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The THEORY of BLACKACK The Compleat Card Counter's Guide to the Casino Game of 21 PETER A.GRIFFIN ::J: c: Â§ z G'.

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

GRIFFIN ::J: c: Â§ 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 Â© 1979, Peter Griffin 2nd Edition Copyright Â© 1981, Peter Griffin 3rd Edition Copyright Â© 1986, Peter Griffin 4th Edition Copyright Â© 1988, Peter Griffin 5th Edition Copyright Â© 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 Why This Book?

Difficulty Interpreting Randomness Blackjack's Uniqueness Use of Computers Cheating Are Card Counters Cheating?

Appendix Bibliography 1 2 3 3 4 5 6 8 8 2 THE BASIC STRATEGy Definition of Basic Strategy Hitting and Standing Doubling Down Pair Splitting Summing Up Condensed Form of Basic Strategy House Advantage Appendix 11 12 12 15 16 16 17 18 20 3 THE SPECTRUM OF OPPORTUNITY An Example Bet Variation Strategy Variation Insurance is 'Linear' Approximating Bet Variation Approximating Strategy Variation How Much Can be Gained by Perfect Play?

Average Disadvantage for Violating Basic Strategy Volatility Appendix 21 22 23 23 24 25 26 28 28 28 32 4 EXPLOITING THE SPECTRUM-SINGLE PARAMETER CARD COUNTING SYSTEMS The Role of the Correlation Coefficient Efficiency Betting Correlation Strategic Efficiency Proper Balance Between Betting and Playing Strength 40 41 42 43 45 47 Simplicity Versus Complexity Appendix 5 MULTIPARAMETER CARD COUNTING SYSTEMS Keeping Track of a Single Denomination The Importance of the Seven When You Have Fourteen Against a Ten Ultimate Human Capability The Effect of Grouping Cards John Henry vs the Steam Engine Appendix 6 TABLES AND APPLICATIONS Insurance and Betting Effects Virtually Complete Strategy Tables How to Use These Tables Quantify the Spectrum of Opportunity at Various Points in the Deck The Normal Distribution of Probability Chance of Being Behind Distribution of a Point Count How Often is Strategy Changed?

Gain From Bet Variation Appendix 48 50 56 57 58 59 60 61 62 69 71 72 86 86 90 90 92 93 94 95 7 ON THE LIKELY CONSEQUENCES OF ERRORS IN CARD COUNTING SYSTEMS Two Types of Error An Exercise in Futility Behavior of Strategic Expectation as the Parameter Changes An Explanation of Errors Appendix 99 106 109 8 MANY DECKS AND DIFFERENT RULES The Effect of Rule Changes Opportunity Arises Slowly in Multiple Decks Betting Gain in Two and Four Decks No Hole Card Surrender Bonus for Multicard Hands Double Exposure Atlantic City Appendix 115 116 117 119 120 120 123 126 127 129 96 97 98 9 MISCELLANY More Than One Betonline blackjack dealer />Shuffling Previous Result's Effect on Next Hand Appendix 131 133 135 137 139 10 CURIOS AND PATHOLOGIES IN THE GAME OF TWENTY-ONE Some Extremely Interesting Facts The Worst Deck Effect of Removal on Dealer's Bust Probability The World's Worst Blackjack Player The Unfinished Hand Appendix 145 147 148 148 150 151 152 11 SOME TECHNIQUES FOR BLACKJACK COMPUTATIONS Dealer's Probabilities Distinguishably Different Subsets Random Subsets Stratified According to Ten Density Stratified Sampling Used to Analyze Expectation in a Particular Deck Use of Infinite Deck Approximations Cascading Process for Determination of Best Strategy Appendix 12 UPDATE - FIFTH NATIONAL CONFERENCE ON GAMBLING Improving Strategy Against the Dealer's Ace A Digression on Precise Pinpointing of Strategic Indices When Reshuffling is Necessary to Finish a Hand Percentage Advantage from Proportional Click the following article Schemes Games Which Have an Advantage for the Full Deck Final Thoughts Appendix 13 REGRESSION IMPLICATIONS FOR BLACKJACK AND BACCARAT The Problem Woolworth Blackjack Digression: The Count of Zero Actual Blackjack, 10,13, and 16 Card Subsets Linear Approximation to the Infinite Deck Blackjack Function 154 158 159 163 167 170 172 179 180 180 183 184 186 188 190 193 203 203 205 208 211 214 Can Baccarat Be Beaten?

