the opportunities we have been discussing will be there whether we perceive them or not.

When we consider the problem of programming the human mind to play black- jack we must abandon the idea of determining instantaneous strategy this web page the exhaustive algorithm described in the earlier parts of the book.

The best we can reasonably expect is that the player be trained to react to the proportions of different denominations remaining in the deck.

Clearly, the information available to mortal card counters will be imperfect; how it can be best obtained and processed for actual play will be the sub- ject of our next chapter.

To build an approximation to what goes on in an arbitrary subset, let's assume a model in which the favorability of hit- ting 16 vs Ten is regarded as a linear function of the cards re- maining in the deck at any instant.

For specificity let there re- main exactly 20 cards in the deck.

Y is the vector of favorabilities associated with each subset of the full deck, X is a matrix each of whose rows con- tains 20 l's and 31 0' s, and the solution, 3, will provide us with our 51x1 vector of desired coefficients.

Run the computer day and night to determine the Y's.

Premultiply a x 51 matrix by its tran,spose.

Multiply the result of b.

Multiply X' by Y and finally e.

Solve the resultant system of 51 equations in 51 unknowns!

The normal equations for the {3j will be +.

Their average value in a given subset is the corresponding estimate of favorability for carrying out the basic strategy.

Other aspects of blackjack, such as the player's expecta- tion itself, or the drawing expectation, or the standing expecta- tion separately, could be similarly treated.

But, since basic strategy blackjack is so well understood it will minimize our error of approximation to use it as a base point, and only estimate the departures from it.

Uniqueness of this solution follows from the non- singularity of a matrix of the form ~ : : ~with a b b b a throughout the main diagonal and b ~ a in every non-diagonal position.

The proof is most easily given by induction.

Let D n,a,b be the determinant of such an n x n matrix.

This derivation has much the flavor of a typical regression problem, but in truth it is not quite of that genre.

Yi is the true conditional check this out for a specified set of our regression variables Xij.

It would be wonderful indeed if Yi were truly the linear conditional mean hypothesized in regression theory, for then our estimation techniques would be perfect.

But here we ap- peal to the method of least squares not to estimate what is assumed to be linear, but rather to best approximate what is almost certainly not quite so.

This emphasizes that Yi is a fixed number we are try- ing to approximate as a linear function of the Xij' and not a particular observation of a random variable as It would be in most least squares fits.

Suppose 0 2 is the variance of the single card payoffs and Il is their full deck average value.

Assume IJ SO and that the card counter only changes strategy or bet when it is favorable to do so.

SO is equivalent to redefining the single card payoffs, if necessary, so they best estimate the favorability of altering the basic strategy.

The Central Limit Theorem appealed to appears in the ex- N ercises of Fisz.

N O,l 37 in distribution.

The applicability is easily verified in our case since the Xi are our "payoffs" and are all bounded.

The "ratio" of b-a to b + c corresponds to efficiency.

It is not really area we should compare here, but it aids understanding to view it that way.

The Einstein count of +1 for 3,4,5, and 6 and -1 for tens results in a correlation of.

The Dubner Hi-Lo extends the Einstein values by counting the 2 as + 1 arid the ace as -1, resulting in a correla- tion coefficient of.

Another system, mentioned in Beat the Dealer by Thorp, extends Dubner's count by counting the 7 as + 1 and the 9 as -1, and has a correlation of.

The assumption that evaluation of card counting systems in terms of their correlation coefficients for the 70 mentioned variations in strategy will be as successful as for the insurance bet is open to question.

The insurance bet is, after all, a truly linear game, while the other variations in strategy involve more complex relations between several cards; these interac- tions are necessarily neglected by the bivariate normal methodology.

There is one interesting comparison which can be made.

Epstein reports a https://allo-hebergeur.com/blackjack/blackjack-charts-strategy.html of seven million hands where variations in strategy were conducted by using the Ten-Count.

An average expectation of 1.

In today's casino conditions the deck will rarely be dealt this deeply, and half the previous figures would be more realistic.

It might also be mentioned blackjack ti 84 plus correlation is undisturbed by the sampling without replacement.

To prove this, let Xi be the payoff associated with the ith card in the deck and Yi be the point value associated with the ith card in the deck by 52 some card counting system.

We n n seek the correlation of ~ X and L:Y for n ~ d subsets.

They then recommend keeping a separate, or "side," count of the aces in order to adjust their primary count for betting purposes.

Let's take a look at how this is done and what the likely effect will be.

Consider the Hi Opt I, or Einstein, count, which has a bet- ting correlation of.

A 23456789 T HiOpt I 0 0 1 1 1 1 0 0 0-1 Betting Effect .

It therefore seems reasonable to regard an excess ace in the deck as meriting a temporary readjustment of the running count for betting purposes only by plus one point.

Similarly, a defi- cient ace should produce a deduction temporary, again of one point.

Should we regard the deck as favorable?

Well, we're shy two aces since the expected distribution is three in 39 cards; therefore we deduct two points to give ourselves a temporary running COtlnt of -1 and regard the deck as probably disadvantageous.

In like fashion, with a count of -1 but all four aces remaining in the last 26 cards we would presume an advantage on the basis of a +1 ad- justed running count.

It can be shown by the mathematics in the appendix that the net effect of this sort of activity will be to increase the system's betting correlation from.

Among these are knowing when to stand with 15 and 16 against a dealer 7 or 8 and knowing when to stand with 12, 13, and 14 against a dealer 9, Ten, or Ace.

Before presenting a method to improve single parameter card counting systems it is useful to look at a quantification of the relative importance of the separate denominations of nontens in the deck.

This quantification can be achieved by calculating the playing efficiency of a card count which assigns one point to each card except the denomination considered, which counts as -12.

The fix- ed sign of the point value obscures this and can only be over- come by assigning the value zero and keeping a separate track of the density of these zero valued cards for reference in ap- propriate situations.

The average effect of removal for the eight cards recogniz- ed by the Einstein count is about 1% and this suggests that, if the deck is one seven short, that should be worth four Einstein points.

The mathematically correct index for standing with 14 against a ten is an average point value above +.

Suppose, however, that there was only one seven left in the deck.

It will save a lot of arguments to keep in mind that a change in strategy can be considered correct from three different perspectives which don't always coincide: it can be mathematically correct with respect to the actual deck com- position confronted; it can be correct blackjack ti 84 plus to the deck composition a card counter's parameter entitles him to presume; and it can be correct depending on what actually hap- pens at the table.

I've seen many poor players insure a pair of tens when the dealer had a blackjack, but I've seen two and a half times more insure when the dealer didn't!

Incorporation of the density of sevens raises our system's correlation from.

We've already seen the importance of the seven for playing 14 v.

Ten in conjunction with the.

The further simplification, "stand if there are no sevens," is almost as effective, being equivalent to the previous rule if less than half the deck remains.

For playing 16 v.

Ten the remarkably elementary direction "stand when there are more sixes than fives remaining, hit otherwise," is more than 60% efficient.

We will see in Chapter Eleven that it consistently out-performs both the Ten Count and Hi Opt I.

Of course, these are highly specialized instruc- tions, without broader applicability, and we should be in no haste to abandon our conventional methods in their favor.

The ability to keep separate densities of aces, sevens, eights, and nines as well as the Einstein point count itself is not beyond a motivated and disciplined intellect.

The memorization of strategy tables for the basic Einstein system as well as proper point values for the separated denominations in different strategic situations should be no problem for an in- dividual who is so inclined.

The increases in playing efficiency and betting correlation are exhibited below.

INCORPORATION OF ZERO VALUED CARDS INTO EINSTEIN SYSTEM Basic System A Cards Incorporated A,7 A,7,8 A,7,8,9 A,7,8,9,2 Playing Efficiency.

It is of little consequence strategically except for doubling down totals of eleven, particularly against a 7, 8, or 9, and totals of ten against a Ten or Ace.

Actually the compleat card counting fanatic who aspires to count separately five zero valued denominations is better off using the Gordon system which differs from Einstein's by counting the 2 rather than the 6.

Although poorer initially than Einstein's system, it provides a better springboard for this level more info ambition.

The Gordon count, fortified with a pro- per valuation of aces, sixes, sevens, eights, and nines, scores.

This may reasonably be supposed to define a possibly realizable upper bound to the ultimate capability of a human being playing an honest game of blackjack from a single deck.

The Effect of Grouping Cards All of the previous discussion has been under the assump- tion that a separate track of each of the zero-valued cards is kept.

David Heath suggested sometime ago a scheme of block- ing the cards into three groups {2,a,4,5}, {6,7,8,9}, and {lO,J, Q,K}.

Using two measures, the differences between the first two groups and the tens, he then created a two dimensional strategy change graphic resembling somewhat a guitar finger- ing chart.

Heath's system is equivalent to fortifying a primary Gor- don count with information provided by the block of "middle" cards, there being no discrimination among these individually.

As we can see from the following table of ef- ficiencies for various blocks of cards properly used in support of the Gordon and Einstein systems, it would have been better to cut down on the number of cards in the blocked group.

IC,D,E Primary Count Auxiliary Grouping Playing Efficiency Gordon { 6,7,8,9 }.

Each of them was analyzed by a computer to determine if basic strategy should have been changed and, if so, how much expectation could have been gained by such appropriate departure.

I, myself, made decisions as to whether I wo'old have altered the conventional basic strategy, using my own version of the system accorded an efficiency of.

The following table displays how much expectation per hand I and the computer gained by our blackjack ti 84 plus changes.

My gain in% appears first, followed by the computer's, the results for which are always at least as good as mine since it was blackjack ti 84 plus ultimate ar- biter as to which decisions were correct and by how much.

Unseen Cards Insurance Gain Non-insurance Gain 8-12.

The discrepancy between this and the theoretical.

This table should be compared with the one on page 28.

The most bizarre change was a double on hard 13 v 6; with three eights, two sixes, sevens, and tens, and one ace, two, three, and four, doubling was 61%better than standing, 18% better than mere- ly drawing.

An indicator count, -12 1 1 1 1 1 1 1 1 1monitors the presence of aces in the deck and will be uncorrelated with the primary one if zero is the assigned point value.

This is because the numerator of the correlation, the inner product between the primary and the ace indicator count, will be zero, merely being the sum of the point values of the primary system assumed to be balanced.

To the degree of validity of the bivariate normal approximation zero correlation is equivalent to independence.

Hence we are justified in taking the square root of the sum of squares of the original systems' correlations as the multiple correlation coefficient.

For the situation discussed, we find the ace indicator count has a.

The seven indicator has a.

The "Six-Five" system for playing 16 v.

T has a correlation of.

We can use the theory of multiple correlation to derive a formula for the appropriate number of points to assign to a block of k zerQ-valued cards when using them to support a primary count system.

However, since the assumption of linearity underlies this theory as well as the artifice of the single card payoffs, the demonstration can be more easily 62 given from the latter vantage point, coupon blackjack apprenticeship only elementary algebra.

We still have 52 cards, but the point count of the deck 13 is L y.

Hence 52 - k 52 - k 63 the removal and replacement of one "blocked" card by a typical unblocked one has altered the full deck total of the payoffs by k k 1 ~ +2!.

It is unrealistic to suppose that such auxiliary point values would be remembered more precisely than to the nearest whole number.

Similar- lyforl6,7,SJ we would use 3 3 3 3 3-10-10-10 33 andfor ~ 6,7} 222 2 2-11-1122 2.

These also will be independent 64 of a primary count which assigns value zero to them, and hence the square root of the sum of the squares of the correlations can be used to find multiple correlation coefficients.

In fact, the original Heath count recommended keeping two counts, what we now call the Gordon 0 1 1 1 1 0 0 0 0 -1 and a "middle against tens" see more 0 0 0 0 0 11 1 1 -1.

These are dependent, having correlation.

There is a subtle difference in the informatIon available from the two approaches which justifies the difference.

Factoring in information from cards already included in, and hence dependent upon, the primary count is usually very difficult to do, and probably not worthwhile.

One case where it works out nicely, however, is in adjusting the Hi Opt I count by the difference of sixes and fives, for playing 16 v Ten.

Both these denominations are included in the primary count, but since it's their difference we are going to be using, our aux- iliary count can be taken as 0 0 001 -1 0000 which is uncor- related with, and effectively independent of, the primary count.

The Chapter Eleven simulations contain data on how well this works out.

Even though it is usually too cumbersome in practice to use multiple correlation with dependent counts, an example will establish the striking accuracy of the method.

It will also illustrate the precise method of determining the expected deck composition subject to certain card counting information.

Let our problem be the following: there are 28 cards left in the deck and a Ten Counter and Hi Lo player pool information.

How many aces should we presume are left in the deck?

The Ten Count sug- gests more than normal, the Hi Lo indicates slightly less than usual.

We can look at this as a multiple regression problem.

Let Xl be the indicator count for aces -12 111111111 ; X2 the Ten Count 4 4 444 4 4 4 4 -9 ; and X3 the HiLo -111111 o0 0 -1.

Hence p 12' the correlation between X1.

VI 0 just click for source 1.

The exact distribution can be found by combinatorial analysis for the 21 cards we are uncertain about.

I had imagined two aces, ten small cards, and nine middle cards would--be represent- ative, but we see the precise average figures are 2.

The only consolation I have is that it was the multivariate methodology which tipped me off to my foolishness.

At no time during the test was any attention paid to whether, in the actual play of the cards, the hand was won or lost.

Had the results been scored on that basis, the statistical variation in a sample of this size would have rendered them almost meaningless.

The estimate, that perfect play gains 3.

I wanted to astonish the spectators by taking senseless chances.

The player's exact gain at any deck level is catalogued completely for a single deck and extensively for two and four decks.

If the remaining number of cards is a multiple of three, add one to it before consulting the charts.

For example, with 36 cards left, the single deck gainis the same as with 37, namely.

There would be 89 unseen cards at a double deck, and full table, first round insurance is worth only.

You can also use these tables to get a reasonable estimate for the total profit available from think, blackjack natural 9 that variations in strategy, not just insurance.

Multiply the insurance gain at the number of unplayed cards you're interested in by seven and that should be reasonably close.

The figures are in %.

EFFECTS OF REMOVAL A 2 3 4 5 - -- 6 789 Sum of.!.

Mean Squares Insur- ance 1.

Nevertheless, finding the Hi-Lo system's -111111 000 -1 insurance correlation will provide a helpful review.

Suppose we see a Reno dealer burn a 2 and a 7.

What is our approximate expec- tation?

If we want to know the effect of removing one card from the deck we merely read it directly from the table.

To practice this, let's find the insurance expectation when the dealer's ace and three other non-tens are removed from the deck.

We adjust the full deck mean of -7.

The full deck expectations for basic strategy are different, however, and this is discussed in Chapter 8.

Very lengthy tables are necessary for a detailed analysis of variations in strategy, and a set as complete as any but the an- tiquarian could desire will follow.

In order to condense the printing, the labeling will be abbreviated and uniform through- out the next several pages.

Each row will present the ten ef- fects of removal for the cards Ace through Ten, full deck favor- ability, m, and sum of squares of effects of removal, ss, for the particular strategy variation considered.

For hard totals of 17 down to 12 we are charting the favorability of drawing over standing, that is, how much bet- ter off we are to draw to the total than to stand with it.

Naturally this will have a negative mean in the eleventh col- umn in many cases, since standing is often the better strategy for the full deck.

Again, in many cases the average favorability for the full deck will be negative, in- dicating the play is probably not basic strategy.

Similarly we present figures for soft doubles, descending from A,9 to A,2showing how much better doubling is than conventional draw- ing strategy.

Finally, the advantage of pair splitting over not splitting will be catalogued.

Not all dealer up cards will have the same set of strategic variations presented, since in many situations like doubling small totals and soft hands v 9,T, or A and split- ting fives there is no practical interest in the matter.

The tables will be arranged by the different dealer up cards and there will be a separate section for the six and ace when the dealer hits soft 17.

There is no appreciable difference in the Charts for 2,3,4, and 5 up in this case.

It's important to remember that the entries in the tables are not expectations, but rather differences in expectation for two separate actions being contemplated.

Once the cards have been dealt the player's interest in his expectation is secondary to his fundamental concern about how to play the hand.

This is resolved by the difference in expectation for the contemplated alternatives.

As a specific example of how to read the table, the arrow on page 76 locates the rowcorresponding to hard 14 v Ten.

The entry in the 11th column, 6.

For the same reason doubling down is not very advan- tageous, even with a total of 11.

The 11th column entries for 1216 are all at least 6.

Because of the increase in busts and fewer 17's produced, standing and doubling both grow in attractiveness.

Note soft 18 is now blackjack ti 84 plus profitable hit.

Not only are they desirable cards for the player to draw, but their removal produces the greatest in- crease in the dealer's chance of busting.

The table also shows that soft 18 with no card higher than a 3 should not be hit.

Since 19 is easier to beat, the player is inclined to hit and double down more often than against a Ten.

A player who split three eights and drew 8,98,7.

Note that otherwise the 9 is almost always a more important high card than the Ten.

On the next page it will be seen to be the correct play when dealer hits soft 17, paradoxically even though.

Standing and soft doubling become more frequent activities.

As mentioned continue reading the previous page, A, 8 is a basic strategy double down, regardless of the number of decks used.

The 11th column full deck advantage figures on pages 74-85 come from exact 52 card calculations, without the dealer's up card or any of the player's cards removed.

The ef- fects of removal first 10 columns are, for hitting totals of 82 DEALER 4 HITTING 17-12 -1.

However, for doubling and splitting removal effects the amount of com- puter time necessary to carry out the calculations exactly would have been excessive; in these situations the removal ef- 83 DEALER 3 HITTING 17-12 -1.

Use of these tables to carry out variations in strategy for the 5,000 hand experiment reported on page 61 resulted in an overall playing efficiency of 98.

The relatively few and inconsequential errors appear more at- tributable to blackjack's essential non-linearity, which is more pronounced deeper in the deck, than to any approximations in the table.

This is done exactly as it was for the insurance and betting effects previously.

Another use is to find some of the "composition" depend- ent departures from the simplified basic strategy defined in Chapter Two.

Should you hit or stand with 4,4,4,4 v 81 To the full deck favorability of 5.

In Chapter Two the question was asked whether one should hit 8,2,2,2 v T after having busted 8,7,7 on the first half of a pair split.

The table for hard 14 against a ten gives the following estimate for the advantage for hitting in this case 6.

Don't forget to remove the dealer's up card as well as the cards in the player's hand, since all of these tables assume a 52 card deck from which dealer's and player's cards have not yet been removed.

Also don't be surprised if you are unable to reproduce exactly the 2.

Quantifying the Spectrum of Opportunity at various Points in the Deck Before we will be able to quantify betting and strategy variations at different points in the deck we'll have to in- 8.

First the table itself.

Corresponding to values of avariable designated by z, which ranges from oto 2.

Unit Normal Linear Loss Integral z.

The following step by step procedure will be used in all such calculations.

Ignore the algebraic sign of m.

Look up in the UNLLI chart the number corresponding to z.

In our case this will be.

Multiply the number found in step 3 by b.

This is the conditional player gain in %assuming the dealer does have an ace showing.

If desired, adjust the figure found in step 4 to reflect the likelihood that the situation will arise.

Repeating the procedure, for two decks, we have ~ 95.

We would interpolate between.

If you're disappointed in the accuracy, there are ways of improving the approximation, principally by adjusting for the dealer's up card.

Removing the dealer's ace changes m, for the single deck, to -7.

Repeating the calculations, 1.

After revising m from .

The exact gain in this situation appears in Chapter Eleven and is 15.

One thing remains, and that is instruction on how to calculate a card counting system's gain, rather than the gain from perfect play.

To do this we must have a preliminary calculation of the correlation of the card counting system and the particular play examined.

Since we already found the cor- relation of the Hi Lo system for insurance to be.

After calculating b in the usual fashion we then 89 multiply it by the card counting system's correlation coeffi- cient and use the resultant product as a revised value of b in all subsequent calculations.

Thus the efficiency of the Hi Lo system, at the 40 card level, is.

The Normal Distribution of Probability The famous normal distribution itself can be used to answer many probabilistic questions with a high degree of ac- curacy.

The table on page 91 exhibits the probability that a "standard normal variable" will have a value between 0 and selected values of z used to designate such a variable from 0 to 3.

Chance of Being behind One type of question that can be answered with this table is "Suppose I have an average advantage of 2% on my big bets; What is the chance that I will be behind on big bets after making 2500 of them?

However, we can take advantage of the sym- metry of the normal curve and determine the area or probabili- ty corresponding to values of z greater than.

We do this by subtracting the tabulated value.

Distribution of a Point Count We can also use the normal distribution to indicate how often different counts will occur for a point count system, pro- viding that the number of cards left in the deck is specified.

The following procedure can be used.

Divide b into one half less than the count value you're interested in.

Divide b into one half more than the count value you're interested in.

The difference between the normal curve areas cor- responding to the two numbers calculated in steps 3 and 4 will be the probability that the particular count value will occur.

As an example, suppose we wish to know the probability that there will be a +3 Hi Opt II count when there are 13 cards left from a single deck.

The area corresponding to.

The precise probability can be found in Appendix A of Chapter Seven, and is.

How often is Strategy changed?

Although our only click to see more interest is in how much can be gained by varying basic strategy, we can also use the normal probability tables to estimate how often it should be done.

To do so is quite simple.

Then we subtract the blackjack ti 84 plus given in our normal curve probability charts corresponding to.

Precise calcula- tions show the answer to be 25%.

Similarly, we find the approximate probability of a favorable hit of hard 12 against the dealer's 6 with five cards left in the deck to be.

The exact probability is found in Chapter Eleven, and is.

To illustrate this, assume single deck play in Reno at a full table, so the player gets only one opportunity to raise his bet.

Following the steps on page 88, we have: 1.

From the UNLLI chart take.

When the player has a basic strategy advantage for the full deck, then this computational technique can be used to measure how much will be saved by each extra unit which is not bet in unfavorable situations.

