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Probability, Decisions and Games
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Table of Contents

Preface xi Acknowledgments xv About the companion website xvii 1 An Introduction to Probability 1 1.1 What is Probability? 1 1.2 Odds and Probabilities 5 1.3 Equiprobable Outcome Spaces and De Mere's Problem 6 1.4 Probabilities for Compound Events 9 1.5 Exercises 12 2 Expectations and Fair Values 15 2.1 Random Variables 15 2.2 Expected Values 16 2.3 Fair Value of a Bet 19 2.4 ComparingWagers 19 2.5 Utility Functions and Rational Choice Theory 23 2.6 Limitations of rational choice theory 24 2.7 Exercises 26 3 Roulette 31 3.1 Rules and Bets 31 3.2 Combining Bets 37 3.3 Biased Wheels 38 3.4 Exercises 42 4 Lotto and Combinatorial Numbers 45 4.1 Rules and Bets 45 4.1.1 The Colorado Lotto 45 4.1.2 The California Superlotto 51 4.2 Sharing Profits: De Mere's Second Problem 52 4.3 Exercises 55 5 The Monty Hall Paradox and Conditional Probabilities 59 5.1 The Monty Hall Paradox 59 5.2 Conditional Probabilities 62 5.3 Independent Events 65 5.4 Bayes Theorem 66 5.5 Exercises 70 6 Craps 75 6.1 Rules and Bets 75 6.1.1 The Pass Line Bet 75 6.1.2 The Don't Pass Line Bet 84 6.1.3 The Come andDon't Come Bets 85 6.1.4 Side Bets 85 6.2 Exercises 86 7 Roulette Revisited 89 7.1 Gambling Systems 89 7.1.1 Martingale Doubling Systems 89 7.1.2 The Labouchere System 92 7.1.3 D'Alembert Systems 94 7.2 You are a BigWinner! 96 7.3 How Long will My Money Last? 97 7.4 IsThisWheel Biased? 101 7.5 Bernoulli Trials 102 7.6 Exercises 103 8 Blackjack 107 8.1 Rules and Bets 107 8.2 Basic Strategy in Blackjack 109 8.3 A Gambling System thatWorks: Card Counting 114 8.4 Exercises 117 9 Poker 121 9.1 Basic Rules 121 9.2 Variants of Poker 123 9.3 Additional Rules 124 9.4 Probabilities of Hands in Draw Poker 124 9.4.1 The Effect of Card Substitutions 127 9.5 Probabilities of Hands in Texas Hold'em 128 9.6 Exercises 132 10 Strategic Zero-Sum Games with Perfect Information 135 10.1 Games with Dominant Strategies 135 10.2 Solving Games with Dominant and Dominated Strategies 139 10.3 General Solutions for Two Person Zero-Sum Games 143 10.4 Exercises 144 11 Rock-Paper-Scissors: Mixed Strategies in Zero-Sum Games 147 11.1 Finding Mixed-Strategy Equilibria 148 11.2 Mixed Strategy Equilibria in Sports 152 11.3 Bluffing as a Strategic Game with a Mixed-Strategy Equilibrium 153 11.4 Exercises 159 12 The Prisoner's Dilemma and Other Strategic Non-zero-sum Games 161 12.1 The Prisoner's Dilemma 161 12.2 The Impact of Communication and Agreements 162 12.3 Which Equilibrium? 164 12.4 Asymmetric Games 169 12.5 Exercises 171 13 Tic-Tac-Toe and Other Sequential Games of Perfect Information 175 13.1 The Centipede Game 175 13.2 Tic-Tac-Toe 178 13.3 The Game of Nim and the First- and Second-Mover Advantages 181 13.4 Can Sequential Games be Fun? 184 13.5 The Diplomacy Game 184 13.6 Exercises 187 A A Brief Introduction to R 191 A.1 Installing R 191 A.2 Simple Arithmetic 192 A.3 Variables 194 A.4 Vectors 195 A.5 Matrices 199 A.6 Logical Objects and Operations 201 A.7 Character Objects 204 A.8 Plots 205 A.9 Iterators 208 A.10 Selection and Forking 211 A.11 OtherThings to Keep in Mind 211 Index 213

About the Author

ABEL RODRIGUEZ, PHD, is Professor in the Department of Applied Mathematics and Statistics at the University of California, Santa Cruz (UCSC), CA, USA. The author of 40 journal articles, his research interests include Bayesian nonparametric methods, machine learning, spatial temporal models, network models, and extreme value theory. BRUNO MENDES, PHD, is Lecturer in the Department of Applied Mathematics and Statistics at the University of California, Santa Cruz, CA, USA.

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