Ultimate Point Counts Appendix 216 219 224 14 POSTSCRIPT 1986 Multiple Deck Strategy Tables Unbalanced Point Counts and the Pivot Volatility of Advantage for Various Rules Some Very Important Information Kelly Criterion Insurance The Small Player Appendix 228 231 233 234 235 236 239 244 SUPPLEMENT I - RULES AND CUSTOMS OF CASINO BLACKJACK Blackjack Insurance The Settlement Hitting and Standing Pair Splitting Doubling Down 248 249 249 249 250 250 251 SUPPLEMENT 11- CARD COUNTING A System Betting by the Count 251 251 252 INDEX 255 INDEX OF CHARTS AND TABLES 261 FOREWORD TO THE READER "You have to be smart enough to understand the game and dumb enough to think it matters.

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 little to them and still profit quite a bit from some comments and material which just seemed to fit more naturally in the Appendices.

Different sections of the Appendices are lettered for convenience 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 terminology 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 edition of this book went to press, the pair was split for the first time under carefully controlled laboratory conditions.

Contrary 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 expectation of precisely .

Only minutes later a triple split of three nines was executed, producing 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 different 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 written 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 gargantuan 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 variation.

My own contribution to the conference, a study of the nature of the relation between the actual opportunity occurring as the blackjack deck is depleted and the approximation 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 conversations helped expand my imagination on the subject; John Christopher, whose proofreading prevented many ambiguities 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 visit web page 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 blackjack continuous shuffling machines card counting play, just a beginner'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 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 ~Jevada 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 continue reading, for today I find myself farther 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 elation 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 Epstein had, themselves, entered into a pact with the casinos to deliberately exaggerate the player's odd~ in the game.

But after renewing my faith by confirming.

Why then should I presume to write a book on this subject?

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 table blackjack poker the theoretical probabilities to share with those who are interested; unfortunately my experience offers no assurance read more 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 appreciation of the possible therapeutic benefits from the intense absorption which overcomes the bettor when awmting the verdict 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 difference, 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 opponent 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 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 an electrician 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 determined a system's effectiveness anyway?

How good were the existing 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 objection 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 approximation 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 insights 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 visit web page 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 legerdemain.

However, you are justified in being more info to accept kt 103 blackjack 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 admitted 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 continue reading 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 doing 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 saying 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 link, 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 personality 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 criticism it provides.

However, reference to any of them is 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 computations were made on desk calculators.

Much of their terminology 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 extensive 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 repla~ement.

It is difficult to read in this untranslated version, and even more difficult to find.

Probability Theory and Mathematical Statistics.

Optimum Strategy in Blackjack.

Claremont 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 Blackjack.

The American Statistician 29 2 :84-88.

The authors, from North Carolina St.

University, present an intriguing and efficient recursive method for finite deck blackjack 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 blackjack.

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.

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.

read article 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 facing 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 example, 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 encounter, but without any reference to any other players' cards or any cards used on a previous round before the deck is reshuffled.

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 relatively among 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 consuming.

This is because, for each of the five distinguishably different 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 "conditional expectation" is.

Some readers may be surprised that a total of 18 is overall a losing hand here.

Note also that check this out 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 computers were deSigned for; limitations on the human life span and supply of paper preclude an individual doing the calculations by hand.

Doubling Down So much for the choice of whether to hit or stand in a particular situation, but how about the decision on whether to double down or not?

In some cases the decision will be obviously 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 advantage.

Hence the decision very blackjack simple math good 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 rais~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 https://veronsmeatmarket.com/blackjack/how-many-blackjack-hands-possible.html closer to the mathematician's ideal of "a fair game" one which has zero expectation for both competitors than the usually hypothesized coin toss, since real coins are flawed and might create a swish blackjack 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 expectation.

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.

Soft drawing and standing: Always draw to 17 and stand with 18, except hit 18 against 9 or T.

Pair Splitting: 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 advantage 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 individual'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 Gain.

Up Card 2 3 4 5 6 7 % Chance of Bust 35 38 40 43 42 26 8 read article T A 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 expectation.

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 recursive.

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 recognizing that, for any first and second hand that can occur, there is an equiprobable reordering of the deck which merely interchanges 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 just click for source 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 permitted, there is an extraordinarily improbable scenario which exhausts 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 significant 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 composition, will have an expectation of 6.