In Chapter Eight we deduce that Atlantic City's six deck game with early surrender gave the basic strategist about a.

The strategy tables presented are not the very best we could come up with in a particular situation.

As mentioned in this chapter more accuracy can be obtained with the normal approximation if we work with a 51 rather than a 52 card deck.

One could even have separate tables of effects for different two card player hands, such as T,6 v T.

Obviously a compromise must be reached, and my motivation has been in the direction of simplicity of exposition and ready applicability to multiple deck play.

You have the usual 16 against the dealer's ubiquitous Ten.

We consider three dif- ferent sets of remaining cards.

Unplayed Residue 4,T 4,4,T,T 4,4,4,T,T,T Favorability of Hitting over Standing -50% 0% +10% From this simple example follow two interesting conclu- sions: 1.

Strategic favorabilities depend not strictly on the pro- portion of different cards in the deck, but really on the absolute numbers.

Every card counting systemever created would misplay at least one of these situations because the value of the card counting parameter would be the same in each case.

The mathematical analysis of blackjack strategies is only in rare instances what might be called an "exact science.

In theory all ques- tions can be so addressed but in practice the required computer time is prohibitive.

We have already, to a reasonable degree, quantified the worth of different systems when played in the error free, tran- sistorized atmosphere of the computer, devoid of the drift of cigar smoke, effects of alcohol, and distracting blandishments of the cocktail waitress.

But what of these real battlefield con- ditions?

To err is human and neither the pit boss, the dealer, nor the cards are divine enough to forgive.

Two Types of Error There are two principal types of error in employing a count strategy: 1 an incorrect measure of the actual parameter which may be due to either an arithmetic error in keeping the running count or an inaccurate assessment of the number of cards remaining in the deck, and 2 an imprecise knowledge of the proper critical index for changing strategy.

It is beyond my scope to comment on the likelihood of numerical or mnemonic errors read more than to suggest they probably occur far more often than people believe, particularly with the more complex point counts.

It strikes me as difficult, for instance, to treat a seven as 7 for evaluating my hand, but as +1 for altering my running count and calling a five 5 for the hand and +4 for the count.

The beauty of simple values like plus one, minus one, and zero is that they amount to mere recognition or non-recognition of cards, with counting for- ward or backwardrather than arithmetic to continuously monitor the deck.

Commercial systems employing so called "true counts" defined as the average number of points per card multiplied by 52 produce both types of error.

Published strategic indices themselves have usually been rounded to the nearest whole number, so a "true count" full deck parameter of 97 5 might have as much as a 10% error in it.

It is the view of the salesmen of such systems that these errors are not serious; it is my suggestion that they probably are.

An Exercise in Futility Even if the correct average number of points in the deck is available, there are theoretical problems in determining critical indices.

When I started to play I faithfully committed to memory all of the change of strategy parameters for the Hi Lo system.

It was not until some years later that I realized that several of them had been erroneously calculated.

For some time, I was firmly convinced that I should stand with 16 v 7 when the average number of points remaining equalled or exceeded.

I now know the proper index should be.

What do you think the consequences of such misinfor- mation would be in this situation?

Not only was I playing the hand worse than a basic strategist, but, with 20 cards left in the deck I would have lost three times as much, at the 30 card level twenty times as much, and at the 40 card level five hundred times as much as knowledge of the cor- rect parameter could have gained me.

The computer technique of altering normal decks so as to produce rich or lean mixtures for investigating different situa- tions has not always incorporated an accurate alteration of conditional probabilities corresponding to the extreme values of the parameter assumed.

The proper approach can be https://allo-hebergeur.com/blackjack/blackjack-zerochan.html from bivariate normal assumptions and consists of maintain- ing the usual density for zero valued cards and displacing the other denominations in proportion to their assigned point values, rather than just their algebraic signs.

Computer averag- ing of all possible decks with this count leaves us with a not surprising "ideal" deck of twelve tens, one each three, four, five, and six, and two of everything else.

There is at present no completely satisfactory resolution of such quan- daries and even the most carefully computerized critical in- dices have an element of faith in them.

Behavior of Strategic Expectation as the Parameter changes The assumption that the favorability for a particular ac- tion is a linear function of the average number of points in the deck is applied to interpolate critical indices and is also a con- sequence of the bivariate normal model used to analyze effi- ciency in terms of correlation coefficients.

How valid is this assumption?

The answer varies, depending on the particular strategic situation considered.

Tables 1 and 2, which present favorabilities for doubling down over drawing with totals of 10 and 11 and hitting over standing for 12 through 16, were pre- pared by using infinite deck analyses of the Hi Opt I and Ten Count strategies.

Critical points interpolated from them should be quite accurate for multiple deck play and incor- porating the effect of removing the dealer's up card permits the adjustment of expectations and indices for a single deck.

The most marked non-linearities are found when the dealer has a 9 or T showing.

This is probably attributable to the fact that the dealer's chance of breaking such a card decreases very more info as the deck gets rich in tens.

Linearity when the dealer shows an ace dealer hits soft 17 is much better because player's and dealer's chance of busting grow apace.

To estimate how much conditional improvement the Hi Opt provides with 20 cards remaining in the deck multiply the Table 1 entries in the second through fifth columns by.

You will observe that many of 99 the albeit technically correct parameters players memorize are virtually worthless.

TABLE 1 STRATEGIC FAVORABILITIES IN% AS A FUNCTION OF HI OPT PARAMETER Hi Opt parameter quoted is average number of points in deck.

Assuming 20 cards left in the deck and that the player holds 14 against a ten, he will gain.

A superstitious player who only counts sevens and stands when all of them are gone will gain 1.

An Explanation of Errors Even if not always realized in practice, the linear assump- tion that the player's conditional gain or loss is a constant times the difference between the proper critical index and the current value of his parameter provides a valuable perspective to illustrate the likely consequences of card counting errors.

Whatever their source the type 1 and 2 errors mentioned earlierthe player will either be changing strategy too often, equivalent to believing the critical index is less extreme than it really is, or not changing strategy enough, equivalent to believing the critical index is more extreme than it actually is.

The subject can perhaps be demystified by appeal to a graphic.

At a certain level of the deck the running count will tend to have a probability distribution like the one below, where the numbers inside the rectangles are the frequencies in % of the different count values.

Only the positive half of the distribution is shown.

This means that there will be neither gain nor loss from changing strategy for a running count of +2, but there will be a conditional loss at any count less than +2 and a conditional gain at any count greater than +2.

The much ban- died "assumption of linearity" means that the gain or loss will be precisely proportional to the distance of the actual running count from the critical count of +2.

Now suppose one was for whatever reason addicted to premature changing of strategy for counts of +1 or higher.

What we see, of course, is that counts closer to zero like +1 are much more likely to occur than the more extreme ones where most of the conditional profit lies.

To fix the idea in your mind try to show, using the diagram, that if the critical threshold value is +3, the player who changes strategy for +2 or above will lose more than the basic strategist who never changesand also will lose more than the perfect employer of the system can gain.

Indeed, the Baldwin group foresaw this in their book: "Ill considered changes will prob- ably do more harm than good.

Many players overemphasize the last few draws and, as a result, make drastic and costly changes in their strategy.

This suggests that it would be a service to both the memory and pocket book to round playing indices to the nearest conveniently remembered and more extreme value.

There is, as in poker, a tendency to "fall in love with one's cards"l which may cause pathologists to linger over unfavorable decks where much of this action is found for the sole purpose of celebrating their knowledge with a bizarre and eye-opening departure play.

This is an under- standable concomitant of the characteristic which best dif- ferentiates the casino blackjack player from the inde- pendent trials gambler, namely a desire to exercise control over his own destiny.

online blackjack pc Optimal Strategy for Pot Limit Poker.

The American Mathematical Monthly, Vol 82, No.

The instantaneous value of any point count system whether it uses +or - 0, 1, 2, 3, 4, 7, 11 etc.

It has already been shown in Chapter Five that cards assigned the value of zero are uncorrelated with the system's parameter and hence tend to have the same neutral distribution regardless of the sign or magnitude of the point count.

We shall now show that more generally, as the count fluctuates, we are entitled to presume a deflection in a card denomination's density proportional to the point value assign- ed to it.

Towards this end we again consider the +1, -12 indicator count for a particular denomination.

Our demonstration is concluded by observing that the deflection of the conditional mean of the indicator count from its overall mean will be proportional to this correlation, and hence proportional to Pk, as promised.

The deflections for negative counts with 39, 26, and 13 cards remaining can be obtained by merely changing the algebraic signs in the 13, 26, and 39 card positive count tables.

Observe that the Band E columns tend to be close in magnitude, but opposite in sign, the C column is generally close to zero, and the D column is about half of E.

This is what the ideal theory suggests will happen.

Table 4 was prepared by a probabilistic analysis of Hi Opt I parameters with 20, 30, and 40 cards left in a single deck.

The lessons to be learned from it would seem to apply to any count system.

Examined critical indices range from.

The body of the table quantifies the player's cumulative gain or loss from changing strategy with possible "action indices" as, or more, extreme than those which https://allo-hebergeur.com/blackjack/juegos-de-blackjack-en-linea-gratis.html in the left hand margin.

The units are arbitrarily scaled to avoid decimals; they would actually depend on the volatility of, and point count's correlation with, the particular situation considered.

For relatively small critical indices such as.

However for larger critical indices the player may lose more from such over- zealousness than someone else playing the system correctly can gain.

For example, it would seem innocuous to mistake a critical index of.

This table can also be used to assess how well a "running count" strategy would fare relative to a strategy based on a "true" knowledge of the average number of points remainingin the deck.

Imagine that the situation with critical index.

If opportunity arises three times, with 20, 30 and 40 cards remaining, the total 112 TABLE 4 ACTION CRITICAL INDICES INDEX.

A "running count" player, making no effort to adjust for depth in the deck, would gain less than this, depending on the critical running count he used.

Furthermore, such numbers, already ingrained in the memory, would not be readily conver- tible for multiple deck play.

Just another two bucks down the tube.

Thus to estimate what expectation our rules would produce for a double deck we would pick .

Likewise we could extrapolate a.

To begin with, almost half of the.

The double down pair often contains two cards the player does not wish to draw and their removal significantly im- proves the chance of a good hand from one deck but is negligi- ble otherwise.

A good example of ~ h i s is doubling nine against See page 170 for explanation of infinite deck.

Presumably the remaining discrepancy reflects the player's gain by judicious standing with stiff totals.

A stiff hand usually contains at least one card, and often several, which would help the dealer's up cards of two through six, against which this option is exercised, and the favorable effect of their removal i.

The Effect of Rule Changes In the next table the effect of some rule changes occasion- ally encountered is given for both one deck and an infinite number of decks.

The reader can use interpolation by the reciprocal of the number of decks to get an estimate of what the effects would be for two and four decks.

For instance, if doubling soft hands is forbidden in a four deck game, take one fourth of the difference between the .

Similarly, we get .

Notice how splitting is more valuable for the infinite deck due to the greater likelihood of pairs being dealt.

Doubling down after pair splitting is worth the same in each case because the reduced frequency of pairs in the single deck is nullified by the increased advantage on double downs.

Each row of the following table provides a comparison of the fluctuations in various numbers of decks by display of the number of remain- ing cards which would have the same degree of fluctuation associated.

If you're playing at that great blackjack table in the sky where St.

Peter deals and you know who is the pit bossyou'll have to wait an eternity, or until 2601 cards are left, before the degree of departure from normal composition is equivalent to that produced by the observation of the burn card from a standard pack of 52.

We see that the last few cards of a multiple deck can be slightly more read more for both betting and playing variations than the corresponding residue from a single deck.

However, it must be kept in mind that such situations are averaged over the entire deck when assessing overall favorability.

An in- teresting consequence of this is that even if one had the time to count down an infinite deck, it would do no good since the slightly spicier situations at the end would still average out to zero.

When we recall that the basic multiple deck games are in- herently less advantageous, the necessity of a very wide bet- ting range must be recognized.

Absolute efficiencies of card counting systems will decrease mildly, perhaps by three per cent for four decks.

Since this decrease will generally be uniform over most aspects of the game, relative standings of zynga poker blackjack nasıl oynanır systems should not differ appreciably from those quoted in Chapter Four.

The next table shows how much profit accrues from betting one ex- tra unit in favorable situations for two and four deck games played according to the rules generally presumed in Chapter Two.

GAIN PER HAND FROM BETTING ONE EXTRA UNIT IN FAVORABLE SITUATIONS % Number of Cards Remaining Double Deck Four Decks 10 2.

If a four 119 deck player's last hand is dealt with 60 cards left, we average all the gains including the.

This is the average profit per hand in %.

Although we've neglected strategy variation this is partially compensated by the assumption that the player diagnoses his basic strategy advantage perfectly.

The rest of the chapter will be devoted to certain uncom- mon but interesting variations in rules.

Since these usually oc- cur in conjunction with four deck games, this will be assumed unless otherwise specified.

No hole Card With "English rules" the dealer does not take a hole card, and in one version, the player who has doubled or split a pair loses the extra bet if the dealer has a blackjack.

In such a case the player minimizes his losses by foregoing eight splitting and doubling on 11 against the dealer's ten and ace and also not splitting aces against an ace.

The primary penalty paid is that the correct basic strategy is not used when the dealer doesn't have blackjack.

In another version, though, the player's built up 21 is allowed to push the dealer's natural; this favors the player by.

Surrender "Surrender" is another, more common, rule.

With this op- tion the player is allowed to give up half his bet without finishing the hand if he doesn't like his to understand blackjack rules />Usually this choice must be made before drawing any cards.

Since the 120 critical expectation for surrendering is .

They will also be useful for discussion of subsequent rule variations.

PLAYER'S EXPECTATION'!

Thus surrendering 16 v T saves the player.

Naturally, the precise saving depends on what cards the player holds and on how many decks are used, but these tables are quite reliable for four deck play.

Some casinos even allow "early surrender", before the dealer has checked his hole card for a blackjack.

This is quite a picnic for the knowledgeable player, particularly against the dealer's ace.

When this is done we get the following table of gain from proper strategy.

When surrender is allowed at any time, and not just on the first two cards, the rule will be worth almost twice as much for conventional surrender and either 10% or 50% more for early surrender depending on whether the dealer shows an ace or a ten.

Bonus for multicard Hands If the Plaza in downtown Las Vegas had had the "Six Card Automatic Winner" rule, I would have been spared the disap- pointment of losing with an eight card 20 to the dealer's three card 21.

Six card hands are not very frequent and the rule is worth about.

The expectation tables suggest a revised five card hitting strategy to cope with the rule in four decks: hit hard 17 v 9, T, and A; hit hard 16 and below v 2 and 3; hit hard 15 and below v 4,5, and 6.

Some Far Eastern casinos have a sort of reverse surrender rule called "Five Card," wherein the player may elect to turn in any five card hand for a payment to him of half his bet.

Again the table of expectations comes in handy, both for decisions on which five card hands to turn in and also for revision of four card hitting strategies.

A five card hand should be' turned in if its expectation is less than +.

A revised and ab- breviated four card strategy is as follows: Hit Soft 19 and Below Against Anything But a 7 or 8 Hit Hard 15 and Below Against a 2 Hit Hard 14 and Below Against a 3 and 4 Hit Hard 13 and Below Against a 5 and 6 Other changes in strategy are to hit all soft 18's against an ace, three card soft 18 against an 8, and hit three card 12 versus a 4.

Obviously there will be many other composition depend- ent exceptions to the conventional basic strategy which are not revealed by the infinite deck approximation to four deck or single deck play.

So the reader feels he's getting his money's worth I will divulge the only four card hard 14 which should be hit against the dealer's five.

In many of the casinos where "Five Card" appears, it col- lides with some of the other rule variations we have already discussed, creating a hydra-headed monster whose expectation cannot be analyzed in a strictly additive fashion.

For instance, if we have already "early surrendered" 14 v dealer Ten, we can neither tie the dealer's natural 21 allowed in Macao nor turn it into a five card situation.

The five card rule is a big money maker, though, being worth about.

This is in Macao, where the-- player can "five card" his way out of some of the dealer's ten- up blackjacks.

The following table gives the frequency of development of five card hands in a four deck game, with the one deck frequen- cy in parentheses next to the four deck figure.

A hand like 3,3,3,3,4with repetition of a particular denomination, will be much 21 como blackjack se juega prob- able for a single deck, but A,2,3,4,5with no repetition, occurs more often in the single deck.

Hands with only one repetition, like 2,3,4,4,5 are almost equally likely in either case and tend to make up the bulk of the distribution anyway.

When a bonus is paid for 6,7,8 of the same suit or 7,7, 7different strategy changes are indicated depending on how much it is.

We can use the infinite deck expectation table to ap- proximate how big a bonus is necessary for 6,7,8 of the same suit in order to induce us to hit the 8 and 6 of hearts against the dealer's two showing.

Suppose B is the bonus paid auto- matically if we get the 7 of hearts in our draw.

We must com- pare our hitting expectation of '" .

The equation B becomes .

Hence, with a 5 to 1 bonus we'd hit, but if it were only 4 to 1 we'd stand.

He gives a strategy for which a player expectation of 2.

Apparently some casino personnel have read Epstein's book, for, in October of 1979, Vegas World introduced "Double Exposure", patterned after zweikartenspiel except that the dealer hits soft 17 and the blackjack bonus has been discon- tinued, although the player's blackjack is an automatic winner even against a dealer natural.

The game is dealt from five decks and has an expectation of about .

In private correspondence about the origin of the game, Epstein "graciously cedes all claim of paternity to Braun.

The dealer stands on soft 17, double after split is permitted, but pairs may blackjack ti 84 plus split only once.

An analysis of the player's expectation for these rules will be useful for illustrating how to employ the information in this chapter.

To begin, we need an estimation of the six deck expec- tation for the typical rules generally presumed in this book.

In- terpolation by reciprocals suggests that the player's expecta- tion will be one sixth of the way between.

The right to double after split is worth.

Early surrender itself provides a gain of.

Summariz- ing, we adjust the previous figure of .

This truly philanthropic state of affairs led to much agony for the New Jersey casino interests!

Not only did the knowledgeable player have an advantage for a complete click of 312 cards, but it turns out that the early surrender rule results in greater fluctuations in the player's ad- vantage as the deck is depleted than those which occur in or- dinary blackjack.

An excess of aces and tens helps the player in the usual fashion when they are dealt to him, but the dealer's more frequent blackjacks are no longer so menacing in rich decks, since the player turns in many of his bad hands for the same constant half unit loss.

The effects of removing a single card of each denomination appear in the next table; even though Atlantic City games are all multiple deck the removals are from a phrase blackjack push ups assured deck so com- parisons can be made with other similar tables and methods presented in the book.

Unfortunately for the less flamboyant players who didn't get barred, a suit requiring casinos to allow card counters to play blackjack was ruled upon favorably by a New Jersey court.

This had as its predictable result the elimination of the surrender option and consequently what had been a favorable game for the player became an unfavorable one.

Under the new set of rules, in effect as of June 1981, the basic strategist's ex- pectation is .

For the correct six deck basic strategy see the end of Chapter Eleven.

The following chart of how much can be gained on each ex- tra unit bet on favorable decks may be of some use to our East Coast brethren for whom "it's the only game in town.

At one time I believed that the frequency of initial two card hands might be responsible for the difference between in- finite and single deck expectations.

However, multiplication of Epstein's single deck expectations by infinite deck pro- babilities of occurrence disabused me of the notion.

One possible justification for the interpolation on the basis of the reciprocal of the number of decks can be obtained by looking at the difference between the infinite deck probability of drawing a second card and the finite deck probability.

The probability of drawing a card of different denomination from one already possessed is 4k for k decks and the 52k-l corresponding chance of getting a card of the same denomina- tion is 4k-l.

The differences between these figures and 52k-l the constant I~which applies to an infinite deck, are I and I 2 respectively.

These differences 13 52k-l 13 52k-l themselves are very nearly proportional to the reciprocal of the number of decks used.

In the last case we 129 find ourselves in the position of.

But because blackjack application samsung the strain That it put on his brain, He chucked math and took up Divinity.

Different equations are necessary to evaluate early sur- render for different hands from one and four decks.

For in- stance, with T,2 v.

The player's loss of ties is greatly offset by aggressive splitting and doubling to exploit the dealer's visible stiff hands.

The magnitudes show Double Exposure to be far more volatile than ordinary blackjack.

There are surprisingly many two card, composition depen- dent, exceptions to the page 126 strategy: stand with A,7 v 8,3 and 7,6 and 8,5 v hard 11, except hit 8,5 v 9,2 ; double 7 v hard 13 other than T,3 ; hit T,6 v 6,2 and 9,7 v hard 7.

Practical casino conditions, however, make this impossible.

For one thing, a negative wager equiva- lent to betting on the house when they have the edge is not permitted.

When I first started playing, I religiously ranged my bets according to Epstein's criterion of survival.

Besides, when the truly degenerate gambler is wiped out of one bank he need only go back to honest work for a few months until he has another.

In my opinion the entire topic has probably been over- worked.

The major reason that such heavy stress has been placed on the problem of optimal betting is that it is one of the few which are easily amenable to solution by existing read article, rather than because of its practical importance.

The game resembles basic strategy blackjack with about 28 cards left in the deck, since for flat bets it is an even game, but every extra unit bet in favorable situations will earn 1.

Now, both Greta and Opie know before each play which situation they will be red redemption blackjack cheat dead />Opie bets optimally, in pro- portion to her advantage, 2 units with a 2% advantage and 6 units with the 6% edge, while Greta bets grossly, 4 units whenever the game is favorable.

Thereby they both achieve the same 3.

Starting with various bank sizes, their goals are to double their stakes without being ruined.

The results of 2000 simulated trials in each circumstance appear below.

