The THEORY of BLACKACK The Compleat Card Counter's Guide to the Casino Game of 21 PETER A. GRIFFIN c:::J: § z G') az ~ Las Vegas, Nevada en The Theory of Blackjack: The Compleat Card Counter's Guide to the Casino Game of21 Published by Huntington Press 3687 South Procyon Avenue Las Vegas, Nevada 89103 (702) 252-0655 vox (702) 252-0675 fax Copyright © 1979, Peter Griffin 2nd Edition Copyright © 1981, Peter Griffin 3rd Edition Copyright © 1986, Peter Griffin 4th Edition Copyright © 1988, Peter Griffin 5th Edition Copyright © 1996, Peter Griffin ISBN 0-929712-12-9 Cover design by Bethany Coffey All rights reserved. No part of this publication may be translated, reproduced, or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without the expressed written permission of the copyright owner. TABLE OF CONTENTS FOREWORD TO THE READER 1 INTRODUCTION 1 Why This Book? 2 Difficulty Interpreting Randomness 3 Blackjack's Uniqueness 3 Use of Computers 4 Cheating 5 Are Card Counters Cheating? 6 Appendix 8 Bibliography 8 2 THE BASIC STRATEGy 11 Definition ofBasic Strategy 12 Hitting and Standing 12 Doubling Down 15 Pair Splitting 16 Summing Up 16 Condensed Form of Basic Strategy 17 House Advantage 18 Appendix 20 3 THE SPECTRUM OF OPPORTUNITY 21 An Example 22 Bet Variation 23 Strategy Variation 23 Insurance is 'Linear' 24 Approximating BetVariation 25 Approximating Strategy Variation 26 How Much Can be Gained by Perfect Play? 28 Average Disadvantage for Violating Basic Strategy 28 Volatility 28 Appendix 32 4 EXPLOITING THE SPECTRUM-SINGLE PARAMETER CARD COUNTING SYSTEMS 40 The Role of the Correlation Coefficient 41 Efficiency 42 Betting Correlation 43 Strategic Efficiency 45 Proper Balance Between Betting and Playing Strength 47 Simplicity Versus Complexity 48 Appendix 50 5 MULTIPARAMETER CARD COUNTING SYSTEMS 56 Keeping Track of a Single Denomination 57 The Importance of the Seven When You Have Fourteen Against a Ten 58 Ultimate Human Capability 59 The Effect of Grouping Cards 60 John Henry vs the Steam Engine 61 Appendix 62 6 TABLES AND APPLICATIONS 69 Insurance and Betting Effects 71 Virtually Complete Strategy Tables 72 How to Use These Tables 86 Quantify the Spectrum of Opportunity at Various Points in the Deck 86 The Normal Distribution of Probability 90 Chance of Being Behind 90 Distribution of a Point Count 92 How Often is Strategy Changed? 93 Gain From Bet Variation 94 Appendix 95 7 ON THE LIKELY CONSEQUENCES OF ERRORS IN CARD COUNTING SYSTEMS 96 Two Types of Error 97 An Exercise in Futility 98 Behavior of Strategic Expectation as the Parameter Changes 99 An Explanation ofErrors 106 Appendix 109 8 MANY DECKS AND DIFFERENT RULES 115 The Effect of Rule Changes 116 Opportunity Arises Slowly in Multiple Decks 117 Betting Gain in Two and Four Decks 119 No Hole Card 120 Surrender 120 Bonus for Multicard Hands 123 Double Exposure 126 Atlantic City 127 Appendix 129 9 MISCELLANY 131 More Than One Hand? 133 Shuffling 135 Previous Result's Effect on Next Hand 137 Appendix 139 10 CURIOS AND PATHOLOGIES IN THE GAME OF TWENTY-ONE 145 Some Extremely Interesting Facts 147 The Worst Deck 148 Effect of Removal on Dealer's Bust Probability 148 The World's Worst Blackjack Player 150 The Unfinished Hand 151 Appendix 152 11 SOME TECHNIQUES FOR BLACKJACK COMPUTATIONS 154 Dealer's Probabilities 158 Distinguishably Different Subsets 159 Random Subsets Stratified According to Ten Density 163 Stratified Sampling Used to Analyze Expectation in a Particular Deck 167 Use of Infinite Deck Approximations 170 Cascading Process for Determination of Best Strategy 172 Appendix 179 12 UPDATE - FIFTH NATIONAL CONFERENCE ON GAMBLING 180 Improving Strategy Against the Dealer's Ace 180 A Digression on Precise Pinpointing of Strategic Indices 183 When Reshuffling is Necessary to Finish a Hand 184 Percentage Advantage from Proportional Betting Schemes 186 Games Which Have an Advantage for the Full Deck 188 Final Thoughts 190 Appendix 193 13 REGRESSION IMPLICATIONS FOR BLACKJACK AND BACCARAT 203 The Problem 203 Woolworth Blackjack 205 Digression: The Count of Zero 208 Actual Blackjack, 10,13, and 16 Card Subsets 211 Linear Approximation to the Infinite Deck Blackjack Function 214 Can Baccarat Be Beaten? 