Player Dealer Hand Up Card Expectation Player Dealer Hand Up Card Expectation 5,6 5,8 5,9 5,T 6,8 8 9 T 6 9 T 6 8 T 6 8 9 5 9 T -- +2 +1 +1 +1 6,9 6,T .

This exploitation of decks favorable for basic strategy will henceforth be referred to as gain from "bet variation.

more info 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 fluctuations 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 illustrate the wild fluctuations associated with extremely depleted decks.

Generally, variations in strategy can 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 pendulum 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 reaso~; 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 complex 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 purposes, 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 accuracy of our approximation called the "method of least squares," it can be shown that the appropriate number~ 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 occasional 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 payoffs corresponding to these cards -19.

It learn more here 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 betting 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 the effect of the removal of each type of card.

These effects are given below, where for convenience 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~st described, of the removal of the ith 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 can be studied further by using the well known normal 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 Unseen Cards Insurance Gain Strategy Gain no Insurance Betting Gain 10 15 20 25 30 35 40 45.

This is consistent, of course, with Thorp and Walden's 'Fundamental Theorem'.

Two other important determiners of how much can be gained from individual strategy variations are also pinpointed by the formula.

Average See more 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 expected 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 nevertheless 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: Full Deck Gain by Hitting Effects of Removal for Hitting 14 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 contradictory 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 approximation 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 blackjack 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 for actual play will be the subject 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 hitting 16 vs Ten is regarded as a linear function of the cards remaining in the deck at any instant.

For specificity let there remain 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 contains 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.

Yi Xij is the total favorability of all k card subsets contain.

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 expectation itself, or the drawing expectation, or the standing expectation 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 nonsingularity 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 appeal 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 trying 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 equivalentÂ· 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 exN 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.

Of course in practical application N is finite rarely exceeding 312 and the proof of the pudding is in the eating.

Another representation of E nwhich will prove more convenient for.

The former relation is consistent with Thorp and Walden's Fundamental Theorem and the latter pinpoints the volatility parameter.

As a specific example of the accuracy of https://veronsmeatmarket.com/blackjack/blackjack-pizza-locations-denver.html normal approximation we will look at the insurance bet.

Because of the probabilistic anomaly that the insurance expectation available with 3k + 1 cards remaining is the same as for3k cards, it.

Figures are presented in the table for 20,21, and 22 cards remaining.

GAIN FROM INSURANCE Cards Remaining Actual Gain Normal Approximation 20 21 22.

The smoothing of the continuous method irons out the discreteness and provides perhaps a more representative answer to a question like "What happens at about the twenty card level?

Out of every 1326 player hands, it was assumed he would face a decision with totals of hard seven through eleven 32, 38, 48, 54, and 64 times respectively, with each soft double 16 times, and with a soft 18-hit on 23 occasions.

Different frequencies were used for the hard totals of twelve through seventeen, depending on the dealer's up card.

For small cards 2-6 these were estimated to be 130, 130, 110, 110, 100, and 100 respectively, while for high cards the figures 150, 155, 160, 165, 165 and 180 were used.

Obviously some dependence is neglected, such as that between the player's hand and dealer's card as well as that if, for example, we make a non-basic stand with fourteen it reduces the frequency of fifteens and sixteens we might stand with.

METER CARD COUNTING SYSTEMS "You count sixteen tens and what do you get?

Another day older and deeper in debt.

Suppose a standard deck.

Â·of cards is dealt through one at a time, without reshuffling.

Before each card is turned the player has the option of wagering, at even money, that the next card will be red.

For a full deck the game has a zero expectation, but after the first card is played the deck will be favorable for the wager on red about half the time.

An optimal card counting strategy is obvious, so for a more interesting illustration we'll assume the player is color blind.

One-can imagine several methods which will show a profit but fall short of optimality.

One idea is to look for an excess of hearts over spades among the unplayed cards.

When this condition obtains, the player should on the average, but not always, have the advantage.

We'll call this system A.

The diamond counter might employ a system B, monitoring the proportion of diamonds in the deck and betting on red when diamonds constitute more than one-fourth of the remaining cards.

Yet go here third possibility would be system C, based on the relative balance between three suits, say clubs, hearts, and diamonds.