NUMBER OF TIMES RUINED TRYING TO DOUBLE A BANK OF Opie Greta 20 877 896 50 668 733 100 438 541 200 135 231 Greta is obviously the more often ruined woman, but since they have the same expectation per play there must be a com- pensating factor.

This is, of course, time-whether double or nothing, Greta usually gets her result more quickly.

This il- lustrates the general truth pointed out by Thorp in his Favorable Games paper that optimal betting systems tend to be "timid", perhaps more so than a person who values her time would find acceptable.

You play every hand as if it's your last, and it might be, if you lose an insurance bet and split four eights in a losing cause!

Another common concern voiced by many players is whether to take more than one hand.

Again, practical con- siderations override mathematical theory since there may be no empty spots available near you.

A bit of rather amusing advice on this matter appeared in a book sold commercially a few years ago.

The author stated that "by taking two hands in a rich situation you reduce the dealer's probability of getting a natura!.

This brings to mind how so many, even well regarded, pundits of subjects such as gambling, sports, economics, etc.

Thus, we have the gambling guru who enjoins us to "bet big when you're winning," the sports announcer who feels compelled to attribute one team's scoring of several con- secutive baskets to the mysterious phantom "momentum," and the stock market analyst who cannot report a fall in price without conjuring up "selling pressure.

A trip to the dictionary confirms that this latter description is probably the most ac- curate in the book.

But to debunk mountebanks is to digress.

Nevertheless, there can be a certain reduction in fluctuations achievable by playing multiple spots.

Suppose we have our choice of playing from one to seven hands at a time, but with the restriction that we have the same amount of action every round every dealer hand.

Then the following table shows the relative fluctuation we could expect in our capital if we follow this pattern over the long haul.

Number of Hands 1 2 3 4 5 6 7 ---- -- Relative Fluctuation 1.

Assuming that we play each of our hands as fast as the dealer does his and ignoring shuffle time, then we can playa single spot on four rounds as often as seven spots on one round.

Similarly three spots could be played twice in the same amount of time.

Now, with our revised criterion of equal total action per time on the clock, our table reads: Number of Hands Relative Fluctuation 1 2 3.

Of course, all this ignores the fact that taking more hands requires more cards and might trigger shuffle up on the dealer's part if he didn't think there were enough cards to complete the round.

Or, sometimes there would be enough cards to deal once to two spots but not twice to one spot.

It's been my observation that when this third round is dealt to five players it's almost always because the first two rounds used very few and predominantly high cards; hence the remainder of the deck is likely to be composed primarily of low ones.

A good exam- ple to illustrate the truncated distribution which results can be obtained by reverting to a simplistic, non-blackjack example.

Consider a deck of four cards, two red and two black.

As in Chapter Four, the dealer turns a card; the player wins if it's red and loses on black.

Ostensibly we have a fair game, but now imagine an oblivious, unsuspecting player and a card-counting, preferentially shuffling dealer.

Initially there are six equally likely orderings of the deck.

RRBB RBRB RBBR BBRR BRBR BRRB Since the dealer is trying to keep winning cards from the player, only the enclosed ones will be dealt.

The effect, we see, is the same as playing one hand from a deck of 14 cards, 9 of which are black.

As an exercise of the same type the reader might start with a five card deck, three red and two black.

The answer depends on how often the deck is reevaluated; blackjack uses typically four to twenty-four cards per round, depending on the number of players.

The following chart shows the percentage of tens that would be dealt as a function of the size of the clump of cards the dealer observes before making his next decision on whether to reshuffle.

Percentage of Tens Played 31 30 29 28 27 26 o 13 26 39 52 Number of Cards in Clump Between Reevaluations of Deck Since about five or six cards are usually used against a single player we can conclude that the dealer could reduce the proportion of tens dealt to this web page 26.

This would give the basic strategist a 1.

By using a better correlated betting count to decide when to reshuffle, the house edge could probably be raised to 2%.

In all honesty, though, I think we must recognize that player card-counting is just the obverse of preferential shuffling-what's sauce for the goose is also for the gander.

While on the subject, it might be surprising that, occa- sionally, the number of times the dealer shuffles may influence 136 the player's expectation.

New decks all seem to be brought to the table with the same arrangement when spread: A23.

If the dealer performs a perfect shuffle of half the deck against the other half, then, of course, the resultant order is deterministic rather than random.

Is it a coincidence that one of the major northern Nevada casinos has a strict procedure calling for five shuffles of a new deck, but three thereafter?

Even experienced dealers would have some difficulty try- ing to perform five perfect shuffles a "magician" demonstrated the skill at the Second Annual Gambling Con- ference sponsored by the University of Nevadabut to get some idea of what might happen if this were attempted, I ask- ed a professional dealer from the Riverside in Reno to try it.

He sent me the resultant orderings for eight such attempts.

Although a result of this sort is not particularly significant in that it, or something worse, would occur about 7% of the time by chance alone, none of the eight decks favored the player.

Previous Result's Effect on next Hand Blackjack's uniqueness is the dependence of results before reshuffling takes place.

While the idea that a previous win or loss will influence the next outcome is manifest nonsense for independent trials gambles like roulette, dice, or keno, it is yet conceivable that in blackjack some way might be found to pro- fitably link the next bet to the result of the previous one.

Wilson discusses the intuition blackjack age florida if the player wins a hand, this is evidence that he has mildly depleted the deck somewhat of the card combinations which are associated with himwinning, and hence he should expect a poorer than average result next time.

My resolution to the question, when it was first broached to me, was to perform a Bayesian analysis 137 through the medium of the Dubner Hi Lo index.

This led to the tentative conclusion that the player's expectation would be reduced by perhaps.

Gwynn also found that a push on the previous hand is apparently a somewhat worse omen for the next one than a win is.

It follows, then, that the player's prospects must improve following a loss, although of course not much, certainly not enough to produce a worthwhile betting strategy.

When all is said and done, the most immediate determiner of the player's advantage is the actual deck composition he'll be facing, and knowledge of whether he won, lost, or pushed the last hand, in itself, really tells us very little about what cards were likely to have left the deck, and implicitly, which ones remain.

Epstein proposes minimizing the probability of ruin sub- ject to achieving an overall positive expectation.

Thus it is generally consistent with the famous Kelly criterion for maximizing the exponential rate of growth.

click here reasonable principle which leads to proportional wagering is that of minimizing the variance of our outcome subject to achieving a fixed expectation per play.

Suppose our game consists of a random collection of subgames indexed by i occurring with probability Pi and having corresponding ex- pectation E i.

Opie's difference equation is of order 12and even more in- tractable.

Increasing their bets effectively diminishes their capital, and when this is taken into account we come up with the following approximations to the ideal fre- quencies of their being ruined, in startling agreement with the simulations.

With this formulation we can approximate gambler's ruin probabilities and also estimate betting frac- tions to optimize average logarithmic growth, as decreed by the Kelly criterion.

In 141 Chapter Eleven the value 1.

It seems doubtful that it would vary appreciably as the deck composition changes within reasonable limits.

Certain properties of long term growth are generally ap- pealed to in order to argue the optimality of Kelly's fixed frac- tion betting scheme, and are based on the assumption of one bankroll, which only grows or shrinks as the result of gambling activity.

The questionable realism in the latter assumption, the upper and lower house limits on wagers, casino scrutiny, and finiteness of human life span all contribute to my lack of enthusiasm for this sort of analysis.

Precisely the suggested scenario could unfold: a hand could be dealt from a residue of one seven; one nine; four threes, eights, and aces each; and ten tens.

This would have a putative advantage of about 12%, and call for a bet in this pro- portion to the player's current capital.

In fact, ignoring the table limits of casinos, the conjec- tured catastrophe would be guaranteed to happen and ruin the player sooner or later.

This is opposed to the Kelly idealization wherein, with only a fixed proportion of capital risked, ruin is theoretically impossible.

From simulated hands I estimate the covariance of two blackjack hands played at the same table to be.

Since the variance of a blackjack hand is about 1.

The first table of relative fluctuation is obtained by multiplying yen by and then taking the square root of the ratio of this quantity to V 7.

One of the problems encountered in approximating black- jack betting situations is that the normal distribution theory assumes that all subsets are equally likely to be encountered at any level of the deck.

An obvious counterexample to this is the 48 card level, which will occur in real blackjack only if the first hand uses exactly four cards.

The only imaginable favorable situation which could occur then would be if the player stood with something like 7, 5 v 6 up, A underneath.

The other problem is the "fixed shuffle point" predicted very well by David Heath in remarks made during the Second Annual Gambling Conference at Harrah's, Tahoe.

Gwynn's simulations, using a rule to shuffle up if 14 or fewer cards re- mained, confirmed Heath's conjecture quite accurately.

Roughly speaking, almost every deck allowed the completion of seven rounds of play, but half the time an eighth hand would be played and it tended to come from a deck poor in high cards, resulting in about a 2.

I resolved it by assuming a bet diagnosis arose every 5.

Preferential shuffling presents an interesting mathematical problem.

For example, if the preferential shuf- fler is trying to keep exactly one card away from the player, he can deal one card and reshuffle if it isn't the forbidden card, but deal the whole deck through if the first one is.

I carried out the Bayesian analysis by using an a priori Hi Lo distribution of points with six cards played and a com- plicated formula to infer hand-winning probabilities for the dif- ferent values of Hi Lo points among the six cards assumed used.

From this was generated an a posteriori distribution of the Hi Lo count, assuming the player did win the hand.

A player win was associated with an average drain of.

This translates into about a.

There is an apparent paradox in that blackjack tips om te winnen cards whose removal most favors the player before the deal are also the cards whose appearance as dealer's up card most favors the player.

Thus an intuitive understanding of the magnitude and direction of the effects is not easy to come by.

The last line tabulates the player's expectation as a function of his own initial card and suggests a partial explanation of the "contradiction", although the question of why the player's first card should be more important than the dealer's is left open.

Let Xl be the number of stiff totals 12-16 which will be made good by the particular denomination considered and X 2 be the number of stiff totals the card will bust.

Finally, to mirror a card's importance in making up a blackjack, define an artificial variable X g to be equal to one for a Ten, four for an Ace, and zero otherwise.

The following equation enables prediction of the ultimate strategy effects with a multiple correlation of.

He wants the deck to be rich in tens, but not too rich.

Some authors, who try to explain why an abundance of tens favors the player, state that the dealer will bust more stiffs with ten rich decks.

This is true, but only up to a point.

The dealer's probability of busting, as a function of ten density, appears to maximize.

This compares with a normal.

The player's advantage, as a function of increasing ten density, behaves in a similar fashion, rising initially, but necessarily returning to zero when there are only tens in the deck and player and dealer automatically push with twenty each.

It reaches its zenith almost 13% when 73% of the cards are tens.

Strangely, a deck with no tens also favors the player who can adjust his strategy with sufficient advantage to overcome the.

Thorp presents the classic example of a sure win with 7,7, 8,8,8 remaining for play, one person opposing the dealer.

This may be the richest highest expec- tation subset of a 52 card deck.

An infinite deck composition of half aces and half tens maximizes the player's chance for blackjack but gives an expectation of only 68% whereas half sevens and half eights will yield an advantage of 164%.

An ordinary pinochle deck would give the player about a 45% advantage with proper strategy, assuming up to four cards could be split.

Insurance would always be taken when of- fered; hard 18 and 19 would be hit against dealer's ten; and, finally, A,9 would be doubled and T,T split regardless of the dealer's up card.

Certainly a deck of all fives would be devastating to the basic strategist who would be forever doubling down and losing, but optimal play would be to draw to twenty and push every hand.

The results of a program written to converge to the worst possible composition of an infinite deck suggest this lower bound can never be achieved.

No odd totals are possible and the only "good" hands are 18 and 20.

The player cannot be dealt hard ten and must "mimic the dealer" with only a few insignificant departures principally standing with 16 against Ten and split- ting sixes against dealer two and six.

The dealer busts with a probability of.

The creation of this pit boss's delight a dealing shoe gaffed in these proportions would provide virtual immunity from the depredations of card counters even if they knew the composition may be thought of as the problemof increasing the dealer's bust probability while simultaneously leaching as many of the player's options from "mimic the dealer" strategy as possible.

It 148 can be verified that proper strategy with a sevenless deck is to stand in this situation and a thought experiment should con- vince the reader that as we add more and more sevens to the deck we will never reach a point where standing would be cor- rect: suppose four million sevens are mixed into an otherwise normal deck.

Then hitting 16 will win approximately four times and tie once out of a million attempts, while standing wins only twice when dealer has a 5 or 6 underneath and never ties!

Calculations assume the occurrence of two non- sevens is a negligible second order possibility.

The addition of a seven decreases the dealer's chance of busting to more than offset the player's gloomier hitting prognosis.

In the following table we may read off the effect of remov- ing a card of each denomination on the dealer's chance of busting for each up-card.

The last line confirms that the removal of a seven increases the chance of busting a ten by.

EFFECT OF REMOVAL ON DEALER'S CHANCE OF BUSTING in % DENOMINATION REMOVED Dealer's Chance Sum of of Up Card A 2 3 4 S 6 7 8 9 T Bust Squares - - - ----- A .

What we learn from the magnitudes of numbers in the "Sumof Squares" column is that the probability of read article tens and nines fluctuates least as the deck is depleted, while the chance of breaking a six or five will vary the most.

This is in keeping with the remarks in Chapter Three about the volatility ex- perienced in hitting and standing with stiffs against large and small cards.

The World's Worst Blackjack Player Ask "who is the best blackjack player?

Watching a hopeless swain stand with 3,2 v T at the Barbary Coast in Las Vegas rekindled my interest in the question "who is the world's worst player and how bad is he?

Penalty in % Always insure blackjack Always insure T,T Always insure anything Stand on stiffs against high cards Hit stiffs against small cards Never double down Double ten v T or A Always split and resplit T,T Always split 4,4 and 5,5 Other incorrect pair splits Failure to hit soft 17 Failure to hit soft 18 v 9 or T Failure to hit A,small 150.

Hence it seems unlikely that any but the deliberately destructive could give the house more than a 15% edge.

This is only a little more than half the keno vigorish of 26%: the dumbest blackjack player is twice as smart as any keno player!

Observations I made in the spring of 1987 showed that the overall casino advantage against a typical customer is about 2%.

The number and cost of players' deviations from basic strategy were recorded for 11,000 hands actually played in Nevada and New Jersey casinos.

The players misplayed about one hand in every 6.

This translates into an expectation 1.

Other findings: Atlantic City players were closer to basic strategy than those in Nevada, by almost.

The casinos probably win less than 1.

Incidentally, standing with A,4 v T is more costly by 13% than standing with 3,2.

It's only because we've grown more accustomed to seeingthe former that we regard the latter as the more depraved act.

One player, when innocently asked why he stood on A,5replied "Evenif I do get a ten emphasis to indicate that he apparently thought this was the best of all possible draws I still would only have 16".

The Unfinished Hand Finally, let the reader be apprised of the possibility of an "unfinished'; blackjack hand.

Imagine a player who splits sixteen tens and achieves a total of twenty-one on each hand by drawing precisely two more cards.

The dealer necessarilyhas an ace up, ace underneath, but cannot complete the hand.

By bouse rules she is condemnedthroughout eternity to a Dante's Inferno task of shuffling the last two aces, offer- ing themto the player for cut, attempting to hit her own hand, and rediscovering that they are the burn and bottom cards, unavailable for play!

Minimization of a function of ten variables is not an easy thing to do.

In this case the ten variables are the densities of the ten distinct denominations of cards and the function is the associated player advantage.

Although I cannot prove this is the worst deck, there are some strong arguments for believing it is: 1.

The minimum of a function of many variables is often found on the boundary and with seven denominations having zero densities we definitely are on a boundary.

To approach, the required bust prob- ability for the theoretically worst deck there would have to be some eights, nines, or tens.

If there are eights or nines, their splitting would probably provide a favorable option to "mimic the dealer" strategy which would reduce the 17% disadvantage from stand- ing with all hands.

Also, if there are nines or tens, the player will occasionally, with no risk of busting, reach good totals in the 17 to 21 range, thus achieving a bet- ter expectation than "never bust" strategy was assumed to yield.

Either way, the theoretical -17% is almost certainly not achievable.

There's an intuitive argument for having only even cards in the "worst deck" - once any odd card is in- troduced then all totals from 17 to 26 can be reached.

Half of these are good and half bad.

But with only even cards you can only reach 18,20,22,24, and 26, three out of five of which are busts.

This reduced flexibility should help in raising the dealer bust probability while simultaneously minimizing the player's options.

Assuming only even numbers, the eights are filtered out because they provide favorable splits for the player.

The fours make good any totals of 14 and 16 and hence lower the dealer bust probability.

The twos are tantalizers in that they bring home only totals of 16 for the dealer, but keep other stiffs stiff for another chance of being busted.

At one of his seminars, the author instructs Sue of the Sacramento Zoo in the art of playing natural 21.

This has been due largely to a concernfor playingthe subsequentlyderivedhands optimally, depending on the cards used on earlier parts of the split.

This begs a distinction between "basic" and "zero-memory" strategy, and will lead us to an algorithm for exact determina- tion of repeated pair splitting expectation with zero-memory strategy.

Imagine you are playing single deck blackjack and have split three deuces against a four.

To each of the first two deuces you draw two sevens, and on the third deuce you receive a ten.

It is basic play to hit T,2 vs.

If you answered yes to the previous question, suppose the first two deuces were busted with two tens each.

You are dealt an ace and a nine to the third deuce.

Now answer the previous question.

Indeed Epstein suggests that zero-memory implies knowledge only of the player's original pair and dealer's up card.

Now, consider splitting eights against a seven in a single deck: A.

Calculate the conditional expectation for starting a hand with an eight against dealer's seven given that 1.

The second condition in A.

The player's expectation from repeated pair splitting is now given by 1081 - 2 A + 1176 90 - 3- B + 1176 155 5 - 4 -C 1176 The three fractions are, of course, the probabilities of splitting two, three, and four eights.

The extension to two and four decks is immediate.

Let E I be the previously described conditional expectation if exactly I cards are split and hence removed and P I be the prob- ability that I cards will be split; then the pair splitting expecta- tion is: I: P I I 1 I ~ 2 The coefficients, I - P Ishrink rapidly and a very satisfac- tory estimate of E J for J ~ 3 could be achieved by extrapola- tion from the calculated value of E 2.

To do this we introduce an artificial E l without any reference to pair splittingas the weighted average expectation of the hands 8,A 8,2.

These expectations would already be available from the general blackjack program and provide us with the base point for our extrapolation.

P 2 : Single Deck 47 46 --- 49 48 Double Deck 95 94 --- 101 100 Four Deck 191 190 ---- 205 204 Infinite Deck 12 12 13 13 R I : 5-I 48-1 9-1 96-1 17-I 192-1 12 53-21}- 52-21 105-21 - 104-21 209-21 - 208-21 169 The factors R I reflect the probability of "opening" drawing a new eight to a split eight and "closing" drawing a non-eight to an already split eight the Ith split card.

To three decimals the P I are I Single Deck Double Deck Four Deck Infinite Deck 2.

This all assumes X,X is being split against Y I: X.

Sim- ple modifications can be made if it is X,X against X.

The following table of the number of distinguishably different drawing sequences for one and many decks suggests the relative amount of computer time required.

Ten One Deck 5995 16390 10509 6359 3904 2255 1414 852 566 288 Many Decks 8497 18721 11125 6589 4024 2305 1441 865 577 289 A program can be written in BASIC in as few as 28 steps to cycle through all of the dealer's drawing sequences and weight the paths for a prescribed up card and deck composi- tion.

Thorp counted the total number of distinguishable blackjack subsets of a single deck as 5 9.

Since there are 2 52 possible subsets there is an average duplication with respect to suit and ten denomination of about 130 million.

The realization that there are only 1993 different subsets of size five was embarrassing to me, since I had simulatedthem2550 times to test the validity of using the normal distribution approximation for the least squares linear estimators of deck favorability for varying basic strategy.

The results provide a worst case evaluation of the ac- curacy of the previously mentioned approximation since the in- teractions neglected by the linear estimates are most severe for small subsets and the normal approximation to their distribution is poorest at the beginning and end of the deck.

The ~ 1 and 5 ; subsets that might be encountered for a given up' card are achieved by weighting the distinguishable 159 subsets properly.

Favorability of hitting over standing was recorded for abstract totals.

The dealer was assumed to stand on soft 17 and the few unresolved situations were completed by a formula which reasonably distributed the dealer's unfinished total on the shuffle up.

Actual frequencies of, and gain from, violating the basic strategy were recorded.

The performances of Hi Opt I, Hi Opt II, and the Ten count were recorded in these situations.

The meaning of the following charts is best explained by example: With five cards left in the deck, perfect knowledge of when to hit hard 14 against a two is worth 16.

The conditional favorability of hitting in those situations where it is ap- propriate is 16.

In parentheses besides these figures appears the corresponding normal approximation estimate of potential gain 15.

The Hi Opt I, Hi Opt II, and Ten Count systems had respective efficiencies of 71, 77, and 68%.

With six cards left in the deck precisely optimal hitting will occur.

The figures do not reflect the likelihood of the dealer having the given up card or the player possessing the particular total.

The greatest condi- tional gain in a hitting situation is the 40.

The greatest conditional standing gain is almost 40%, with 16 vs 7.

Hitting hard 17 is the most important variation against an eight and a system which counts A, 2, 3, 4 low and 6, 7, 8, 9 high would be nearly.

Random Subsets stratified according to Ten Density To examine the behavior of the normal approximation estimates for larger subsets, 3000 each of sizes 10 through 23 were simulated by controlling the number of tens in each subset to reflect actual probabilities.

The only up-card con- sidered was the ten because of the rapidity of resolution of the dealer's hand.

The effect of this stratification could thus be expected to be a reduction in the variance of the sample distributions pro- portional to the square of the Ten Count's correlation coeffi- cients for the six situations examined.