216 Ultimate Point Counts 219 Appendix 224 14 POSTSCRIPT 1986 228 Multiple Deck Strategy Tables 231 Unbalanced Point Counts and the Pivot 233 Volatility ofAdvantage for Various Rules 234 Some Very Important Information 235 Kelly Criterion Insurance 236 The Small Player 239 Appendix 244 SUPPLEMENT I- RULES AND CUSTOMS OF CASINO BLACKJACK 248 Blackjack 249 Insurance 249 The Settlement 249 Hitting and Standing 250 Pair Splitting 250 Doubling Down 251 SUPPLEMENT 11- CARD COUNTING 251 A System 251 Betting by the Count 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." Eugene McCarthy, on the similarity between politicians and football coaches. 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." For this reason I assume that all readers have an understanding of the rudiments of probability, at least to the extent of multiplying and adding appropriate fractions. 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." My purpose in doing this is not to dissuade those without knowledge of advanced mathematics orstatistics from reading the Appendix, butrather to forewarn them that the arguments presented may occasionally tran­ scend theirlevel of preparation. Thus advised, they will then be able to skim over the formulas and derivations which mean lit­ tle to them and still profit quite a bit from some comments and material which just seemed to fit more naturally in the Appen­ dices. Different sections of the Appendices are lettered for con­ venience and follow the development within the chapter itself. The Appendix to Chapter One will consist of a bibliography of all books or articles referred to later. When cited in subsequent chapters only the author's last name will be mentioned, unless this leads to ambiguity. For the intrepid soul who disregards my warning and insists on plowing forward without the slightest knowledge of blackjack at all, I have included two Supplements, the first to acquaint him with the rules, practices, and terminol­ ogy of the game and the second to explain the fundamental principles and techniques of card counting. These will be found at the end of the book. Revised Edition On November 29, 1979, at 4:30 PM, just after the first edi­ tion of this book went to press, the pair was split for the first time under carefully controlled laboratory conditions. Con­ trary to original fears there was only an insignificant release of energy, and when the smoke had cleared I discovered that splitting exactly two nines against a nine yielded an expecta­ tion of precisely -.0531 on each of the split cards. Only minutes later a triple split of three nines was executed, produc­ ing an expectation of -.0572 on each card. Development of an exact, composition dependent strategy mechanism as well as an exact, repeated pair splitting algorithm now enables me to update material in Chapters Six, Eight, and, particularly, Eleven where I present correct basic strategy recommendations for any number of decks and dif­ ferent combinations of rules. There is new treatment of Atlantic City blackjack in Chapters Six and Eight. In addition the Chapter Eight analysis of Double Exposure has been altered to reflect rule changes which have occurred since the original material was written. A fuller explication of how to approximate gambler's ruin probabilities for blackjack now appears in the Appendix to Chapter Nine. A brand new Chapter Twelve has been writ­ ten to bring the book up to date with my participation in the Fifth National Conference on Gambling. Elephant Edition In December, 1984, The University of Nevada and Penn State jointly sponsored the Sixth National Conference on Gambling and Risk Taking in Atlantic City. The gar- gantuan simulation results of my colleague Professor John Gwynn of the Computer Science Department at California State University, Sacramento were by far the most significant presentation from a practical standpoint and motivated me to adjust upwards the figures on pages 28 and 30, reflecting gain from computer-optimal strategy varia­ tion. My own contribution to the conference, a study of the nature of the relation between the actual opportunity occur­ ring as the blackjack deck is depleted and the approxima­ tion provided by an ultimate point count, becomes a new Chapter Thirteen. In this chapter the game of baccarat makes an unexpected appearance, as a foil to contrast with blackjack. 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.