Since on the average there are twice as many red cards as clubs, the deck should tend to be favorable whenever 40 the remaining red cards are more than twice as numerous as the clubs.

All three of these card counting methods can be carried out by assigning point values to the cards remaining in the deck, which point values would be opposite in algebraic sign to the numbers counted and continuously added as the cards are removed from the deck.

The appropriate point values for the three systems discussed, as well as the payoffs for the game itself, are given below.

Sum of System Spade Heart Diamond Club Squares Correlation A -1 1 0 0 2 B -1 -1 3 -1 12 COl Payoffs -1 1 1 1 -2 -1 6 4.

To my way of thinking the exÂ· ample had two advantages.

First of all, I could program the computer to determine precisely how much could be gained at any deck level with the three systems, as well as with the optimal color dependent strategy given by the payoffs themse~ves-there would be no sampling error since exact probabilities would be used.

please click for source second advantage was that the very simplicity in structure might make evident the direction to pursue in analyzing the manifoldly more complex game of blackjack.

For some reason, which I can no longer recollect, I had already calculated what, in statistics, is called the correlation coefficient between the point values of the card counting systems arid the payoff for the game itself.

This is done by dividing the sum of the products of the respective values assigned to each suit and the payoffs for the suit by the square root of the product of the sum of squares of values for the card counting system and sum of squares of the payoffs.

Relative Amount of Total Profit Gained by Red-Black Systems Number of Cards Left A B C 9 18 27 36 45.

This derivation appears in the Chapter Appendix.

With this in mind it seems natural to define the efficiency of a card counting system to be the ratio of the profit accruing from using the system to the total gain possible from perfect knowledge and interpretation of the unplayed set of cards.

What we learn from the mathematics, then, is that efficiency is directly related, and in some cases equal, to 'the correlation between the point values of the card counting system and the single card payoffs approximating th~ blackjack situation considered.

In blackjack we have one card counting system which may be used for a variety of purposes; first of all to determine if the deck is favorable to the player or not, and secondly to conduct any of more than a hundred different variations in strategy which might arise after the hand is dealt.

We can consider the card counting system to be an assignment go here point values to the cards remaining in or deleted from the deck, at our convenience.

In theory any assignment of points is permissible, but simple integers are more tractable for the human memory.

In addition it is desirable to have the restriction that the count be balanced in that the sum of the point values for a full deck be zero.

This way the direction of deflection of the deck from normal is instantaneously evident from the algebraic sign of the running count, regardless of depth in the deck.

Betting Correlation As a first example of the efficiency of a blackjack system, we will look at the most frequent and important decision, namely whether to bet extra money on the hand about to be dealt.

The index of this capability to diagnose favorable decks will be the correlation coefficient between the point values of the count system and the best linear estimates of deck favorability mentioned in the previous chapter.

To calculate, for instance, the betting strength of the HiOpt II system, we have Sum of A Hi Opt II Point Values Effects of Removal 0 23456789 2 2 o 0 T Squares -2 .

The correlation for 8.

A single deck was dealt down to the 14 card level and all systems were evaluated according to their simultaneous diagnosis of the samehanq.

If the pre-deal count for a system suggested an advantage, one unit was bet, otherwise nothing.

The systems are described in the table by their ten point values, ace through ten from left to right.

This is followed by the betting correlation, the Las Vegas simulation yield in % of units gained per hand played, the predicted yield in parenthesisthe Reno simulation yield, and again the predicted yield using the methods of this section.

Slightly altered param44 eters are necessary to approximate the Reno game, but these correlations are not presented since they differ very little from those for Las Vegas.

Las Correlation System Vegas Gain Reno Gain.

However, increased scrutiny by casino personnel makes wide variation in wagers impractical and there has evolved a secondary concern for how effective these systems would be for just varying strategy, particularly in single deck games.

Reliable simulation estimates of this capability are extraordinarily time consuming, so the correlation method of analysis proves ideal for getting a fix on how much can be gained by these tactics.

With this in mind, a program was written to converge to the optimal point values for conducting the 70 variations of strategy associated with hard totals of 10 through 16, along with the insurance wager.

Relative results for any card counting systems do not differ appreciably whether there are 10,20,30, or 40 cards remaining.

The initial optimization was to maximize overall strategic efficiency subject to a point value of -180 assigned to the tens.

Subsequent optimization was conducted at different levels of complexity, level of complexity being defined as the maximum of the absolute values of the points assigned.