In addition to this reduction in variance of typically 40%, there wouldbe the added bonus of saving computer time by not having to select the ten- valued cards using random numbers.

The results provide the continuum necessary to compare different card counting systems.

Again, the following charts are best explained by example: with 10 cards left in the deck it was proper to stand with twelve in.

The gain over basic strategy was 3.

The Ten Count was 28% efficient, and a "special" system based on go here density understand real steel blackjack toy right! the sevens, eights, and nines scored an impressive 78%.

The loss shown for the Ten counter playing a total of twelve with 21 cards left indicates the critical subsets with ex- actly 10 tens in them probably had an unduly large number of sevens, eights, and nines.

A basic strategist who always hits twelve would have done better in this instance.

II Special "6-5" 10.

Extreme discontinuities in efficiencies as a function of the number of cards in the subset can usually be explained by one of the system's realizable values being very close to its critical change of strategy parameter.

For example, the Ten Count's critical change ratio for standing with 15 is close to 2 others to 1 ten, and efficiencies take a noticeable dip with 12, 15, 18, and 21 cards in the deck.

In such cases the card counting system, whether it suggests a change in strategy or not, is us- ing up a considerable part of its probability distribution in very marginal situations.

Stratified Sampling used to analyze Expectation in a particular Deck The following approximate computations show that the variance of a blackjack hand result is about 1.

Player Approximate Squared Result Probability Result 2.

The sample sum will have a variance of 13 1.

It is worthwhile to study the consequences of stratified, rather than random, sampling.

Let the thirteen hands now be played against each of the denominations ace through king as dealer up-card.

Then the of the sum would obey 13 13 13 Var.

Thus the average variance for these stratified sample observations has been reduced from 1.

Using Epstein's tables of player expectation as a function of dealer up-card, we find this average square to be.

The same principle, albeit with more elaborate symbolism, can be used to show that controlling the player's first card as 168 well as the dealer's up-card will reduce variance by.

The average squared expectation for three card situations, where player's hand and dealer's up card are specified, is.

To get an approximation to how much variance reduction would result if four or more cards were forced to obey exact probability laws in the sample, we can assume the resolved hands have the same 1.

Then, employing some of Gwynn's computer results which showthat about 17% of all hands require four cards, 40% five cards, 28% six cards, 11% seven cards, and 4% eight or more cards, we complete the following table: Number of Cards Controlled Precisely o 1 2 3 4 5 6 7 8 Average Squared Expectation.

Beyond this, however, lurk even greater savings in computer time since the number of cards actually simulated with random numbers would be very few.

This would have the same variance as a purely random sample of about 25 million hands.

Moreover, only about 5 million cards would have to be generated to complete the 8.

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Generally considered the bible for serious blackjack players, Peter Griffin's classic work provides insight into the methods and numbers behind the development https://allo-hebergeur.com/blackjack/blackjack-sls-c-1-fr-s-p.html today's card-counting systems.

It contains the most complete and accurate basic strategy.

Generally considered the bible for serious blackjack players, Peter Griffin's classic work provides insight into the source and numbers behind the development of today's card-counting systems.

It contains the most complete and accurate basic strategy.

Generally considered the bible for serious blackjack players, Peter Griffin's classic work provides insight into the methods and numbers behind the development of today's card-counting systems.

It contains the most complete and accurate basic strategy.

The THEORY of BLACKACK The Compleat Card Counter's Guide to the Casino Game of 21 PETERA.

GRIFFIN ::J: c: z G' a z ~ en Las Vegas, Nevada The Theory of Blackjack: The Compleat Card Counter's Guide to the Casino Game of 21 Published by Huntington Press 3687 South Procyon Avenue Las Vegas, Nevada 89103 702 252-0655 vox 702 252-0675 fax Copyright 1979, Peter Griffin 2nd Edition Copyright 1981, Peter Griffin 3rd Edition Copyright 1986, Peter Griffin 4th Edition Copyright 1988, Peter Griffin 5th Edition Copyright 1996, Peter Griffin ISBN 0-929712-12-9 Cover design by Bethany Coffey All rights reserved.

No part of this publication may be translated, reproduced, or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without the expressed written permission of the copyright owner.

TABLE OF CONTENTS FOREWORD TO THE READER 1 INTRODUCTION 1 Why This Book?

This book will not teach you how to play blackjack; I assume you already know how.

Individuals who don't possess an acquaintance with Thorp's Beat The Dealer, Wilson's Casino Gambler's Guide, or Epstein's Theory of Gambling and Statistical Logic will probably find it inadvisable to begin their serious study of the mathematics of blackjack here.

This is because I envision my book as an extension, rather than a repetition, of these excellent works.

Albert Einstein once said "everything should be made as simple as possible, but no simpler.

However, I recognize that the readers will have diverse backgrounds and accordingly I have divided each chapter into two parts, a main body and a subsequent, parallel, "mathematical appendix.

Thus advised, they will then be able to skim over the formulas and derivations which mean lit- tle to them and still profit quite a bit from some comments and material which just seemed to fit more naturally in the Appen- dices.

Different sections of the Appendices are lettered for con- venience and follow the development within the chapter itself.

The Appendix to Chapter One will consist of a bibliography of all books or articles referred to later.

When cited in subsequent chapters only the author's last name will be mentioned, unless this leads to ambiguity.

For the intrepid soul who disregards my warning and insists on plowing forward without the slightest knowledge of blackj ack at all, I have included two Supplements, the first to acquaint him with the rules, practices, and terminol- ogy of the game and the second to explain the fundamental principles and techniques of card counting.

These will be found at the end of the book.

Revised Edition On November 29, 1979, at 4:30 PM, just after the first edi- tion of this book went to press, the pair was split for the first time under carefully controlled laboratory conditions.

Con- trary to original fears there was only an insignificant release of energy, and when the smoke had cleared I discovered that splitting exactly two nines against a nine yielded an expecta- tion of precisely .

Only minutes later a triple split of three nines was executed, produc- ing an expectation of .

Development of an exact, composition dependent strategy mechanism as well as an exact, repeated pair splitting algorithm now enables me to update material in Chapters Six, Eight, and, particularly, Eleven where I present correct basic strategy recommendations for any number of decks and dif- ferent combinations of rules.

There is new treatment of Atlantic City blackjack in Chapters Six and Eight.

In read article the Chapter Eight analysis of Double Exposure has been altered to reflect rule changes which have occurred since the original material was written.

A fuller explication of how to approximate gambler's ruin probabilities for blackjack now appears in the Appendix to Chapter Nine.

A brand new Chapter Twelve has been writ- ten to bring the book up to date with my participation in the Fifth National Conference on Gambling.

Elephant Edition In December, 1984, The University of Nevada and Penn State jointly sponsored the Sixth National Conference on Gambling and Risk Taking in Atlantic City.

The gar- gantuan simulation results of my colleague Professor John Gwynn of the Computer Science Department at California State University, Sacramento were by far the most significant presentation from a practical standpoint and motivated me to adjust upwards the figures on pages 28 and 30, reflecting gain from computer-optimal strategy varia- tion.

My own contribution to the conference, a study of the nature of the relation between the actual opportunity occur- ring as the blackjack deck is depleted and the approxima- tion provided by an ultimate point count, becomes a new Chapter Thirteen.

In this chapter the game of baccarat makes an unexpected appearance, as a foil to contrast with blackj ack.

Readers interested in baccarat will be rewarded with the absolutely most powerful card counting methods available for that game.

Loose ends are tied together in Chapter Fourteen where questions which have arisen in the past few years are answered.

Perhaps most importantly, the strategy tables of Chapter Six are modified for use in any number of decks.

This chapter concludes with two sections on the increasingly popular topic of risk minimization.

It is appropriate here to acknowledge the valuable assistance I have received in writing this book.

Thanks are due to: many individuals among whom John Ferguson, Alan Griffin, and Ben Mulkey come to mind whose conver- sations helped expand my imagination on the subject; John Christopher, whose proofreading prevented many ambigui- ties and errors; and, finally, readers Wong, Schlesinger, Bernhardt, Gwynn, French, Wright, Early, and especially the eagle-eyed Speer for pointing out mistakes in the earlier editions.

Photographic credits go to Howard Schwartz, John Christopher, Marcus Marsh, and the Sacramento Zoo.

To John Luckman "A merry old soul was he" Las Vegas will miss him, and so will I.

Rosenbaum I played my first blackjack in January, 1970, at a small club in Yerington, Nevada.

Much to the amusement of a local Indian and an old cowboy I doubled down on A,9 and lost.

No, it wasn't a knowledgeable card counting play, just a begin- ner's mistake, for I was still struggling to learn the basic strategy as well as fathom the ambiguities of the ace in "soft" and "hard" hands.

The next day, in Tonopah, I proceeded to top this gaffe by standing with 5,4 against the dealer's six showing; my train of thought here had been satisfaction when I first picked up the hand because I remembered what the basic strategy called for.

I must have gotten tired of waiting for the dealer to get around to me at the crowded table since, after the dealer made 17 and turned over my cards, there, much to everyone's surprise, was my pristine total of nine!

At the check this out, I was preparing to give a course in The Mathematics of Gambling which a group of upper division math majors had petitioned to have offere4.

It had occurred to me, after agreeing to teach it, that I had utterly no gambling experience at all; whenever travelling through ~ J e v a d a with friends I had always stayed outside in the casino parking lot to avoid the embarrassment of witnessing their foolishness.

But now I had an obligation to know first hand about the subject I was going to teach.

An excellent mathematical text, R.

Epstein's Theory of Gambling and Statistical Logic, had come to my attention, but to adequately lead the discussion of our supplementary reading, Dostoyevsky's The Gambler, I clearly had to share this experience.

What the text informed me was that, short of armed robbery or counterfeiting chips and I had considered thesethere was only one way to get my money back.

Soon, indeed, I had recouped my losses and was playing with their money, but it wasn't long before the pendulum swung the other way again.

Although this book should prove interesting to those who hope to profit from casino blackjack, I can offer them no encouragement, for today I find myself far- ther behind in the game than I was after my original odyssey in 1970.

I live in dread that I may never again be able to even the score, since it may not be possible to beat the hand held game and four decks bore me to tears.

My emotions have run the gamut from the inebriated ela- tion following a big win which induced me to pound out a chorus of celebration on the top of an occupied Reno police car to the frustrated depths of biting a hole through a card after picking up what seemed my 23rd consecutive stiff hand against the dealer's ten up card.

My playing career has had a sort of a Faustian aspect to it, as I began just click for source explore the mysteries of the game I began to lose, and the deeper Go here delved, the more I lost.

There was even a time when I wondered if Messrs.

Thorp, Wilson, Braun, and Eps- tein had, themselves, entered into a pact with the casinos to deliberately exaggerate the player' s o d d ~ in the game.

But after renewing my faith by confirming.

Why then should I presume to write a book on this sub- ject?

Perhaps, like Stendahl, "I prefer the pleasure of writing all sorts of foolishness to that of wearing an embroidered coat 2 costing 800 francs.

But I do have a knowledge of the theoretical probabilities to share with those who are in- terested; unfortunately my experience offers no assurance that these will be realized, in the short or the long run.

Shaw's insight: If you can do something, then you do it; if you can't, you teach others to do it; if you can't teach, you teach people to teach; and if you can't do that, you administrate.

I must, I fear, like Marx, relegate myself to the role of theoretician rather than active revolutionary.

Long since disabused of the notion that I can win a fortune in the game, my lingering addiction is to the pursuit of solutions to the myriad of mathematical questions posed by this intriguing game.

Difficulty interpreting Randomness My original attitude of disapproval towards gambling has been mitigated somewhat over the years by a growing ap- preciation of the possible therapeutic benefits from the intense absorption which overcomes the bettor when awmting the ver- dict of.

Indeed, is there anyone who, with a wager at stake, can avoid the trap of trying to perceive patterns when confronting randomness, of seeking "purpose where there is only process?

Not long ago a Newsweek magazine article described Kirk Kerkorian as "an expert crapshooter.

Nevertheless, while we can afford to be a bit more sympathetic to those who futilely try to impose a system on dice, keno, or roulette, we should not be less impatient in urging them to turn their attention to the dependent trials of blackjack.

Blackjack's Uniqueness This is because blackjack is unique among all casino games in that it is a game in which skill should make a dif- ference, even-swing the odds in the player's favor.

Some will also enjoy the game for its solitaire-like aspect; since the dealer has no choices it's like batting a ball against a wall; there is no oppo- nent and the collisions of ego which seem to characterize so many games of skill, like bridge and tournament sands reno blackjack, do not occur.

Use of Computers Ultimately, all mathematical problems related to card counting are Bayesian; they involve conditional probabilities subject to information provided by a card counting parameter.

It took me an inordinately long time to realize this when I was pondering how to find the appropriate index for insurance with the Dubner HiLo system.

Following several months of wasted bumbling I finally realized that the dealer's conditional probability of blackjack could be calculated for each value of the HiLo index by simple enumerative techniques.

My colleague, Professor John Christopher, wrote a computer program which provided the answer and also introduced me to the calculating power of the device.

To him lowe a great debt for his patient and priceless help in teaching me how to master the machine myself.

More than once when the computer rejected or otherwise played havoc with one of my programs he counseled me to look for a logical error rather than to persist in my demand that blackjack rules card 7 elec- trician be called in to check the supply of electrons for purity.

After this first problem, my interest became more general.

Why did various count strategies differ occasionally in their recommendations on how to play some hands?

What determin- ed a system's effectiveness anyway?

How good were the ex- isting systems?

Could they be measureably improved, and if so, how?

Although computers are a sine qua non for carrying out lengthy blackjack calculations, I am not as infatuated by them as many of my colleagues in education.

It's quite fashionable these days to orient almost every course toward adaptability to the computer.

To this view I raise the anachronistic objec- tion that one good Jesuit in our schools will accomplish more than a hundred new computer terminals.

In education the means is the end; how facts and calculations are produced by our students is more important than how many or how precise they are.

Fascinated by Buck Rogers gadgetry, they look forward to wiring themselves up like bombs and stealthily plying their trade under the very noses of the casino personnel, fueled by hidden power sources.

For me this removes the element of human challenge.

The only interest I'd have in this machine a very good approxima- tion to which could be built with the information in Chapter Six of this book is in using it as a measuring rod to compare how well I or others could play the game.

Indeed one of the virtues I've found in not possessing such a contraption, from which answers come back at the press of a button, is that, by having to struggle for and check approximations, I've developed in- sights which I otherwise might not have achieved.

Cheating No book on blackjack seems complete without either a warning about, or whitewashing of, the possibility of being cheated.

I'll begin my comments with the frank admission that I am completely incapable of detecting the dealing of a second, either by sight or by sound.

Nevertheless I know I have been cheated on some occasions and find myself wondering just how often it takes place.

The best card counter can hardly expect to have more than a two percent advantage over the house; blackjack system peach icandy travel if he's cheated more than one hand out of fifty he'll be a loser.

I say I know I've been cheated.

I'll recite only the obvious cases which don't require proof.

I lost thirteen hands in a row to a dealer before I realized she was deliberately interlacing the cards in a high low stack.

Another time I drew with a total of thirteen against the dealer's three; I thought I'd busted until I realized the dealer had delivered two cards to me: the King that broke me and, underneath it, the eight she was clumsily trying to hold back for herself since it probably would fit so well with her three.

I had a dealer shuffle up twice during a hand, both times with more than twenty unplayed cards, because she could tell that the card she just brought off the deck would have helped me: "Last card" she said with a quick turn of the wrist to destroy the evidence.

Then she either didn't or did have blackjack depending it seems, on whether they did or didn't insure; unfortunately the last time when she turned over her blackjack there was also a four hiding underneath with the ten!

As I mentioned earlier, I had been moderately successful playing until the "pendulum swung.

The result of my sample, that the dealers had 770 tens or aces out of 1820 hands played, was a statistically significant indication of some sort of legerde- main.

However, you are justified in being reluctant to accept this conclusion since the objectivity of the experimenter can be called into question; I produced evidence to explain my own long losing streak as being the result of foul play, rather than my own incompetence.

An investigator for the Nevada Gaming Commission ad- mitted point blank at the 1975 U.

I find little solace in this view that Nevada's country bumpkins are less trustworthy but more dextrous than their big city cousins.

I am also left wondering about the responsibility of the Gaming Commission since, if they knew the allegation was true why didn't they close the places, and if they didn't, why would their representative have made such a statement?

One of the overlooked motivations for a dealer to cheat is not financial at all, but psychological.

The dealer is compelled by the rules to function like an automaton and may be inclined, either out of resentment toward someone the card counter do- ing something of which he's incapable or out of just plain boredom, to substitute his own determination for that of fate.

Indeed, I often suspect that many dealers who can't cheat like to suggest they're in control of the game by cultivation of what they imagine are the mannerisms of a card-sharp.

The best cheats, I assume, have no mannerisms.

Are Card Counters Cheating?

Credit for one of the greatest brain washing achievements must go to the casino industry for promulgation of the notion 6 that card counting itself is a form of cheating.

Not just casino employees, but many members of the public, too, will say: "tsk, tsk, you're not supposed to keep track of the cards", as if there were some sort of moral injunction to wear blinders when entering a casino.

Robbins Cahill, director of the Nevada Resort Association, was quoted in the Las Vegas Review of August 4,1976 link say- ing that most casinos "don't really like the card counters because they're changing the natural odds of the game.

Card counters are no more changing the odds than a sunbather alters the weather by staying inside on rainy days!

And what are these "natural odds"?

Is not this, too, as "unnatural" an act as standing on 4,4,4,4 against the dealer's ten after you've seen another player draw four fives?

Somehow the casinos would have us believe the former is acceptable but the latter is ethically suspect.

It's certainly understandable that casinos do not welcome people who can beat them at their own game; particularly, I think, they do not relish the reversal of roles which takes place where they become the sucker, the chump, while the card counter becomes the casino, grinding them down.

The paradox is that they make their living encouraging people to believe in systems, in luck, cultivating the notion that some people are better gamblers than others, that there is a savvy, macho per- sonality that can force dame fortune to obey his will.

How much more sporting is the attitude of our friends to the North!

Consider the following official policy statement of the Province of Alberta's Gaming Control Section of the Department of the Attorney General: "Card counters who obtain an honest advantage over the house through a playing strategy do not break any law.

Gaming supervisors should ensure that no steps are taken to discourage any player simply because he is winning.

Books of a less technical nature I deliberately do not mention.

There are many of these, of varying degrees of merit, and one can often increase his general awareness of blackjack by skimming even a bad book on the subject, if only for the exercise in criticismit provides.

However, reference to any of themis unnecessary for my purposes and I will confine my bibliography to those which have been of value to me in developing and corroborating a mathematical theory of blackjack.

An Introduction to Multivariate Statistical Analysis, Wiley, 1958.

This is a classical reference for multivariate statistical methods, such as those used in Chapter Five.

BALDWIN, CANTEY, MAISEL, and McDERMOTT.

Joumal of the American Statistical Association, Vol.

It is remarkably accurate considering that the com- putations were made on desk calculators.

Much of their ter- minology survives to this day.

Playing Blackjack to Win, M.

Barrons and Company, 1957.

This whimsical, well written guide to the basic strategy also contains suggestions on how to vary strategy depending upon cards observed during play.

This may be the first public mention of the possibilities of card counting.

Unfortunately it is now out of print and a collector's item.

Braun presents the results of several million simulated hands as well as a meticulous explanation of many of his computing techniques.

Theory of Gambling and Statistical Logic.

New York: Academic Press, rev.

There is also a complete version of two different card counting strategies and extensive simulation results for the ten count.

What is here, and not found anywhere else, is the ex- tensive table of player expectations with each of the 550 initial two card situations in blackjack for single deck play.

There is a wealth of other gambling and probabilistic information, with a lengthy section on the problem of optimal wagering.

ERD jS and RENYI.

On the Central Limit Theorem for Samples from a Finite Population.

Conditions are given to justify asymptotic normality when sampling without It is difficult to read in this untranslated version, and even more difficult to find.

Probability Theory and Mathematical Statistics.

Optimum Strategy in Blackjack.

Clare- mont Economic Papers; Claremont, Calif.

This contains a useful algorithm for playing infinite deck blackjack.

Experimental Comparison of Blackjack Betting Systems.

Paper presented to the Fourth Conference on Gambling, Reno, 1978, sponsored by the University of Nevada.

People who distrust theory will have to believe the results of Gwynn's tremendous simulation study of basic strategy blackjack with bet variations, played on his efficient "table driven" computer program.

Algorithms for Computations of Blackjack Strategies, presented to the Second Conference on Gambling, sponsored by the University of Nevada, 1975.

This contains a good exposition of an infinite deck computing algorithm.

MANSON, BARR, and GOODNIGHT.

Optimum Zero Memory Strategy and Exact Probabilities for 4-Deck Black- jack.

The American Statistician 29 2 :84-88.

The authors, from North Carolina St.

University, present an in- triguing and efficient recursive method for finite deck black- jack calculations, as well as a table of four deck expectations, most of which are exact and can be used as a standard for checking other blackjack programs.

New York: Vintage Books, 1966.

If I were to recommend one book, and no other, on the subject, it would be this original and highly successful popularization of the opportunities presented by the game of casino blackj ack.

Optimal Gambling Systems for Favorable Games.

Review of the International Statistics Institute, Vol.

This contains a good discussion of the gambler's ruin problem, as well as an analysis of several casino games from this standpoint.

The Fundamental Theorem of Card Counting.

International J oumal of Game Theory, Vol.

This paper, presumably an outgrowth of the authors' work on baccarat, is important for its combinatorial demonstration that the spread, or variation, in player expectation for any fixed strategy, played against a diminishing and unshuffled pack of cards, must increase.

The Casino Gambler's Guide.

This is an exceptionally readable more info which lives up to its title.

Wilson's blackjack coverage is excellent.