Blackjack gurus seem unanimous in the opinion that the ace should be valued as zero since it behaves like a small card for strategic variations and a big card for betting strategy; this optimization also presumed kt 103 blackjack a value.

The following table presents the champions of their respective divisions.

Also, bigger is not necessarily better; the level four system narrowly edges the level five system.

For the evaluated systems all decisions are made on the basis of a single parameter, the average number of points remaining in the deck.

Evidently the maximum efficiency possible for strategic variation with a single parameter system is of the order of 70% and one can come quite close to that without going beyond the third level.

Although the Ten Count, when parameterized as a point count, uses the numbers 4 and -9, it is certainly not ,t the 9th level of mental gymnastics-one keeps track of the proportion of tens by counting off the tens and non-tens as they leave the deck.

They gave very similar relative results for all the systems' strategy gains reported here, with only an occasional interchange of the order of two systems whose efficiencies differ only in the third digit after the decimal point.

Proper Balance between Betting and Playing Strength The proper relative importance to attach to betting efficiency and playing efficiency depends on several factors: depth 47 of penetration, permissible increase in bet, and playing efficiency restricted to favorable decks.

Assuming the same penetration used in the previously mentioned Gwynn simulations the following empirical formula provides such a weighting by estimating the average profit available in terms of the basic betting unit.

One should allow about 20% more for Las Vegas rules and 10% less for Reno.

The formula suggests the two efficiencies are almost equally important for a 1 to 4 betting scale and that betting efficiency is rarely more than one and a half times as important as playing efficiency.

In summary, then, the- player who is shopping around for a best single parameter card counting system has a choice between Strategy Efficiency Betting Efficiency Best Strategy System Best Betting System 70% 90% 55% 100% Simplicity versus Complexity "It is my experience that it is rather more difficult to recapture directness and simplicity than to advance in the direction of evermore sophistication and complexity.

Any third-rate engineer or researcher can increase complexity; but it takes a flair of real insight to make things simple again.

Schumacher, Small is Beautiful Now, when you build a better mousetrap the world will beat a path to your door, just is a reputation as a blackjack "expert" entitles one to crÂ˘k letters on the topic "what system should I use?

Using the best one-level system you can achieve either a 64% playing efficiency or a 97% betting efficiency, and so the small sacrifice seems justified when we consider the ease on the memory as well as the source likelihood of error.

In addition, the simple plus or minus one systems are much more easily modified by inclusion of other information.

We will see in the next chapter that to raise strategy efficiency above 70% one must invoke separate parameters, and one of the easiest ways to do this is to use knowledge of the uncounted zero-valued cards not recognized by a simple level one system.

Defining the efficiency of a card counting system to be the ratio of profit from using the system to total profit Possible.

In any case h p can be shown to be an increasing function of p by the same argument which established E n to be an increasing function of a.

Further, rewriting dy 51 we may invoke the argument that E n decreases as J j.

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 correlation 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.

JÂŁL 20 21 22.

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 interactions 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 apologise, blackjack ballroom email sorry 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.

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 betting 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 deficient 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 login pokerstars these passwords with on the basis of a + 1 adjusted 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 stars blackjack shuffle considered, which counts as -12.

The fixed sign of the point value obscures this and can only be overcome by assigning the value zero and keeping a separate track of the density of these zero valued cards for reference in appropriate situations.

The average effect of removal for the eight cards recognized 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 link 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 composition 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 happens 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 instructions, 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 https://veronsmeatmarket.com/blackjack/blackjack-dc-metro.html denominations in different strategic situations should be no problem for an individual who is so inclined.

The increases in playing efficiency and betting correlation are exhibited below.

INCORPORATION OF ZERO VALUED CARDS INTO EINSTEIN SYSTEM Cards Incorporated Basic System A 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 proper 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 continue reading has been under the assumption that a separate track of each of the zero-valued cards is kept.

David Heath suggested sometime ago a scheme of blocking 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 fingering chart.

Heath's system is equivalent to fortifying a primary Gordon count with information provided by the block of "middle" cards, ~6,7,8,9}, there being no discrimination among these car~s individually.

As we can see from the following table of efficiencies 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 Auxiliary Grouping Playing Efficiency Gordon Gordon Gordon Primary Count { 6,7,8,9 } {6,7,8 } {6,7}.