In addition, any elementary statistics text may prove helpful for understanding the probability, normal curve, and regression theory which is appealed to.

I make no particular recommendations among them.

When I had an ace and jack I heard him say again, "If you draw another card It will not be a ten; You'll wish you hadn't doubled And doubtless you will rue.

Shameless Plagiarism of A.

Housman Unless otherwise specified, all subsequent references will be to single deck blackjack as dealt on the Las Vegas Strip: dealer stands on soft 17, player may double on any two initial cards, but not after splitting pairs.

Furthermore, although it is contrary to almost all casino practices, it will be assumed, when necessary to illustrate general principles of probability, that all 52 cards will be dealt before reshuffling.

The first questions to occur to a mathematician when fac- ing a game of blackjack are: 1 How should I play to maximize my expectation?

The answer to the first determines the answer to the second, and the answer to the second determines whether the mathematician is interested in playing.

It is even conceivable, if not probable, that nobody, experts included, knows precisely what the basic strategy is, if we pursue the definition to include instructions on how to play the second and subsequent cards of a split depending on what cards were used on the earlier parts.

For ex- ample, suppose we split eights against the dealer's ten, busting the first hand 8,7,7 and reaching 8,2,2,2 on the second.

Quickly now, do we hit or do we stand with the 141 You will be able to find answers to such questions after you have mastered Chapter Six.

The basic strategy, then, constitutes a complete set of decision rules covering all possible choices the player may en- counter, but without any reference to any other players' cards or any cards used on a previous round before the deck is reshuf- fled.

These choices are: to split or not to split, to double down or not to double down, and to stand or to draw another card.

Some of them seem self evident, such as always drawing another card to a total of six, never drawing to twenty, and not splitting a pair of fives.

But what procedure must be used to assess the correct action in more marginal cases?

While relativelyamong the simplest borderline choices to analyze, we will see that precise resolution of the matter requires an extraordinary amount of arithmetic.

If we stand on our 16, we will win or lose solely on the basis of whether the dealer busts; there will be no tie.

In Chapter Eleven there will be found just such a program.

Once the deed has been done we find the dealer's exact chance of busting is.

This has been the easy part; analysis of what happens when we draw a card will be more than fivefold more time con- suming.

This is because, for each of the five distinguishably dif- ferent cards we can draw without busting A,2,3,4,5the dealer's probabilities of making various totals, and not just of busting, must be determined separately.

For instance, if we draw a two we have 18 and presumably would stand with it.

We must go back to our dealer probability routine and play out the dealer's hand again, only now from a 48 card residue our deuce is unavailable to the dealer rather than the 49 card remainder used previously.

Once this has been done we're interested not just in the dealer's chance of busting, but also specifically in how often he comes up with 17,18,19,20,and 21.

The result is found in the third line of the next table.

Hence our "condi- tional expectation" is.

Some readers may be surprised that a total of 18 is overall a losing hand here.

Note also that the dealer's chance of busting increased slightly, but not significantly, when he couldn't use "our" deuce.

Similarly we find all other conditional expectations.

Since a loss of 48 cents by drawing is preferable to one of 54 cents from standing, basic strategy is to draw to T,6 v 9.

Note that it was assumed that we would not draw a card to T,6,AT,6,2etc.

This decision would rest on a previous and similar demonstration that it was not in our interest to do so.

All this is very tedious and time consuming, but necessary if the exact player expectation is sought.

This, of course, is what com- puters were deSigned for; limitations on the human life span and supply of paper preclude an individual doing the calcula- tions by hand.

Doubling Down So much for the choice of whether to hit or stand in a par- ticular situation, but how about the decision on whether to double down or not?

In some cases the decision will be obvious- ly indicated by our previous calculations, as in the following example.

Suppose we have A,6 v dealer 5.

Any two card total of hard 10 or 11 would illustrate the situation equally well against the dealer's up card of 5.

We know three things: 1.

We want to draw another card, it having already been determined that drawing is preferable to standing with soft 17.

We won't want a subsequent card no matter what we draw for instance, drawing to.

A,6,5 would be about 7% worse than standing.

Our overall expectation from drawing one card is positive-that is, we have the advan- tage.

Hence the decision is clear; by doubling down we make twice as much money as by conducting an undoubled draw.

The situation is not quite so obvious when contemplating a double of 8,2 v 7.

Conditions 1 and 3 above still hold, but if we receive a 2,3,4,5, or click here in our draw we would like to draw another card, which is not permitted if we have selected the double down option.

Therefore, we must compare the amount we lose by forfeiting the right to draw another card with the amount gained by doubling our bet on the one card draw.

It turns out we give up about 6% by not drawing a card to our 15 subsequently developed stiff hands, but the advantage on our extra, doubled, dollar is 21%.

Since our decision to double r a i s ~ s expectation it becomes part of the basic strategy.

The Baldwin group pointed out in their original paper that most existing recommendations at the time hardly suggested doubling at all.

Probably the major psychological reason for such a conservative attitude is the sense of loss of control of the hand, since another card cannot be requested.

Doubling on small soft totals, like A,2heightens this feeling, because one could often make a second draw to the hand with no risk of busting whatsoever.

But enduring this sense of helplessness, like taking a whiff of ether before necessary surgery, is sometimes the preferable choice.

Pair Splitting Due to their infrequency of occurrence, decisions about pair splitting are less important, but unfortunately much more complicated to resolve.

Imagine we have 7,7 blackjack tips te winnen />The principal ques'tion facing us is whether playing one fourteen is better than playing two, or more, sevens in what is likely to be a losing situation.

Determination of the exact splitting expectation requires a tortuous path.

First, the exact probabilities of ending up with two, three, and four sevens would be calculated.

Then the player's expectation starting a hand with a seven in each of the three cases would be determined by the foregoing methods.

The overall expectation would result from adding the product of the probabilities of splitting a particular number of cards and the associated expectations.

The details are better reserved for Chapter Eleven, where a computer procedure for pair splitting is outlined.

This, of course, is for the set of rules and single deck we assumed.

It's not inconceivable that this highly complex game is closer to the mathematician's ideal of "a fair game" one which has zero ex- pectation for both competitors than the usually hypothesized coin toss, since real coins are flawed and might create a greater bias than the fourth decimal of the blackjack expectation, whatever it may be.

Condensed Form of the Basic Strategy By definition, the description of the basic strategy is "composition" dependent rather than "total" dependent in that some card combinations which have the same total, but unlike compositions, require a different action to optimize ex- pectation.

This is illustrated by considering two distinct three card 16's to be played against the dealer's Ten as up card: with 7,5,4 the player is 4.

Notwithstanding these many "composition" dependent exceptions which tax the memory and can be ignored at a total cost to the player of at most.

Hit stiff totals 12 to 16 against high cards 7,8,9,T,Abut stand with them against small cards 2,3,4,5,6except hit 12 against a 2 or 3.

Always draw to 17 and stand with 18, ex- cept hit 18 against 9 or T.

Never split 4,45,5or T,Tbut always split 8,8 and A,A.

Split 9,9 against 2 through 9, except not against a 7.

Split the others against 2 through 7, except hit 6,6 v 7, 2,2 and 3,3 v 2, and 3,3 v 3.

Hard Doubling: Always double 11.

Double 10 against all cards except T or A.

Double 9 against 2 through 6.

Double 8 against 5 and 6.

Soft Doubling: Double 13 through 18 against 4,5, and 6.

Double 17 against 2 and 3.

Double 18 against 3.

Double 19 against 6.

House Advantage If you ask a casino boss how the house derives its advan- tage in blackjack he will probably reply "The player has to draw first and if he busts, we win whether we do or not.

Being ignorant of our basic strategy, such an in- dividual's inclination might not unnaturally be to do what the Baldwin group aptly termed "Mimicking the Dealer"-that is hitting all his hands up to and including 16 without any discrimination of the dealer's up card.

This "mimic the dealer" strategy would give the house about a 5.

How can the basic strategist whittle this 5.

The following chart of departures from "mimic the dealer" is a helpful way to understand the nature of the basic strategy.

DEPARTURES FROM "MIMIC THE DEALER" Option Proper pair splitting Doubling down Hitting soft 17,18 Proper standing Gain.

Up Card 2 3 4 5 6 7 8 9 T A %Chance of Bust 35 38 40 43 42 26 24 -23 21 11 Note that most of the aggressive actions, like doubling and splitting, are taken when the dealer shows a small card, and these cards bust most often overall, about 40% of the time.

Incidentally, I feel the quickest way to determine if somebody is a bad player is to watch whether his initial eye contact is with his own, or the dealer's first card.

The really unskilled function as if the laws of probability had not yet been discovered and seem to make no distinction between a five and an ace as dealer up card.

The interested reader can profit by consulting several other sources about the mathematics of basic strategy.

Wilson has a lengthy section on how he approached the problem, as well as a unique and excellent historical commentary about the various attempts to assess the basic strategy and its expecta- tion.

The Baldwin group's paper is interesting in this light.

Manson et alii present an almost exact determination of 4 deck basic strategy, and it is from their paper that I became aware of the exact recursive algorithm they use.

They credit Julian Braun with helping them, and I'm sure some of my own procedures are belated germinations of seeds planted when I read various versions of his monograph.

Infinite deck algorithms were presented at the First and Second Gambling Conferences, respectively by Edward Gordon and David Heath.

These, of course, are totally recur- sive.

Their appeal stems ironically from the fact that it takes far less time to deal out all possible hands from an infinite deck of cards than it does from one of 52 or 2081 B.

The two card, "composition" dependent, exceptions are standing with 7,7 v T, standing with 8,4 and 7,5 v 3, hitting T,2 v 4 and 6, hitting T,3 v 2, and not doubling 6, 2 v 5 and 6.

The multiple card exceptions are too numerous to list, although most can be deduced from the tables in Chapter Six.

The decision not to double 6,2 v 5 must be the closest in basic strategy blackjack.

The undoubled expectation is.

The poor blackjack deck is being stripped naked of all her secrets.

This is easily proven by imagining all possible permutations of the deck and recogniz- ing that, for any first and second hand that can occur, there is an equiprobable reordering of the deck which merely inter- changes the two.

For example, it is just as likely that the player will lose the first hand 7,5 to 6,A and push the second 9,8 to 3,4,T as that he pushes the first one 9,8 to 3,4,T and loses the next 7,5 to 6,A.

If resplitting pairs were prohibited there would always be enough cards for four hands before reshuffling and that would guarantee an identical expectation for basic strategy play on all four hands.

Unfortunately, with multiple splitting permit- ted, there is an extraordinarily improbable scenario which ex- hausts the deck before finishing the third hand and denies us the luxury of asserting the third hand will have precisely the same expectation as the first two: on the first hand split 6, 5, T6, 5, T6, 5, Tand 6, 5, T versus dealer 2, 4, T, T ; second hand, split 3, 9, 4, T3, 9, 4, T3, 9, 4, Tand 3, A, T, 7 against dealer 7, 9, T ; finally, develop 8, 7, T8, 7, T8, 2, 2, A, A, A, Tand unfinished 8, 2, 1 in the face of dealer's T, T.

Gwynn's simulation study showed no statistically signifi- cant difference in basic strategy expectation among the first seven hands dealt from a' full pack and only three times in 8,000,000 decks was he unable to finish four hands using 38 cards.

Thus, as a matter of practicality, we may assume the first several hands have the same basic strategy expectation.

This realization leads us to consider what Thorp and Walden termed the "spectrum of opportunity" in their paper The Fundamental Theorem of Card Counting wherein they proved that the variations in player expectation for a fixed strategy must become increasingly spread out as the deck is depleted.

Notice there are no pair splits possible and the 38 total pips available guarantee that all hands can be resolved without reshuffling.

The basic strategist, while perhaps unaware of this composi- tion, will have an expectation of 6.

Player Dealer Player Dealer Hand Up Card Expectation Hand Up Card Expectation -- 5,6 8 +2 6,9 5 +1 9 +1 8 0 T +1 T 0 5,8 6 +1 final, blackjack hole card rules not 21 sympathise 5 +1 9 .

This exploitation of decks favorable for basic strategy will henceforth be referred to as gain from "bet variation.

Strategy Variation Another potential source of profit is the recognition of when to deviate from the basic strategy.

Keep in mind that, by definition, basic strategy is optimal for the full deck, but not necessarily for the many subdecks like the previous five card example encountered before reshuffling.

Basic strategy dictates hitting 5,8 v.

If we survive our hit we only get a push, while a successful stand wins.

Similarly it's better to see more with 5,8 v T, 6,8 v 9, and 6,8 v T, for the same reason.

In each of the four cases we are 50% better off to violate the basic strategy, and if we had been aware of this we could have raised our basic strategy edge of 6.

This extra gain occasionally available from appropriate departure from basic strategy, in response to fluc- tuations in deck composition which occur before reshuffling, will be attributed to "variations in strategy.

Some of the 23 departures from basic strategy are eye opening indeed and il- lustrate the wild fluctuations associated with extremely de- pleted decks.

Generally, variations in strategycan mitigate the disadvantage for compositions unfavorable for basic strategy, or make more profitable an already rich deck.

This is a seldom encountered case in that variation in strategy swings the pen- dulum from unfavorable to favorable.

Since these examples are exceedingly rare, the presumption that the only decks worth raising our bet on are those already favorable for basic strategy, although not entirely true, will be useful to maintain.

Insurance is "linear" A simple illustration of how quickly the variations can arise is the insurance bet.

Insurance is interesting for another it is the one situation in blackjack which is truly "linear," being resolved by just one card the dealer's hole card rather than by a com- plex interaction of possibly several cards whose order of appearance could be vital.

From the standpoint of settling the insurance bet, we might as well imagine that the value -1 has been painted on 35 cards in the deck and +2 daubed on the other 16 of them.

The player's insurance expectation for any subdeck is then just the sum of these "payoffs" divided by the number of cards left.

This leads to an extraordinarily simple mathematical solution to any questions about how much money can be made from the insurance bet if every player in Nevada made perfect insurance bets it might cost the casinos about 40 million dollars a yearbut unfortunately other manifestations of the spectrum of opportunity are not so uncomplicatedly linear.

Our problem is to select these 52 numbers which will replace, for our immediate pur- poses, the original denominations of the cards so that the average value of the remaining payoffs will be very nearly equal to the true basic strategy expectation for any particular subset.

Using a traditional mathematical measurement of the ac- curacy of our approximation called the "method of least squares," it can be shown that the appropriate are, as intuition would suggest, the same for all cards of the same denomination: Best Linear Estimates of Deck Favorability in % A 2 3 4 5 6 7 8 9 T 31.

To assert that these are "best" estimates under the criterion of least squares means that, although another choice might work better in oc- casional situations, this selection is guaranteed to minimize the overall average squared discrepancy between the true expectation and our estimate of it.

We add the six https://allo-hebergeur.com/blackjack/blackjack-holdings.html corresponding to these cards -19.

It is the ensemble of squared differences between numbers like -6.

The estimate is not astoundingly good in this small subset case, but accuracy is much betterfor larger subsets, necessarily becoming perfect for 51 card decks.

Approximating Strategy Variation The player's many different possible variations in strategy can be thought of as many embedded subgames, and they too are just click for source to this sort of linearizing.

Precisely which choices of strategy may confront the player will not be known, of course, until the hand is dealt, and this is in contrast to the bet- ting decision which is made before every hand.

Consider the player who holds a total of 16 when the dealer shows a ten.

The exact cards the player's total comprises are important only as they reveal information about the remaining cards in the deck, so suppose temporarily that the player possesses a piece of paper on which is written his current total of 16, and that the game of "16 versus Ten" is played in can blackjack win you online a 51 card deck.

Computer calculations show that the player who draws a card to such an abstract total of 16 has an expectation of .

Suppose now that it is known that one five has been removed from the deck.

Faced with this reduced 50 card deck 26 the player's expectation by drawing is .

In this case, he should stand on 16; the effect of the removal of one five is a reduction of the original.

In similar fashion one can determine the effect of the removal of each type of card.

These effects are given below, where for con- venience of display we switch from decimals to per cent.

Effects of Removal on Favorability of hitting 16 v.

Now we construct a one card payoff game of the type already mentioned, where the player's payoff is given by E i is the effect, j ~ s t described, of the removal of the i th card.

Approximate determination of whether the blackjack player should hit or stand for a particular subset of the deck can be made by averaging these payoffs.

Their average value for any subset is our "best linear estimate" of how much in % would be gained or lost by hitting.

Similarly, any of the several hundred playing decisions can be approximated by assigning appropriate single card payoffs to the distinct denominations of the blackjack deck.

The distribution of favorability for changing violating basic strategy blackjack counters be studied further by using the well known nor- mal distribution of traditional statistics to determine how often the situations arise and how much can be gained when they do.

Derivation of this method is also reserved for the Appendix.

Number of Insurance Strategy Gain Betting Unseen Cards Gain no Insurance Gain 10.

This is consistent, of course, with Thorp and Walden's 'Fundamental Theorem'.

Two other important deter- miners of how much can be gained from individual strategy variations are also pinpointed by the formula.

Average Disadvantage for Violating Basic Strategy In general, the greater the loss from violating the basic strategy for a full deck, the less frequent will be the opportunity for a particular strategy change.

For example, failure to double down 11 v 3 would cost the player 29% with a full deck, while hitting a total of 13 against the same card would carry only a 4% penalty.

Hence, the latter change in strategy can be ex- pected to arise much more quickly than the former, sometimes as soon as the second round of play.

Volatility Some not blackjack hit split variant are quite unfavorable for a full deck, but never- theless possess a great "volatility" which will overcome the 28 previous factor.

Consider the effects of removal on, and full deck gain from, hitting 14 against a four and also against a nine: Effects of Removal for Hitting 14 Full Deck Gain by Hitting A 2 3 4 S 6 7 8 9 T -- - - - - - v.

This is because large effects of removal are characteristic of hitting stiff hands against small cards and hence these plays can become quite valuable deep in the deck despite being very unfavorable initially.

This is not true of the option of standing with stiffs against big cards, which plays tend to be associated with small effects.

In the first case an abundance of small cards favors both the player's hitting and the dealer's hand, doubly increasing the motivation to hit the stiff against a small card which the dealer is unlikely to break.

In the second case an abundance of high cards is unfavorable for the player's hitting, but is favorable for the dealer's hand; these contradic- tory effects tend to mute the gain achievable by standing with stiffs against big cards.

We can liken the full deck loss from violating basic strategy to the distance that has to be traveled before the threshold of strategy change is reached.

The effects of removal or more precisely their squares, as we shall learn are the forces which can produce the necessary motion.

The following table breaks down strategy variation into each separate component and was prepared by the normal ap- proximation methods.

This should roughly approximate dealing three quarters of the deck, shuffling up with 13 or fewer cards remaining before the start of a hand, but otherwise finishing a hand in progress.

Gain from the latter activity is perhaps unfairly recorded in the 12-17 rows.

Similarly the methodology incorrectly assesses situations where drawing only one card is dominated by a standing strategy, but drawing more than once is preferable to both.

An example of this could arise when the player has 13 against the dealer's ten and the remaining six cards consist of commit spongebob blackjack thank 4's and two tens.

The expectation by drawing only one card is .

The next higher step of approximation, an interactive model of blackjack, would pick this sort of thing up, but it's doubtful that the minuscule increase in accuracy would be balanced by the difficulty of developing and applying the theory.

Remember, the opportunities we have been discussing will be there whether we perceive them or not.

When we consider the problem of programming the human mind to play black- jack we must abandon the idea of determining instantaneous strategy by the exhaustive algorithm described in the earlier parts of the book.

The best we can reasonably expect is that the player be trained to react to the proportions of different denominations remaining in the deck.

Clearly, the information available to mortal card counters will be imperfect; how it can be best obtained and processed blackjack ti 84 plus actual play will be the sub- ject of our next chapter.

To build an approximation to what goes on in an arbitrary subset, let's assume a model in which the favorability of hit- ting 16 vs Ten is regarded as a linear function of the cards re- maining in the deck at any instant.

For specificity let there re- main exactly 20 cards in the deck.

Y is the vector of favorabilities associated with each subset of the full deck, X is a matrix each of whose rows con- tains 20 l's and 31 0' s, and the solution, 3, will provide us with our 51x1 vector of desired coefficients.

Run the computer day and night to determine the Y's.

Premultiply a x 51 matrix by its tran,spose.

Multiply the result of b.

Multiply X' by Y and finally e.

Solve the resultant system of 51 equations in 51 unknowns!

The normal equations for the {3j will be +.

Their average value in a given subset is the corresponding estimate of favorability for carrying out the basic strategy.

Other aspects of blackjack, such as the player's expecta- tion itself, or the drawing expectation, or the standing expecta- tion separately, could be similarly treated.

But, since basic strategy blackjack is so well understood it will minimize our error of approximation to use it as a base point, and only estimate the departures from it.

Uniqueness of this solution follows from the non- singularity of a matrix of the form ~ : : ~with a b b b a throughout the main diagonal and b ~ a in every non-diagonal position.

The proof is most easily given by induction.

Let D n,a,b be the determinant of such an n x n matrix.

This derivation has much the flavor of a typical regression problem, but in truth it is not quite of that genre.

Yi is the true conditional mean for a specified set of our regression variables Xij.

It would be wonderful indeed if Yi were truly the linear conditional mean hypothesized in regression theory, for then our estimation techniques would be perfect.

But here we ap- peal to the method of least squares not to estimate what is assumed to be linear, but rather to best approximate what is almost certainly not quite so.

This emphasizes that Yi is a fixed number we are try- ing to approximate as a linear function of the Xij' and not a particular observation of a random variable as It would be in most least squares fits.

Suppose 0 2 is the variance of the single card payoffs and Il is their full deck average value.

Assume IJ SO and that the card counter only changes strategy or bet when it is favorable to do so.

SO is equivalent to redefining the single card payoffs, if necessary, so they best estimate the favorability of altering the basic strategy.

The Central Limit Theorem appealed to appears in the ex- N ercises of Fisz.

N O,l 37 in distribution.

The applicability is easily verified suggest blackjack rules casino style know our case since the Xi are our "payoffs" and are all bounded.

The "ratio" of b-a to b + c corresponds to efficiency.

It is not really area we should compare here, but it aids understanding to view it that way.

The Einstein count of +1 for 3,4,5, and 6 and -1 for tens results in a correlation of.

The Dubner Hi-Lo extends the Einstein values by counting the 2 as + 1 arid the ace as -1, resulting in a correla- tion coefficient of.

Another system, mentioned in Beat the Dealer by Thorp, extends Dubner's count by counting the 7 as + 1 and the 9 as -1, and has a correlation of.

The assumption that evaluation of card counting systems in terms of their correlation coefficients for the 70 mentioned variations in strategy will be as successful as for the insurance bet is open to question.

The insurance bet is, after all, a truly linear game, while the other variations in strategy involve more complex relations between several cards; these interac- tions are necessarily neglected by the bivariate normal methodology.

There is one interesting comparison which can be made.

Epstein reports a simulation of seven million hands where variations in strategy were conducted by using the Ten-Count.

An average expectation of 1.

In today's casino conditions the deck will rarely be dealt this deeply, and half the previous figures would be more realistic.

It might also be mentioned that correlation is undisturbed by the sampling without replacement.

To prove this, let Xi be the payoff associated with the ith card in the deck and Yi be the point value associated with the ith card in the deck by 52 some card counting system.

We n n seek the correlation of ~ X and L:Y for n ~ d subsets.

They then recommend keeping a separate, or "side," count of the aces in order to adjust their primary count for betting purposes.

Let's take a look at how this is done and what the likely effect will be.

Consider the Hi Opt I, or Einstein, count, which has a bet- ting correlation of.

A 23456789 T HiOpt I 0 0 1 1 1 1 0 0 0-1 Betting Effect .

It therefore seems reasonable to regard an excess ace in the deck as meriting a temporary readjustment of the running count for betting purposes only by plus one point.

Similarly, a defi- cient ace should produce a deduction temporary, again of one point.

Should we regard the deck as favorable?

Well, we're shy two aces since the expected distribution is three in 39 cards; therefore we deduct two points to give ourselves a temporary running COtlnt of -1 and regard the deck as probably disadvantageous.

In like fashion, with a count of -1 but all four aces remaining in the last 26 cards we would presume an advantage on the basis of a +1 ad- justed running count.

It can be shown by the mathematics in the appendix that the net effect of this sort of activity will be to increase the system's betting correlation from.

Among these are knowing when to stand with 15 and 16 against a dealer 7 or 8 and knowing when to stand with 12, 13, and 14 against a dealer 9, Ten, or Ace.

Before presenting a method to improve single parameter card counting systems it is useful to look at a quantification of the relative importance of the separate denominations of nontens in the deck.

This quantification can be achieved by calculating the playing efficiency of a card count which assigns one point to each card except the denomination considered, which counts as -12.

The fix- ed sign of the point value obscures this and can only be over- come by assigning the value zero and keeping a separate track of the density of these zero valued cards for reference in ap- propriate situations.

The average effect of removal for the eight cards recogniz- ed by the Einstein count is about 1% and this suggests that, if the deck is one seven short, that should be worth four Einstein points.

The mathematically correct index for standing with 14 against a ten is an average point value above +.

Suppose, however, that there was only one seven left in the deck.

It will save a lot of arguments to keep in mind that a change in strategy can be considered correct from three different perspectives which don't always coincide: it can be mathematically correct with respect to the actual deck com- position confronted; it can be correct according to the deck composition a card counter's parameter entitles him to presume; and it can be correct depending on what actually hap- pens at the table.

I've seen many poor players insure a pair of tens when the dealer had a blackjack, but I've seen two and a half times more insure when the dealer didn't!

Incorporation of the density of sevens raises our system's correlation from.

We've already seen the importance of the seven for playing 14 v.

Ten in conjunction with the.

The further simplification, "stand if there are no sevens," is almost as effective, being equivalent to the previous rule if less than half the deck remains.

For playing 16 v.

Ten the remarkably elementary direction "stand when there are more sixes than fives remaining, hit otherwise," is more than 60% efficient.

We will see in Chapter Eleven that it consistently out-performs both the Ten Count and Hi Opt I.

Of course, these are highly specialized instruc- tions, without broader applicability, and we should be in no haste to abandon our conventional methods in their favor.

The ability to keep separate densities of aces, sevens, eights, and nines as well as the Einstein point count itself is not beyond a motivated and disciplined intellect.

The memorization of strategy tables for the basic Einstein system as well as proper point values for the separated denominations in different strategic situations should be no problem for an in- dividual who is so inclined.

The increases in playing efficiency and betting correlation are exhibited below.

INCORPORATION OF ZERO VALUED CARDS INTO EINSTEIN SYSTEM Basic System A Cards Incorporated A,7 A,7,8 A,7,8,9 A,7,8,9,2 Playing Efficiency.

It is of little consequence strategically except for doubling down totals of eleven, particularly against a 7, 8, or 9, and totals of ten against a Ten or Ace.

Actually the compleat card counting fanatic who aspires to count separately five zero valued denominations is better off using the Gordon system which differs from Einstein's by counting the 2 rather than the 6.

Although poorer initially than Einstein's system, it provides a better springboard for this level of ambition.

The Gordon count, fortified with a pro- per valuation of aces, sixes, sevens, eights, and nines, scores.

This may reasonably be supposed to define a possibly realizable upper bound to the ultimate capability of a human being playing an honest game of blackjack from a single deck.

The Effect of Grouping Cards All of the previous discussion has been under the assump- tion that a separate track of each of the zero-valued cards is kept.

David Heath suggested sometime ago a scheme of block- ing the cards into three groups {2,a,4,5}, {6,7,8,9}, and {lO,J, Q,K}.

Using two measures, the differences between the first two groups and the tens, he then created a two dimensional strategy change graphic resembling somewhat a guitar finger- ing chart.

Heath's system is equivalent to fortifying a primary Gor- don count with information provided by the block of "middle" cards, there being no discrimination among these individually.

As we can see from the following table of ef- ficiencies for various blocks of cards properly used in support of the Gordon and Einstein systems, it would have been better to cut down on the number of cards in the blocked group.

IC,D,E Primary Count Auxiliary Grouping Playing Efficiency Gordon { 6,7,8,9 }.

Each of them was analyzed by a computer to determine if basic strategy should have been changed and, if so, how much expectation could have been gained by such appropriate departure.

I, myself, made decisions as to whether I wo'old have altered the conventional basic strategy, using my own version of the system accorded an efficiency of.

learn more here following table displays how much expectation per hand I and the computer gained by our strategy changes.

My gain in% appears first, followed by the computer's, the results for which are always at least as good as mine since it was the ultimate ar- biter as to which decisions were correct and by how much.

Unseen Cards Insurance Gain Non-insurance Gain 8-12.

The discrepancy between this and the theoretical.

go here table should be compared with the one on page 28.

The most bizarre change was a double on hard 13 v 6; with three eights, two sixes, sevens, and tens, and one ace, two, three, and four, doubling was 61%better than standing, 18% better than mere- ly drawing.

An indicator count, -12 1 1 1 1 1 1 1 1 1monitors the presence of aces in the deck and will be uncorrelated with the primary one if zero is the assigned point value.

This is because the numerator of the correlation, the inner product between the primary and the ace indicator count, will be zero, merely being the sum of the point values of the primary system assumed to be balanced.

To the degree of validity of the bivariate normal approximation zero correlation is equivalent to independence.

Hence we are justified in taking the square root of the sum of squares of the original systems' correlations as the multiple correlation coefficient.

For the situation discussed, we find the ace indicator count has a.

The seven indicator has a.

The "Six-Five" system for playing 16 v.

T has a correlation of.

We can use the theory of multiple correlation to derive a formula for the appropriate number of points to assign to a block of k zerQ-valued cards when using them to support a primary count system.

However, since the assumption of linearity underlies this theory as well as the artifice of the single card payoffs, the demonstration can be more easily 62 given from the latter vantage point, using only elementary algebra.

We still have 52 cards, but the point count of the deck 13 is L y.

Hence 52 - k 52 - k 63 the removal and replacement of one "blocked" card by a typical unblocked one has altered the full deck total of the payoffs by k k 1 ~ +2!.

It is unrealistic to suppose that such auxiliary point values would be remembered more precisely than to the nearest whole number.

Similar- lyforl6,7,SJ we would use 3 3 3 3 3-10-10-10 33 andfor ~ 6,7} 222 2 2-11-1122 2.

These also will be independent 64 of a primary count which assigns value zero to them, and hence the square root of the sum of the squares of the correlations can be used to find multiple correlation coefficients.

In fact, the original Heath count recommended keeping two counts, what we now call the Gordon 0 1 1 1 1 0 0 0 0 -1 and a "middle against tens" count 0 0 0 0 0 11 1 1 -1.

These are dependent, having correlation.

There is a subtle difference in the informatIon available from the two approaches which justifies the difference.

Factoring in information from cards already included in, and hence dependent upon, the primary count is usually very difficult to do, and probably not worthwhile.

One case where it works out nicely, however, is in adjusting the Hi Opt I count by the difference of sixes and fives, for playing 16 v Ten.

Both these denominations are included in the primary count, but since it's their difference we are going to be using, our aux- iliary count can be taken as 0 0 001 -1 0000 which is uncor- related with, and effectively independent of, the primary count.

The Chapter Eleven simulations contain data on how well this works out.

Even though it is usually too cumbersome in practice to use multiple correlation with dependent counts, an example will establish the striking accuracy of the method.

It will also illustrate the precise method of determining the expected deck composition subject to certain card counting information.

Let our problem be the following: there are 28 cards left in the deck and a Ten Counter and Hi Lo player pool information.

How many aces should we presume are left in the deck?

The Ten Count sug- gests more than normal, the Hi Rules exposure blackjack double indicates slightly less than usual.

We can look at this as a multiple regression problem.

Let Xl be the indicator count for aces -12 111111111 ; X2 the Ten Count 4 4 444 4 4 4 4 -9 ; and X3 the HiLo -111111 o0 0 -1.

Hence p 12' the correlation between X1.

VI 0 468 1.

The exact distribution can be found by combinatorial analysis for the 21 cards we are uncertain about.

I had imagined two aces, ten small cards, and nine middle cards would--be represent- ative, but we see the precise average figures are 2.

The only consolation I have is that it was the multivariate methodology which tipped me off to my foolishness.

At no time during the test was any attention paid to whether, in the actual play of the cards, the hand was won or lost.

Had the results been scored on that basis, the statistical variation in a sample of this size would have rendered them almost meaningless.

The estimate, that perfect play gains 3.

I wanted to astonish the spectators by taking senseless chances.

The player's exact gain at any deck level is catalogued completely for a single deck and extensively for two and four decks.

If the remaining number of cards is a multiple of three, add one to it before consulting the charts.

For example, with 36 cards left, the single deck gainis the same as with 37, namely.

There would be 89 unseen cards at a double deck, and full table, first round insurance is worth only.

You can also use these tables to get a reasonable estimate for the total profit available from all variations in strategy, not just insurance.

Multiply the insurance gain at the number of unplayed cards you're interested in by seven and that should be reasonably close.

The figures are in %.

EFFECTS OF REMOVAL A 2 3 4 5 - -- 6 789 Sum of.!.

Mean Squares Insur- ance 1.

Nevertheless, finding the Hi-Lo system's -111111 000 -1 insurance correlation will provide a helpful review.

Suppose we see a Reno dealer burn a 2 and a 7.

What is our approximate expec- tation?

If we want to know the effect of removing one card from the deck we merely read it directly from the table.

To practice this, let's find the insurance expectation when the dealer's ace and three other non-tens are removed from the deck.

We adjust the full deck mean of -7.

The full deck expectations for basic strategy are different, however, and this is discussed in Chapter 8.

Very lengthy tables are necessary for a detailed analysis of variations in strategy, and a set as complete as any but the an- tiquarian could desire will follow.

In order to condense the printing, the labeling will be abbreviated and uniform through- out the next several pages.

Each row will present the ten ef- fects of removal for the cards Ace through Ten, full deck favor- ability, m, and sum of squares of effects of removal, ss, for the particular strategy variation considered.

For hard totals of 17 down to 12 we are charting the favorability of drawing over standing, that is, how much bet- ter off we are to draw to the total than to stand with it.

Naturally this will have a negative mean in the eleventh col- umn in blackjack ti 84 plus cases, since standing is often the better strategy for the full deck.

Again, in many cases the average favorability for the full deck will be negative, in- dicating the play is probably not basic strategy.

Similarly we present figures for soft doubles, descending from A,9 to A,2showing how much better blackjack money online is than conventional draw- ing strategy.

Finally, the advantage of pair splitting over not splitting will be catalogued.

Not all dealer up cards will have the same set of strategic variations presented, since in many situations like doubling small totals and soft hands v 9,T, or A and split- ting fives there is no practical interest in the matter.

The tables will be arranged by the different dealer up cards and there will be a separate section for the six and ace when the dealer hits soft 17.

There is no appreciable difference in the Charts for 2,3,4, and 5 up in this case.

It's important to remember that the entries in the tables are not expectations, but rather differences in expectation for two separate actions being contemplated.

Once the cards have been dealt the player's interest in his expectation is secondary to his fundamental concern about how to play the hand.

This is resolved by the difference in expectation for the contemplated alternatives.

As a specific example of how to read the table, the arrow on page 76 locates the rowcorresponding to hard 14 v Ten.

The entry in the 11th column, 6.

For the same reason doubling down is not very advan- tageous, even with a total of 11.

The 11th column entries for 1216 are all at least 6.

Because of the increase in busts and fewer 17's produced, standing and doubling both grow in attractiveness.

Note soft 18 is now a profitable hit.

Not only are they desirable cards for the player to draw, but their removal produces the greatest in- crease in the dealer's chance of busting.

The table also shows that soft 18 with no card higher than a 3 should not be hit.

Since 19 is easier to beat, the player is inclined to hit and double down more often than against a Ten.

A player who split three eights and drew 8,98,7.

Note that otherwise the 9 is almost always a more important high card than the Ten.

On the next page it will be seen to be the correct play when dealer hits soft 17, paradoxically even though.

Standing and soft doubling become more frequent activities.

As mentioned on the previous page, A, 8 is a basic strategy double down, regardless of the number of decks used.

The 11th column full deck advantage figures on pages 74-85 come from exact 52 card calculations, without the dealer's up card or any of the player's cards removed.

The ef- fects of removal first 10 columns are, for hitting totals of 82 DEALER 4 HITTING 17-12 -1.

However, for doubling and splitting removal effects the amount of com- puter time necessary to carry out the calculations exactly would have been excessive; in these situations the removal ef- 83 DEALER 3 HITTING 17-12 -1.

Use of these tables to carry out variations in strategy for the 5,000 hand experiment reported on page 61 resulted in an overall playing efficiency of 98.

The relatively few and inconsequential errors appear more at- tributable to blackjack's essential non-linearity, which is more pronounced deeper in the deck, than to any approximations in the table.

This is done exactly as it was for the insurance and betting effects previously.

Another use is to find some of the "composition" depend- ent departures from the simplified basic strategy defined in Chapter Two.

Should you hit or stand with 4,4,4,4 v 81 To the full deck favorability of 5.

In Chapter Two the question was asked whether one should hit 8,2,2,2 v T after having busted 8,7,7 on the first half of a pair split.

The table for hard 14 against a ten gives the following estimate for the advantage for hitting in this case 6.

Don't forget to remove the dealer's up card as well as the cards in the player's hand, since all of these tables assume a 52 card deck from which dealer's and player's cards have not yet been removed.

Also don't be surprised if you are unable to reproduce exactly the 2.

Quantifying the Spectrum of Opportunity at various Points in the Deck Before we will be able to quantify betting and strategy variations at different points in the deck we'll have to in- 8.

First the table itself.

Corresponding to values of avariable designated by z, which ranges from oto 2.

Unit Normal Linear Loss Integral z.

The following step by step procedure will be used in all such calculations.

Ignore the algebraic sign of m.

Look up in the UNLLI chart the number corresponding to z.

In our case this will be.

Multiply the number found in step 3 by b.

This is the conditional player gain in %assuming the dealer does have an ace showing.

If desired, adjust the figure found in step 4 to reflect the likelihood that the situation will arise.

Repeating the procedure, for two decks, we have ~ 95.

We would interpolate between.

If you're disappointed in the accuracy, there are ways of improving the approximation, principally by adjusting for the dealer's up card.

Removing the dealer's ace changes m, for the single deck, to -7.

Repeating the calculations, 1.

After revising m from .

The exact gain in this situation appears in Chapter Eleven and is 15.

One thing remains, and that is instruction on how to calculate a card counting system's gain, rather than the gain from perfect play.

To do this we must have a preliminary calculation of the correlation of the card counting system and the particular play examined.

Since we already found the cor- relation of the Hi Lo system for insurance to be.

After calculating b in the usual fashion we then 89 multiply it by the card counting system's correlation coeffi- cient and use the resultant product as a revised value of b in all subsequent calculations.

Thus the efficiency of the Hi Lo system, at the 40 card level, is.

The Normal Distribution of Probability The famous normal distribution itself can be used to answer many probabilistic questions with a high degree of ac- curacy.

The table on page 91 exhibits the probability that a "standard normal variable" will have a value between 0 and selected values of z used to designate such a variable from 0 to 3.

Chance of Being behind One type of question that can be answered with this table is "Suppose I have an average advantage of 2% on my big bets; What is the chance that I will be behind on big bets after making 2500 of them?

However, we can take advantage of the sym- metry of the normal curve and determine the area or probabili- ty corresponding to values of z greater than.

We do this by subtracting the tabulated value.

Distribution of a Point Count We can also use the normal distribution to indicate how often different counts will occur for a point count system, pro- viding that the number of cards left in the deck is specified.

The following procedure can be used.

Divide b into one half less than the count value you're interested in.

Divide b into one half more than the count value you're interested in.

The difference between the normal curve areas cor- responding to the two numbers calculated in steps 3 and 4 will be the probability that the particular count value will occur.

As an example, suppose we wish to know the probability that there will be a +3 Hi Opt II count when there are 13 cards left from a single deck.

The area corresponding to.

The precise probability can be found in Appendix A of Chapter Seven, and is.

How often is Strategy changed?

Although our only practical interest is in how much can be gained by varying basic strategy, we can also use the normal probability tables to estimate how often it should be done.

To do so is quite simple.

Then we subtract the area given in our normal curve probability charts corresponding to.

Precise calcula- tions show the answer to be 25%.

Similarly, we find the approximate probability of a favorable hit of hard 12 against the dealer's 6 with five cards left in the deck to be.

The exact probability is found in Chapter Eleven, and is.

To illustrate this, assume single deck play in Reno at a full table, so the player gets only one opportunity to raise his bet.

Following the steps on page 88, we have: 1.

From the UNLLI chart take.

When the player has a basic strategy advantage for the full deck, then this computational technique can be used to measure how much will be saved by each extra unit which is not bet in unfavorable situations.

In Chapter Eight we deduce that Atlantic City's six deck game with early surrender gave the basic strategist about a.

The strategy tables presented are not the very best we could come up with in a particular situation.

As mentioned in this chapter more accuracy can be obtained with the normal approximation if article source work with a 51 rather than a 52 card deck.

One could even have separate tables of effects for different two card player hands, such as T,6 v T.

Obviously a compromise must be reached, and my motivation has been in the direction of simplicity of exposition and ready applicability to multiple deck play.

You have the usual 16 against the dealer's ubiquitous Ten.

We consider three dif- ferent sets of remaining cards.

Unplayed Residue 4,T 4,4,T,T 4,4,4,T,T,T Favorability of Hitting over Standing -50% 0% +10% From this simple example follow two interesting conclu- sions: 1.

Strategic favorabilities depend not strictly on the pro- portion of different cards in the deck, but really on the absolute numbers.

Every card counting systemever created would misplay at least one of these situations because the value of the card counting parameter would be the same in each case.

The mathematical analysis of blackjack strategies is only in rare instances what might be called an "exact science.

In theory all ques- tions can be so addressed but in practice the required computer time is prohibitive.

We have already, to a reasonable degree, quantified the worth of different systems when played in the error free, tran- sistorized atmosphere of the computer, devoid of the drift of cigar smoke, effects of alcohol, and distracting blandishments of the cocktail waitress.

But what of these real battlefield con- ditions?

To err is human and neither the pit boss, the dealer, nor the cards are divine enough to forgive.

Two Types of Error There are two principal types of error in employing a count strategy: 1 an incorrect measure of the actual parameter which may be due to either an arithmetic error in keeping the running count or an inaccurate assessment of the number of cards remaining in the deck, and 2 an imprecise knowledge of the proper critical index for changing strategy.

It is beyond my scope to comment on the likelihood of numerical or mnemonic errors other than to suggest they probably occur far more often than people believe, particularly with the more complex point counts.

It strikes me as difficult, for instance, to treat a seven as 7 for evaluating my hand, but as +1 for altering my running count and calling a five 5 for the hand and +4 for the count.

The beauty of simple values like plus one, minus one, and zero is that they amount to mere recognition or non-recognition of cards, with counting for- ward or backwardrather than arithmetic to continuously monitor the deck.

Commercial systems employing so called "true counts" defined as the average number of points per card multiplied by 52 produce both types of error.

Published strategic indices themselves have usually been rounded to the nearest whole number, so a "true count" full deck parameter of 97 5 might have as much as a 10% error in it.

It is the view of the salesmen of such systems that these errors are not serious; it is my suggestion that they probably are.

An Exercise in Futility Even if the correct average number of points in the deck is available, there are theoretical problems in determining critical indices.