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 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 arbiter 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, blackjack what mean does sixes, sevens, and tens, and one ace, two, three, and four, doubling was 61 % better than standing, 18% better than merely 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 L1 y.

It is unrealistic to suppose that such auxiliary point values would be remembered more precisely than to the nearest whole number.

Blocks of cards, like t6,7,8,9~.

Similarlyforl6,7,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 auxiliary count can be taken as 0 0 001 -1 0000 which is uncorrelated 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 suggests 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 o 0 0 -1.

Hence p 12' the correlation between X 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 representative, 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 gain is 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 Insur- 2 4 5 -3 - 6.!.

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 expectation?

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 antiquarian could desire will follow.

In order to condense the printing, the labeling will be abbreviated and uniform throughout the next several pages.

Each row will present the ten method blackjack counting of removal for the cards Ace through Ten, full deck favorability, 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 better off we are to draw to the total than to stand with it.

Naturally this will have a negative mean in the eleventh column 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, indicating 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 drawing 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 splitting 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 row corresponding to hard 14 v Ten.

The entry in the 11th column, 6.

For the same reason doubling down is not very advantageous, even with a total of 11.

The 11th column entries for 12Â·16 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 increase 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 effects 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 computer 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 attributable 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" dependent 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 2 -4.

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 in8.

First the table itself.

Corresponding to values of a variable designated by z, which ranges from o to 2.

Unit Normal Linear Loss Integral -z 0.

The following step by step procedure will be used in all such calculations.

Ignore the algebraic sign of m.

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 1 Â· 2.

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 click the following article, 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 correlation 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 coefficient 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 accuracy.

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 symmetry of the normal curve and determine the area or probability 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, providing that the number of cards left in the deck is specified.

The following procedure can be used.

Calculate the sum of squares of the point values https://veronsmeatmarket.com/blackjack/what-is-a-blackjack-pill.html to the thirteen denominations, calling it 2.

SSe where N is the number 13 N-I of cards in the full deck and n is the number of cards remaining.

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 calculations show the answer to be 25%.

Similarly, we find the approximate probability of a learn more here 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.

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 different sets of remaining cards.

Unplayed Residue Favorability of Hitting over Standing 4,T 4,4,T,T 4,4,4,T,T,T -50% 0% +10% From this simple example follow two interesting conclusions: 1.

Strategic favorabilities depend not strictly on the proportion of different cards in the deck, but really on the absolute numbers.

Every card counting system ever 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 questions 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, transistorized 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 conditions?

To err is human and neither the pit boss, the blackjack club, nor the cards are divine enough to forgive.

Two Types of Error There are two principal types of error in employing a count nonsense! ko blackjack trainer well 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 click the following article 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 forward 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 misinformation 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 correct parameter could have gained me.

The computer technique of altering normal decks so as to produce rich or lean mixtures for investigating different situations 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 new jersey rules consists of maintaining 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 averaging 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 quandaries and even the most carefully computerized critical indices have an element of faith in them.

Behavior of Strategic Expectation as the Parameter changes The assumption that the favorability for a particular action is a linear function of the average number of points in the deck is applied to interpolate critical indices and is also a consequence of the bivariate normal model used to analyze efficiency 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 prepared 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 incorporating 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 casino tournament hollywood perryville blackjack grow apace.

To estimate how much conditional blackjack table limits atlantic city 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 assumption 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 bandied "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 probably 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 click to see more 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 understandable concomitant of the characteristic which best differentiates the casino blackjack player from the independent 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 assigned 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 units are arbitrarily scaled to avoid decimals; they would actually depend on the volatility of, and point count's correlation with, the particular situation source />For relatively small critical indices such as.

However for larger critical indices the player may lose more from such overzealousness 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 remaining in 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 INDEX CRITICAL INDICES.

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 convertible 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 improves the chance of a good hand from one deck but is negligible otherwise.

A good example of ~his is doubling nine against Â·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 occasionally encountered is given visit web page 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 https://veronsmeatmarket.com/blackjack/blackjack-world-tour-app.html .

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 remaining 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 interesting 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 inherently less advantageous, the necessity of a very wide betting 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 extra 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 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 uncommon but interesting variations in rules.

Since these usually occur 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 click to see more 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 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 continue reading 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 disappointment of losing with an click 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 abbreviated four card strategy is as follows: Hit Hit Hit Hit Soft 19 and Below Against Anything But a 7 or 8 Hard 15 and Below Against a 2 Hard 14 and Below Against a 3 and 4 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 dependent 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 collides 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 tenup blackjacks.