When I started to play I faithfully committed to memory all of the change of strategy parameters for the Hi Lo system.

It was not until some years later that I realized that several of them had been erroneously calculated.

For some time, I was firmly convinced that I should stand with 16 v 7 when the average number of points remaining equalled or exceeded.

I now know the proper index should be.

What do you think the consequences of such misinfor- mation would be in this situation?

Not only was I playing the hand worse than a basic strategist, but, with 20 cards left in the deck I would have lost three times as much, at the 30 card level twenty times as much, and at the 40 card level five hundred times as much as knowledge of the cor- rect parameter could have gained me.

The computer technique of altering normal decks so as to produce rich or lean mixtures for investigating different situa- tions has not always incorporated an accurate alteration of conditional probabilities corresponding to the extreme values of the parameter assumed.

The proper approach can be derived from bivariate normal assumptions and consists of maintain- ing the usual density for zero valued cards and displacing the other denominations in proportion to their assigned point values, rather than just their algebraic signs.

Computer averag- ing of all possible decks with this count leaves us with a not surprising "ideal" deck of twelve tens, one each three, four, five, and six, and two of everything else.

There is at present no completely satisfactory resolution of such quan- daries and even the most carefully computerized critical in- dices have an element of faith in them.

Behavior of Strategic Expectation as the Parameter changes The assumption that the favorability for a particular ac- tion is a linear function of the average number of points in the deck is applied to interpolate critical indices and is also a con- sequence of the bivariate normal model used to analyze effi- ciency in terms of correlation coefficients.

How valid is this assumption?

The answer varies, depending on the particular strategic situation considered.

Tables 1 and 2, which present favorabilities for doubling down over drawing with totals of 10 and 11 and hitting over standing for 12 through 16, were pre- pared by using infinite deck analyses of the Hi Opt I and Ten Count strategies.

Critical points interpolated from them should be quite accurate for multiple deck play and incor- porating the effect of removing the dealer's up card permits the adjustment of expectations and indices for a single deck.

The most marked non-linearities are found when the dealer has a 9 or T showing.

This is probably attributable to the fact that the dealer's chance of breaking such a card decreases very rapidly as the deck gets rich in tens.

Linearity when the dealer shows an ace dealer hits soft 17 is much better because player's and dealer's chance of busting grow apace.

To estimate how much conditional improvement the Hi Opt provides with 20 cards remaining in the deck multiply the Table 1 entries in the second through fifth columns by.

You will observe that many of 99 the albeit technically correct parameters players memorize are virtually worthless.

TABLE 1 STRATEGIC FAVORABILITIES IN% AS A FUNCTION OF HI OPT PARAMETER Hi Opt parameter quoted is average number of points in deck.

Assuming 20 cards left in the deck and that the player holds 14 against a https://allo-hebergeur.com/blackjack/blackjack-online-vs-friends.html, he will gain.

A superstitious player who only counts sevens and stands when all of them are gone will gain 1.

An Explanation of Errors Even if not always realized in practice, the linear assump- tion that the player's conditional gain or loss is a constant times the difference between the proper critical index and the current value of his parameter provides a valuable perspective to illustrate the likely consequences of card counting errors.

Whatever their source the type 1 and 2 errors mentioned earlierthe player will either be changing strategy too often, equivalent to believing the critical index is less extreme than it really is, or not changing strategy enough, equivalent to https://allo-hebergeur.com/blackjack/my-tournament-tickets-pokerstars.html the critical index is more extreme than it actually is.

The subject can perhaps be demystified by appeal to a graphic.

At a certain level of the deck the running count will tend to have a probability distribution like the one below, where the numbers inside the rectangles are the frequencies in % of the different count values.

Only the positive half of the distribution is shown.

This means that there will be neither gain nor loss from changing strategy for a running count of +2, but there will be a conditional loss at any count less than +2 and a conditional gain at any count greater than +2.

The much ban- died "assumption of linearity" means that the gain or loss will be precisely proportional to the distance of the actual running count from the critical count of +2.

Now suppose one was for whatever reason addicted to premature changing of strategy for counts of +1 or higher.

What we see, of course, is that counts closer to zero like +1 are much more likely to occur than the more extreme ones where most of the conditional profit lies.

To fix the idea in your mind try to show, using the diagram, that if the critical threshold value is +3, the player who changes strategy for +2 or above will lose more than the basic strategist who never changesand also will lose more than the perfect employer of the system can gain.

Indeed, the Baldwin group foresaw this in their book: "Ill considered changes will prob- ably do more harm than good.

Many players overemphasize the last few draws and, as a result, make drastic and costly changes in their strategy.

This suggests that it would be a service to both the memory and pocket book to round playing indices to the nearest conveniently remembered and more extreme value.

There is, as in poker, a tendency to "fall in love with one's cards"l which may cause pathologists to linger over unfavorable decks where much of this action is found for the sole purpose of celebrating their knowledge with a bizarre and eye-opening departure play.

This is an under- standable concomitant of the characteristic which best dif- ferentiates the casino blackjack player from the inde- pendent trials gambler, namely a desire to exercise control over his own destiny.

An Optimal Strategy for Pot Limit Poker.

The American Mathematical Monthly, Vol 82, No.

The instantaneous value of any point count system whether it uses +or - 0, 1, 2, 3, 4, 7, 11 etc.

It has already been shown in Chapter Five that cards assigned the value of zero are uncorrelated with the system's parameter and hence tend to have the same neutral distribution regardless of the sign or magnitude of the point count.

We shall now show that more generally, as the count fluctuates, we are entitled to presume a deflection in a card denomination's density proportional to the point value assign- ed to it.

Towards this end we again consider the +1, -12 indicator count for a particular denomination.

Our demonstration is concluded by observing that the deflection of the conditional mean of the indicator count from its overall mean will be proportional to this correlation, and hence proportional to Pk, as promised.

The deflections for negative counts with 39, 26, and 13 cards remaining can be obtained by merely changing the algebraic signs in the 13, 26, and 39 card positive count tables.

Observe that the Band E columns tend to be close in magnitude, but opposite in sign, the C column is generally close to zero, and the D column is about half of E.

This is what the ideal theory suggests will happen.

Table 4 was prepared by a probabilistic analysis of Hi Opt I parameters with 20, 30, and 40 cards left in a single deck.

The lessons to be learned from it would seem to apply to any count system.

Examined critical indices range from.

The body of the table quantifies the player's cumulative gain or loss from changing strategy with possible "action indices" as, or more, extreme than those which appear in the left hand margin.

The units are arbitrarily scaled to avoid decimals; they would actually depend on the volatility of, and point count's correlation with, the particular situation considered.

For relatively small critical indices such as.

However for larger critical indices the player may lose more from such over- zealousness than someone else playing the system correctly can gain.

For example, it would seem innocuous to mistake a critical index of.

This table can also be used to assess how well a "running count" strategy would fare relative to a strategy based on a "true" knowledge of the average click of points remainingin the deck.

Imagine that the situation with critical index.

If opportunity arises three times, with 20, 30 and 40 cards remaining, the total 112 TABLE 4 ACTION CRITICAL INDICES INDEX.

A "running count" player, making no effort to adjust for depth in the deck, would gain less than this, depending on the critical running count he used.

Furthermore, such numbers, already ingrained in the memory, would not be readily conver- tible for multiple deck play.

Just another two bucks down the tube.

Thus to estimate what expectation our rules would produce for a double deck we would pick .

Likewise we could extrapolate a.

To begin with, almost half of the.

The double down pair often contains two cards the player does not wish to draw and their removal significantly im- proves see more chance of a good hand from one deck but is negligi- ble otherwise.

A good example of ~ h i s is doubling nine against See page 170 for explanation of infinite deck.

Presumably the remaining discrepancy reflects the player's gain by judicious standing with stiff totals.

A stiff hand usually contains at least one card, and often several, which would help the dealer's up cards of two through six, against which this option is exercised, and the favorable effect of their removal i.

The Effect of Rule Changes In the next table the effect of some rule changes occasion- ally encountered is given for both one deck and an infinite number of decks.

The reader can use interpolation by the reciprocal of the number of decks to get an estimate of what the effects would be for two and four decks.

For instance, if doubling soft hands is forbidden in a four deck game, take one fourth of the difference between the .

Similarly, we get .

Notice how splitting is more valuable for the infinite deck due to the greater likelihood of pairs being dealt.

Doubling down after pair splitting is worth the same in each case because the reduced frequency of pairs in the single deck is nullified by the increased advantage on double downs.

Each row of the following table provides a comparison of the fluctuations in various numbers of decks by display of the number of remain- ing cards which would have the same degree of fluctuation associated.

If you're playing at that great blackjack table in the sky where St.

Peter deals and you know who is the pit bossyou'll have to wait an eternity, or until 2601 cards are left, before the degree of departure from normal composition is equivalent to that produced by the observation of the burn card from a standard pack of 52.

We see that the last few cards of a multiple deck can be slightly more favorable for both betting and playing variations than the corresponding residue from a single deck.

However, it must be kept in mind that such situations are averaged over the entire deck when assessing overall favorability.

An in- teresting consequence of this is that even if one had the time to count down an infinite deck, it would do no good since the slightly spicier situations at the end would still average out to zero.

When we recall that the basic multiple deck games are in- herently less advantageous, the necessity of a very wide bet- ting range must be recognized.

Absolute efficiencies of card counting systems will decrease mildly, perhaps by three per cent for four decks.

Since this decrease will generally be uniform over most aspects of the game, relative standings of different systems should not differ appreciably from those quoted in Chapter Four.

The next table shows how much profit accrues from betting one ex- tra unit in favorable situations for two and four deck games played according to the rules generally presumed in Chapter Two.

GAIN PER HAND FROM BETTING ONE EXTRA UNIT IN FAVORABLE SITUATIONS % Number of Cards Remaining Double Deck Four Decks 10 2.

If a four 119 deck player's last hand is dealt with 60 cards left, we average all the gains including the.

This is the average profit per hand in %.

Although we've neglected strategy variation this is partially compensated by the assumption that the player diagnoses his basic strategy advantage perfectly.

The rest of the chapter will be devoted to certain uncom- mon but interesting variations in rules.

Since these usually oc- cur in conjunction with four deck games, this https://allo-hebergeur.com/blackjack/casino-blackjack-scene.html be assumed unless otherwise specified.

No hole Card With "English rules" the dealer does not take a hole card, and in one version, the player who has doubled or split a pair loses the extra bet if the dealer has a blackjack.

In such a case the player minimizes his losses by foregoing eight splitting and doubling on 11 against the dealer's ten and ace and also not splitting aces against an ace.

The primary penalty paid is that the correct basic strategy is not used when the dealer doesn't have blackjack.

In another version, though, the player's built up 21 is allowed to push the dealer's natural; this favors the player by.

Surrender "Surrender" is another, more common, rule.

With this op- tion the player is allowed to give up half his bet without finishing the hand if he doesn't like his prospects.

Usually this choice must be made before drawing any cards.

Since the 120 critical expectation for surrendering is .

They will also be useful for discussion of subsequent rule variations.

PLAYER'S EXPECTATION'!

Thus surrendering 16 v T saves the player.

Naturally, the precise saving depends on what cards the player holds and on how many decks are used, but these tables are quite reliable for four deck play.

Some casinos even allow "early surrender", before the dealer has checked his hole card for a blackjack.

This is quite a picnic for the knowledgeable player, particularly against the dealer's ace.

When click to see more is done we get the following table of gain from proper strategy.

When surrender is allowed at any time, and not just on the first two cards, the rule will be worth almost twice as much for conventional surrender and either 10% or 50% more for early surrender depending on whether the dealer shows an ace or a ten.

Bonus for multicard Hands If the Plaza in downtown Las Vegas had had the "Six Card Automatic Winner" rule, I would have been spared the disap- pointment of losing with an eight card 20 to the dealer's three card 21.

Six card hands are not very frequent and the rule is worth about.

The expectation tables suggest a revised five card hitting strategy to cope with the rule in four decks: hit hard 17 v 9, T, and A; hit hard 16 and below v 2 and 3; hit hard 15 and below v 4,5, and 6.

Some Far Eastern casinos have a sort of reverse surrender rule called "Five Card," wherein the player may elect to turn in any five card hand for a payment to him of half his bet.

Again the table of expectations comes in handy, both for decisions on which five card hands to turn in and also for revision of four card hitting strategies.

A five card hand should blackjack ti 84 plus turned in if its expectation is less than +.

A revised and ab- breviated four card strategy is as follows: Hit Soft 19 and Below Against Anything But a 7 or 8 Hit Hard 15 and Below Against a 2 Hit Hard 14 and Below Against a 3 and 4 Hit Hard 13 and Below Against a 5 and 6 Other changes in strategy are to hit all soft 18's against an ace, three card soft 18 against an 8, and hit three card 12 versus a 4.

Obviously there will be many other composition depend- ent exceptions to the conventional basic strategy which are not revealed by the infinite deck approximation to four deck or single deck play.

So the reader feels he's getting his money's worth I will divulge the only four card hard 14 which should be hit against the dealer's five.

In many of the casinos where "Five Card" appears, it col- lides with some of the other rule variations we have already discussed, creating a hydra-headed monster whose expectation cannot be analyzed in a strictly additive fashion.

For instance, if we have already "early surrendered" 14 v dealer Ten, we can neither tie the dealer's natural 21 allowed in Macao nor turn it into a five card situation.

The five card rule is a big money maker, though, being worth about.

This is in Macao, where the-- player can "five card" his way out of some of the dealer's ten- up blackjacks.

The following table gives the frequency of development of five card hands in a four deck game, with the one deck frequen- cy in parentheses next to the four deck figure.

A hand like 3,3,3,3,4with repetition of a particular denomination, will be much less prob- able for a single deck, but A,2,3,4,5with no repetition, occurs more often in the single deck.

Hands with only one repetition, like 2,3,4,4,5 are almost equally likely in either case and tend to make up the bulk of the distribution anyway.

When a bonus is paid for 6,7,8 of the same suit or 7,7, 7different strategy changes are blackjack flowering time depending on how much it is.

We can use the infinite deck expectation table to ap- proximate how big a bonus is necessary for 6,7,8 of the same suit in order to induce us to hit the 8 and 6 of https://allo-hebergeur.com/blackjack/live-casino-blackjack-online.html against the dealer's two showing.

Suppose B is the bonus paid auto- matically if we get the 7 of hearts in our draw.

We must com- pare our hitting expectation of '" .

The equation B becomes .

Hence, with a 5 to 1 bonus we'd hit, but if it were only 4 to 1 we'd stand.

He gives a strategy for which a player expectation of 2.

Apparently some casino personnel have read Epstein's book, for, in October of 1979, Vegas World introduced "Double Exposure", patterned after zweikartenspiel except that the dealer hits soft 17 and the blackjack bonus has been discon- tinued, although the player's blackjack is an automatic winner even against a dealer natural.

The game is dealt from five decks and has an expectation of about .

In private correspondence about the origin of the game, Epstein "graciously cedes all claim of paternity to Braun.

The dealer stands on soft 17, double after split is permitted, but pairs may be split only once.

An analysis of the player's expectation for these rules will be useful for illustrating how to employ the information in this chapter.

To begin, we need an estimation of the six deck expec- tation for the typical rules generally presumed in this book.

In- terpolation by reciprocals click at this page that the player's expecta- tion will be one sixth of the way between.

The right to double after split is worth.

Early surrender itself provides a gain of.

Summariz- ing, we adjust the previous figure of .

This truly philanthropic state of affairs led to much agony for the New Jersey casino interests!

Not only did the knowledgeable player have an advantage for a complete pack of 312 cards, but it turns out that the early surrender rule results in greater fluctuations in the player's ad- vantage as the deck is depleted than those which occur in or- dinary blackjack.

An excess of aces and tens helps the player in the usual fashion when they are dealt to him, but the dealer's more frequent blackjacks are no longer so menacing in rich decks, since the player turns in many of his bad hands for the same constant half unit loss.

The effects of removing a single card of each denomination appear in the next table; even though Atlantic City games are all multiple deck the removals are from a single deck so com- parisons can be made with other similar tables and methods presented in gaming blackjack online book.

Unfortunately for the less flamboyant players who didn't get barred, a suit requiring casinos to allow card counters to play blackjack was ruled upon favorably by a New Jersey court.

This had as its predictable result the elimination of the surrender option and consequently what had been a favorable game for the player became an unfavorable one.

Under the new set of rules, in effect as of June 1981, the basic strategist's ex- pectation is .

For the correct six deck basic strategy see the end of Chapter Eleven.

The following chart of how much can be gained on each ex- tra unit bet on favorable decks may be of some use to our East Coast brethren for whom "it's the only game read article town.

At one time I believed that the frequency of initial two card hands might be responsible for the difference between in- finite and single deck expectations.

However, multiplication of Epstein's single deck expectations by infinite deck pro- babilities of occurrence disabused me of the notion.

One possible justification for the interpolation on the basis of the reciprocal of the number of decks can be obtained by looking at the difference between the infinite deck probability of drawing a second card and the finite deck probability.

The probability of drawing a card of different denomination from one already possessed is 4k for k decks and the 52k-l corresponding chance of getting a card of the same denomina- tion is 4k-l.

The differences between these figures and 52k-l the constant I~which applies to an infinite deck, are I and I 2 respectively.

These differences 13 52k-l 13 52k-l themselves are very nearly proportional to the reciprocal of the number of decks used.

In the last case we 129 find ourselves in the position of.

But because of the strain That it put on his brain, He chucked math and took up Divinity.

Different equations are necessary to evaluate early sur- render for different hands from one and four decks.

For in- stance, with T,2 v.

The player's loss of ties is greatly offset by aggressive splitting and doubling to exploit the dealer's visible stiff hands.

The magnitudes show Double Exposure to be far more volatile than ordinary blackjack.

There are surprisingly many two card, composition depen- dent, exceptions to the page 126 strategy: stand with A,7 v 8,3 and 7,6 and 8,5 v hard 11, except hit 8,5 v 9,2 ; double 7 v hard 13 other than T,3 ; hit T,6 v 6,2 and 9,7 v hard 7.

Practical casino conditions, however, make this impossible.

For one thing, a negative wager equiva- lent to betting on the house when they have the edge is not permitted.

When I first started playing, I religiously ranged my bets according to Epstein's criterion of survival.

Besides, when the truly degenerate gambler is wiped out of one bank he need only go back to honest work for a few months until he has another.

In my opinion the entire topic has probably been over- worked.

The major reason that such heavy stress has been placed on the problem of optimal betting is that it is one of the few which are easily amenable to solution by existing mathematics, rather than because of its practical importance.

The game resembles basic strategy blackjack with about 28 cards left in the deck, since for flat 9 blackjack natural it is an even game, but every extra unit bet in favorable situations will earn 1.

Now, both Greta and Opie know before each play which situation they will be confronting.

Opie bets optimally, in pro- portion to her advantage, 2 units with a 2% advantage and 6 units with the 6% edge, while Greta bets grossly, 4 units whenever the game is favorable.

Thereby they both achieve the same 3.

Starting with various bank sizes, their goals are to double their stakes without being ruined.

The results of 2000 simulated trials in each circumstance appear below.

NUMBER OF TIMES RUINED TRYING TO DOUBLE A BANK OF Opie Greta 20 877 896 50 668 733 100 438 541 200 135 231 Greta is obviously the more often ruined woman, but since they have the same expectation per play there must be a com- pensating factor.

This is, of course, time-whether double or nothing, Greta usually gets her result more quickly.

This il- lustrates the general truth pointed out by Thorp in his Favorable Games paper that optimal betting systems tend to be "timid", perhaps more so than a person who values her time would find acceptable.

You play every hand as if it's your last, and it might be, if you lose an insurance bet and split four eights in a losing cause!

Another common concern voiced by many players is whether to take more than one hand.

Again, practical con- siderations override mathematical theory since there may be no empty spots available near you.

A bit of rather amusing advice on this matter appeared in a book sold commercially a few years ago.

The author stated that "by taking two hands in a rich situation you reduce the dealer's probability of getting a natura!.

This brings to mind how so many, even well regarded, pundits of subjects such as gambling, sports, economics, etc.

Thus, we have the gambling guru who enjoins https://allo-hebergeur.com/blackjack/blackjack-tire-repair-napa.html to "bet big when you're winning," the sports announcer who feels compelled to attribute one team's scoring of several con- secutive baskets to the mysterious phantom "momentum," and the stock market analyst who cannot report a fall in price without conjuring up "selling pressure.

A trip to the dictionary confirms that this latter description is probably the most ac- curate in the book.

But to debunk mountebanks is to digress.

Nevertheless, there can be a certain reduction in fluctuations achievable by playing multiple spots.

Suppose we have our choice of playing from one to seven hands at a time, but with the restriction that we have the same amount of action every round every dealer hand.

Then the following table shows the relative fluctuation we could expect in our capital if we follow this pattern over the long haul.

Number of Hands 1 2 3 4 5 6 7 ---- -- Relative Fluctuation 1.

Assuming that we play each of our hands as fast as the dealer does his and ignoring shuffle time, then we can playa single spot on four rounds as often counting blackjack 2020 seven spots on one round.

Similarly three spots could be played twice in the same amount of time.

Now, with our revised criterion of equal total action per time on the clock, our table reads: Number of Hands Relative Fluctuation 1 2 3.

Of course, all this ignores the fact that taking more hands requires more cards and might trigger shuffle up on the dealer's part if he didn't think there were enough cards to complete the round.

Or, sometimes there would be enough cards to deal once to two spots but not twice to one spot.

It's been my observation that when this third round is dealt to five players it's almost always because the first two rounds used very few and predominantly high cards; hence the remainder of the deck is likely to be composed primarily of low ones.

A good exam- ple to illustrate the truncated distribution which results can be obtained by reverting to a simplistic, non-blackjack example.

Consider a deck of four cards, two red and two black.

As in Chapter Four, the dealer turns a card; the player wins if it's red and loses on black.

Ostensibly we have a fair game, but now imagine an oblivious, unsuspecting player and a card-counting, preferentially shuffling dealer.

Initially there are six equally likely orderings of the deck.

RRBB RBRB RBBR BBRR BRBR BRRB Since the dealer is trying to keep winning cards from the player, only the enclosed ones will be dealt.

The effect, we see, is the same as playing one hand from a deck of 14 cards, 9 of which are black.

As an exercise of the same type the reader might start with a five card deck, three red click to see more two black.

The answer depends on how often the deck is reevaluated; blackjack uses typically four to twenty-four cards per round, depending on the number of players.

join. blackjack west palm beach congratulate following chart shows the percentage of tens that would be dealt as a function of the size of the clump of cards the dealer observes before making his next decision on whether to reshuffle.

Percentage of Tens Played 31 30 29 28 27 26 o 13 26 39 52 Number of Cards in Clump Between Reevaluations of Deck Since about five or six cards are usually used against a single player we can conclude that the dealer could reduce the proportion of tens dealt to about 26.

This would give the basic strategist a 1.

By using a better correlated betting count to decide when to reshuffle, the house edge could probably be raised to 2%.

In all honesty, though, I think we must recognize that player card-counting is just the obverse of preferential shuffling-what's sauce for the goose is also for the gander.

While on the subject, it might be surprising that, occa- sionally, the number of times the dealer shuffles may influence 136 the player's expectation.

New decks all seem to be brought to the table with the same arrangement when spread: A23.

If the dealer performs a perfect shuffle of half the deck against the other half, then, of course, the resultant order is deterministic rather than random.

Is it a coincidence that one of the major northern Nevada casinos has a strict procedure calling for five shuffles of a new deck, but three thereafter?

Even experienced dealers would have some difficulty try- ing to perform five perfect shuffles a "magician" demonstrated the skill at the Second Annual Gambling Con- ference sponsored by the University of Nevadabut to get some idea of what might happen if this were attempted, I ask- ed a professional dealer from the Riverside in Reno to try it.

He sent me the resultant orderings for eight such attempts.

Although a result of this sort is not particularly significant in that it, or something worse, would occur about 7% of the time by chance alone, none of the eight decks favored the player.

Previous Result's Effect on next Hand Blackjack's uniqueness is the dependence of results before reshuffling takes place.

While the idea that a previous win or loss will influence the next outcome is manifest nonsense for independent trials gambles like roulette, dice, or keno, it is yet conceivable that in blackjack some way might be found to pro- fitably link the next bet to the result of the previous one.

Wilson discusses the intuition that if the player wins a hand, this is evidence that he has mildly depleted the deck somewhat of the card combinations which are associated with himwinning, and hence he should expect a poorer than average result next time.

My resolution to the question, when it was first broached to me, was to perform a Bayesian analysis 137 through the medium of the Dubner Hi Lo index.

This led to the tentative conclusion that the player's expectation would be reduced by perhaps.

Gwynn also found that a push on the previous hand is apparently a somewhat worse omen check this out the next one than a win is.

It follows, then, that the player's prospects must improve following a loss, although of course not much, certainly not enough to produce a worthwhile betting strategy.

When all is said and done, the most immediate determiner of the player's advantage is the actual deck composition he'll be facing, and knowledge of whether he won, lost, or pushed the last hand, in itself, really tells us very little about what cards were likely to have left the deck, and implicitly, which ones remain.

Epstein proposes minimizing the probability of ruin sub- ject to achieving an overall positive expectation.

Thus it is generally consistent with the famous Kelly criterion for maximizing the exponential rate of growth.

Another reasonable principle which leads to proportional wagering is that of minimizing the variance of our outcome subject to achieving a fixed expectation per play.

Suppose our game consists of a random collection of subgames indexed by i occurring with probability Pi and having corresponding ex- pectation E i.

Opie's difference equation is of order 12and even more in- tractable.

Increasing their bets effectively diminishes their capital, and when this is taken into account we come up with the following approximations to the ideal fre- quencies of their being ruined, in startling agreement with the simulations.

With this formulation we can approximate gambler's ruin probabilities and also estimate betting frac- tions to optimize average logarithmic growth, as decreed by the Kelly criterion.

In 141 Chapter Eleven the value 1.

It seems doubtful that it would vary appreciably as the deck composition changes within reasonable limits.

Certain properties of long term growth are generally ap- pealed to in order to argue the optimality of Kelly's fixed frac- tion betting scheme, and are based on the assumption of one bankroll, which only grows or shrinks as the result of gambling activity.

The questionable realism in the latter assumption, the upper and lower house limits on wagers, casino scrutiny, and finiteness of human life span all contribute to my lack of enthusiasm for this sort of analysis.

Precisely the suggested scenario could unfold: a hand could be dealt from a residue of one seven; one nine; four threes, eights, and aces each; and ten tens.

This would have a putative advantage learn more here about 12%, and call for a bet in this pro- portion to the player's current capital.

In fact, ignoring the table limits of casinos, the conjec- tured catastrophe would be guaranteed to happen and ruin the player sooner or later.

This is opposed to the Kelly idealization wherein, with only a fixed proportion of capital risked, ruin is theoretically impossible.

From simulated hands I estimate the covariance of two blackjack hands played at the same table to be.

Since the variance of a blackjack hand is about 1.

The first table of relative fluctuation is obtained by multiplying yen by and then taking the square root of the ratio of this quantity to V 7.

One of the problems encountered in approximating black- jack betting situations is that the normal distribution theory assumes that all subsets are equally likely to be encountered at any level of the deck.

An obvious counterexample to this is the 48 card level, which will occur in real blackjack only if the first hand uses exactly four cards.

The only imaginable favorable situation which could occur then would be if the player stood with something like 7, 5 v 6 up, A underneath.

The other problem is the "fixed shuffle point" predicted very well by David Heath in remarks made during the Second Annual Gambling Conference at Harrah's, Tahoe.

Gwynn's simulations, using a rule to shuffle up if 14 or fewer cards re- mained, confirmed Heath's conjecture quite accurately.

Roughly speaking, almost every deck allowed the completion of seven rounds of play, but half the time an eighth hand would be played and it tended to come from a deck poor in high cards, resulting in about a 2.

I resolved it by assuming a bet diagnosis arose every 5.

Preferential shuffling presents an interesting mathematical problem.

For example, if the preferential shuf- fler is trying to keep exactly one card away from the player, he can deal one card and reshuffle if it isn't the forbidden card, but deal the whole deck through if the first one is.

I carried out the Bayesian analysis by using an a priori Hi Lo distribution of points with six cards played and a com- plicated formula to infer hand-winning probabilities for the dif- ferent values of Hi Lo points among the six cards assumed used.

From this was generated an a posteriori distribution of the Hi Lo count, assuming the player did win the hand.

A player win was associated with an average drain of.

This translates into about a.

There is an apparent paradox in that the cards whose removal most favors the player before the deal are also the cards whose appearance as dealer's up card most favors the player.

Thus an intuitive understanding of the magnitude and direction of the effects is not easy to come by.

The last line tabulates the player's expectation as a function of his own initial card and suggests a partial explanation of the "contradiction", although the question of why the player's first card should be more important than the dealer's is left open.

Let Xl be the number of stiff totals 12-16 which will be made good by the particular denomination considered and X 2 be the number of stiff totals the card will bust.

Finally, to mirror a card's importance in making up a blackjack, define an artificial variable X g to be equal to one for a Ten, four for an Ace, and zero otherwise.

The following equation enables prediction of the ultimate strategy effects with a multiple correlation of.

He wants the deck to be rich in tens, but not too rich.

Some authors, who try to explain why an abundance of tens favors the player, state that the dealer will bust more stiffs with ten rich decks.

This is true, but only up to a point.

The dealer's probability of busting, as a function of ten density, appears to maximize.

This compares with a normal.

The player's advantage, as a function of increasing ten density, behaves in a similar fashion, rising initially, but necessarily returning to zero when there are only tens in the deck and player and dealer automatically push with twenty each.

It reaches its zenith almost 13% when 73% of the cards are tens.

Strangely, a deck with no tens also favors the player who can adjust his strategy with sufficient advantage to overcome the.

Thorp presents the classic example of a sure win with 7,7, 8,8,8 remaining for play, one person opposing the dealer.

This may be the richest highest expec- tation subset of a 52 card deck.

An infinite deck composition of half aces and half tens maximizes the player's chance for blackjack but gives an expectation of only 68% whereas half sevens and half eights will yield an advantage of 164%.

An ordinary pinochle deck would give the player about a 45% advantage with proper strategy, assuming up to four cards could be split.

Insurance would always be taken when of- fered; hard 18 and 19 would be hit against dealer's ten; and, finally, A,9 would be doubled and T,T split regardless of the dealer's up card.

Certainly a deck of all fives would be devastating to the basic strategist who would be forever doubling down and losing, but optimal play would be to draw to twenty and push every hand.

The results of a program written to converge to the worst possible composition of an infinite deck suggest this lower bound can never be achieved.

No odd totals are possible and the only "good" hands are 18 and 20.

The player cannot be dealt hard ten and must "mimic the dealer" with only a few insignificant departures principally standing with 16 against Ten and split- ting sixes against dealer two and six.

The dealer busts with a probability of.

The creation of this pit boss's delight a dealing shoe gaffed in these proportions would provide virtual immunity from the depredations of card counters even if they knew the composition may be thought of as the problemof increasing the dealer's bust probability while simultaneously leaching as many of the player's options from "mimic the dealer" strategy as possible.

It 148 can be verified that proper strategy with a sevenless deck is to stand in this situation and a thought experiment should con- vince the reader that as we add more and more sevens to the deck we will never reach a point where standing would be cor- rect: suppose four million sevens are mixed into an otherwise normal deck.

Then hitting 16 will win approximately four times and tie once out of a blackjack card counting money management attempts, while standing wins only twice when dealer has a 5 or 6 underneath and never ties!

Calculations assume the occurrence of two non- sevens is a negligible second order possibility.

The addition of a seven decreases the dealer's chance of busting to more than offset the player's gloomier hitting prognosis.

In the following table we may read off the effect of remov- ing a card of each denomination on the dealer's chance of busting for each up-card.

The last line confirms that the removal of a seven increases the chance of busting a ten by.

EFFECT OF REMOVAL ON DEALER'S CHANCE OF BUSTING in % DENOMINATION REMOVED Dealer's Chance Sum of of Up Card A 2 3 4 S 6 7 8 9 T Bust Squares - - - ----- A .

What we learn from the magnitudes of numbers in the "Sumof Squares" column is that the probability of busting tens and nines fluctuates least as the deck is depleted, while the chance of breaking a six or five will vary the most.

This is in keeping with the remarks in Chapter Three about the volatility ex- perienced in hitting and standing with stiffs against large and small cards.

The World's Worst Blackjack Player Ask "who is the best blackjack player?

Watching a hopeless swain stand with 3,2 v T at the Barbary Coast in Las Vegas rekindled my interest in the question "who is the world's worst player and how bad is he?

Penalty in % Always insure blackjack Always insure T,T Always insure anything Stand on stiffs against high cards Hit stiffs against small cards Never double down Double ten v T or A Always split and resplit T,T Always split 4,4 and 5,5 Other incorrect pair splits Failure to hit soft 17 Failure to hit soft 18 v 9 or T Failure to hit A,small 150.

Hence it seems unlikely that any but the deliberately destructive could give the house more than a 15% edge.

This is only a little more than half the keno vigorish of 26%: the dumbest blackjack player is twice as smart as any keno player!

Observations I made in the spring of 1987 showed that the overall casino advantage against a typical customer is about 2%.

The number and cost of players' deviations from basic strategy were recorded for 11,000 hands actually played in Nevada and New Jersey casinos.

The players misplayed about one hand in every 6.

This translates into an expectation 1.

Other findings: Atlantic City players were closer to basic strategy than those in Nevada, by almost.

The casinos probably win less than 1.

Incidentally, standing with A,4 v T is more costly by 13% than standing with 3,2.

It's only because we've grown more accustomed to seeingthe former that we regard the latter as the more depraved act.

One player, when innocently asked why he stood on A,5replied "Evenif I do get a ten emphasis to indicate that he apparently thought this was the best of all possible draws I still would only have 16".

The Unfinished Hand Finally, let the reader be apprised of the possibility of an "unfinished'; blackjack hand.

Imagine a player who splits sixteen tens and achieves a total of twenty-one on each hand by drawing precisely two more cards.

The dealer necessarilyhas an ace up, ace underneath, but cannot complete the hand.

By bouse rules she is condemnedthroughout eternity to a Dante's Inferno task of shuffling the last two aces, offer- ing themto the player for cut, attempting to hit her own hand, and rediscovering that they are the burn and bottom cards, unavailable for play!

Minimization of a function of ten variables is not an easy thing to do.

In this case the ten variables are the densities of the ten distinct denominations of cards and the function is the associated player advantage.

Although I cannot prove this is the worst deck, there are some strong arguments for believing it is: 1.

The minimum of a function of many variables is often found read more the boundary and with seven denominations having zero densities we definitely are on a boundary.

To approach, the required bust prob- ability for the theoretically worst deck there would have to be some eights, nines, or tens.

If there are eights or nines, their splitting would probably provide a favorable option to "mimic the dealer" strategy which would reduce the 17% disadvantage from stand- ing with all hands.

Also, if there are nines or tens, the player will occasionally, with no risk of busting, reach good totals in the 17 to 21 range, thus achieving a bet- ter expectation than "never bust" strategy was assumed to yield.

Either way, the theoretical -17% is almost certainly not achievable.

There's an intuitive argument for blackjack ti 84 plus only even cards in the "worst deck" - once any odd card is in- troduced then all totals from 17 to 26 can be reached.

Half of these are good and half bad.

But with only even cards you can only reach 18,20,22,24, and 26, three out of five of which are busts.

This reduced flexibility should help in raising the dealer bust probability while simultaneously minimizing the player's options.

Assuming only even numbers, the eights are filtered out because they provide favorable splits for the player.

The fours make good any totals of 14 and 16 and hence lower the dealer bust probability.

The twos are tantalizers in that they bring home only totals of 16 for the dealer, but keep other stiffs stiff for another chance of being busted.

At one of his seminars, the author instructs Sue of the Sacramento Zoo in the art of playing natural 21.

This has been due largely to a concernfor playingthe subsequentlyderivedhands optimally, depending on the cards used on earlier parts of the split.

This begs a distinction between "basic" and "zero-memory" strategy, and will lead us to an algorithm for exact determina- tion of repeated pair splitting expectation with zero-memory strategy.

Imagine you are playing single deck blackjack and have split three deuces against a four.

To each of the first two deuces you draw two sevens, and on the third deuce you receive a ten.

It is basic play to hit T,2 vs.

If you answered yes to the previous question, suppose the first two deuces were busted with two tens each.

You are dealt an ace and a nine to the third deuce.

Now answer the previous question.

Indeed Epstein suggests that zero-memory implies knowledge only of the player's original pair and dealer's up card.

Now, consider splitting eights against a seven in a single deck: A.

Calculate the conditional expectation for starting a hand with an eight against dealer's seven given that 1.

The second condition in A.

The player's expectation from repeated pair splitting is now given by 1081 - 2 A + 1176 90 - 3- B + 1176 155 5 - 4 -C 1176 The three fractions are, of course, the probabilities of splitting two, three, and four eights.

The extension to two and four decks is immediate.

Let E I be the logic blackjack described conditional expectation if exactly I cards are split and hence removed and P I be the prob- ability that I cards will be split; then the pair splitting expecta- tion is: I: P I I 1 I ~ 2 The coefficients, I - P Ishrink rapidly and a very satisfac- tory estimate of E J for J ~ 3 could be achieved by extrapola- tion from the calculated value of E 2.

To do this we introduce an artificial E l without any reference to pair splittingas the weighted average expectation of the hands 8,A 8,2.

These expectations would already be available from the general blackjack program and provide us with the base point for our extrapolation.

P 2 : Single Deck 47 46 --- 49 48 Double Deck 95 94 --- 101 100 Four Deck 191 190 ---- 205 204 Infinite Deck 12 12 13 13 R I : 5-I 48-1 9-1 96-1 17-I 192-1 12 53-21}- 52-21 105-21 - 104-21 209-21 - 208-21 169 The factors R I reflect the probability of "opening" drawing a new eight to a split eight and "closing" drawing a non-eight to an already split eight the Ith split card.

To three decimals the P I are I Single Deck Double Deck Four Deck Infinite Deck 2.

This all assumes X,X is being split against Y I: X.

Sim- ple modifications can be made if it is X,X against X.

The following table of the number of distinguishably different drawing sequences for one and many decks suggests the relative amount of computer time required.

Ten One Deck 5995 16390 10509 6359 3904 2255 1414 852 566 288 Many Decks 8497 18721 11125 6589 4024 2305 1441 865 577 289 A program can be written in BASIC in as few as 28 steps to cycle through all of the dealer's drawing sequences and weight the paths for a prescribed up card and deck composi- tion.

Thorp counted the total number of distinguishable blackjack subsets of a single deck as 5 9.

Since there are 2 52 possible subsets there is an average duplication with respect to suit and ten denomination of about 130 million.

The realization that there are only 1993 different subsets of size five was embarrassing to me, since I had simulatedthem2550 times to test the validity of using the normal distribution approximation for the least squares linear estimators of deck favorability for varying basic strategy.

The results provide a worst case evaluation of the ac- curacy of the previously mentioned approximation since the in- teractions neglected by the linear estimates are most severe for small subsets and the normal approximation to their distribution is poorest at the beginning and end of the deck.

The ~ 1 and 5 ; subsets that might be encountered for a given up' card are achieved by weighting the distinguishable 159 subsets properly.

Favorability of hitting over standing was recorded for abstract totals.

The dealer was assumed to stand on soft 17 and the few unresolved situations were completed by a formula which reasonably distributed the dealer's unfinished total on the shuffle up.

Actual frequencies of, and gain from, violating the basic strategy were recorded.

The performances of Hi Opt I, Hi Opt II, and the Ten count were recorded in these situations.

The meaning of the following charts is best explained by example: With five cards left in the deck, perfect knowledge of when to hit hard 14 against a two is worth 16.

The conditional favorability of hitting in those situations where it is ap- propriate is 16.

In parentheses besides these figures appears the corresponding normal approximation estimate of potential gain 15.

The Hi Opt I, Hi Opt II, and Ten Count systems had respective efficiencies of 71, 77, and 68%.

With six cards left in the deck precisely optimal hitting will occur.

The figures do not reflect the likelihood of the dealer having the given up card or the player possessing the particular total.

The greatest condi- tional gain in a hitting situation is the 40.

The greatest conditional standing gain is almost 40%, with 16 vs 7.

Hitting hard 17 is the most important variation against an eight and a system which counts A, 2, 3, 4 low and 6, 7, 8, 9 high would be nearly.

Random Subsets stratified according to Ten Density To examine the behavior of the normal approximation estimates for larger subsets, 3000 each of sizes 10 through 23 were simulated by controlling the number of tens in each subset to reflect actual probabilities.

The only up-card con- sidered was the ten because of the rapidity of resolution of the dealer's hand.

The effect of this stratification could thus be expected to be a reduction in the variance of the sample distributions pro- portional to the square of the Ten Count's correlation coeffi- cients for the six situations examined.

In addition to this reduction in variance of typically 40%, there wouldbe the added bonus of saving computer time by not having to select the ten- valued cards using random numbers.

The results provide the continuum necessary to compare different card counting systems.

Again, the following charts are best explained by example: with 10 cards left in the deck it was proper to stand with twelve in.

The gain over basic strategy was 3.

The Ten Count was 28% efficient, and a "special" system based on the density of the sevens, eights, and nines scored an impressive 78%.

The loss shown for the Ten counter playing a total of twelve with 21 cards left indicates the critical subsets with ex- actly 10 tens in them probably had an unduly large number of sevens, eights, and nines.

A basic strategist who always hits twelve would have done better in this instance.

II Special "6-5" 10.

Extreme discontinuities in efficiencies as a function of the number of cards in the subset can usually be explained by one of the system's realizable values being very close to its critical change of strategy parameter.

For example, the Ten Count's critical change ratio for standing with 15 is close to 2 others to 1 ten, and efficiencies take a noticeable dip with 12, 15, 18, and 21 cards in the deck.

In such cases the card counting system, whether it suggests a change in strategy or not, is us- ing up a considerable part of its probability distribution in very marginal situations.

Stratified Sampling used to analyze Expectation in a particular Deck The following approximate computations show that the variance of a blackjack hand result is about 1.

Player Approximate Squared Result Probability Result 2.

The sample sum will have a variance of 13 1.

It is worthwhile to study the consequences of stratified, rather than random, sampling.

Let the thirteen hands now be played against each of the denominations ace through king as dealer up-card.

Then the of the sum would obey 13 13 13 Var.

Thus the average variance for these stratified sample observations has been reduced from 1.

Using Epstein's tables of player expectation as a function of dealer up-card, we find this average square to be.

The same principle, albeit with more elaborate symbolism, can be used to show that controlling the player's first card as 168 well as the dealer's up-card will reduce variance by.

The average squared expectation for three card situations, where player's hand and dealer's up card are specified, is.

To get an approximation to how much variance reduction would result if four or more cards were forced to obey exact probability laws in the sample, we can assume the resolved hands have the same 1.

Then, employing some of Gwynn's computer results which showthat about 17% of all hands require four cards, 40% five cards, 28% six cards, 11% seven cards, and 4% eight or more cards, we complete the following table: Number of Cards Controlled Precisely o 1 2 3 4 5 6 7 8 Average Squared Expectation.

Beyond this, however, lurk even greater savings in computer time since the number of cards actually simulated with random numbers would be very few.

This would have the same variance as a purely random sample of about 25 million hands.

Moreover, only about 5 million cards would have to be generated to complete the 8.

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