The following table gives the frequency of development of five card hands in a just click for source deck game, with the one deck frequency in parentheses next to the four beatable blackjack switch figure.

A hand like 3,3,3,3,4with repetition of a particular denomination, will be much less probable 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 approximate 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 automatically if we get the 7 of hearts in our draw.

We must compare our hitting expectation of 'Â±" .

The equation becomes B .

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 discontinued, 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 expectation for the typical rules generally presumed in this book.

Interpolation by reciprocals suggests that the player's expectation will be one sixth of the way between.

The right to double after split is worth.

Early surrender itself provides a gain of.

Summarizing, 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 advantage as the deck is depleted than those which occur in ordinary 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 comparisons 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 expectation 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 extra 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 read article infinite and single deck expectations.

However, multiplication of Epstein's single deck expectations by infinite deck probabilities 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 52k-l for k decks and the corresponding chance of getting a card of the same denomination is 4k-l.

The differences between these figures and 52k-l the constant I 13 52k-l and I~which applies to an infinite deck, are I 2 13 52k-l respectively.

These differences themselves are very nearly proportional to the reciprocal of the number of decks used.

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 surrender for different hands from one and four decks.

For instance, 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 dependent, exceptions to the page 126 strategy: stand with A,7 v 8,3 and 7,6 and 8,5 v hard 11, except hit automatic shuffler 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 equivalent 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 overworked.

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 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 confronting.

Opie bets optimally, in proportion 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 20 50 100 200 Opie 877 668 Greta 896 733 438 541 135 231 Greta is obviously the more often ruined woman, but since they have the same expectation per play there must be a compensating factor.

This is, of course, time-whether double or nothing, Greta usually gets her result more quickly.

This illustrates 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 considerations 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 consecutive 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 accurate 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 https://veronsmeatmarket.com/blackjack/simple-blackjack-system.html 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 vegas blackjack rule 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 example 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.

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, occasionally, 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 trying to perform five perfect shuffles a "magician" demonstrated the skill at the Second Annual Gambling Conference sponsored by the University of Nevadabut to get some idea of what might happen if this were attempted, I asked a professional dealer from the Riverside in Reno read more 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 profitably 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 him winning, 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 subject to achieving an overall positive expectation.

Thus it click to see more 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 P i and having corresponding expectation Ei.

J ~ WÂ·PÂ·EÂ· 1 1 subiect.

Opie's difference equation is of order 12and even more intractable.

Increasing their bets effectively diminishes their capital, and when this is taken into account we come up with the following approximations to the ideal frequencies of their being ruined, in startling agreement with the simulations.

With this formulation we can approximate gambler's ruin probabilities and also estimate betting fractions to optimize blackjack en ligne gratuit 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-Â· pealed to in order to argue the optimality of Kelly's fixed fraction 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 proportion to the player's current capital.

In fact, ignoring the table limits of casinos, the conjectured 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 ~2 and then taking the square root of the ratio of this click to V 7.

One of the problems encountered in approximating blackjack 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 remained, 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 shuffler 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 complicated formula to infer hand-winning probabilities for the different 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 please click for source 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 removed parx casino blackjack the, 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 just click for source 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 expectation 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 offered; 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 splitting 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 problem of 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 convince the reader that as we add more and more sevens to the deck we will never reach a point where standing would be correct: suppose four million sevens are mixed into an otherwise normal deck.

Then hitting 16 will win approximately four times and tie read more 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 nonsevens 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 removing 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 Up Card A 2 3 4 5 6 7 8 9 T A 2 .

What we learn from the magnitudes of numbers in the "Sum of 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 experienced 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 Â·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 theÂ· 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 seeing the former that we regard the latter as the more depraved act.

One player, when innocently asked why he stood on A,5replied "Even if 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 https://veronsmeatmarket.com/blackjack/como-se-juega-el-blackjack.html 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 necessarily has an ace up, ace underneath, but cannot complete the hand.

By bouse rules she is condemned throughout eternity to a Dante's Inferno task of shuffling the last two aces, offering them to 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 more info denominations having zero densities we definitely are on a boundary.

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 standing 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 better 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 introduced then all totals from 17 to 26 can be reached.

Half of these are good and half bad.

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