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Probability, Decisions and Games: A Gentle Introduction using R

Probability, Decisions and Games: A Gentle Introduction using R

Abel Rodríguez, Bruno Mendes

ISBN: 978-1-119-30262-9

Mar 2018

240 pages


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Introduces the fundamentals of probability, statistics, decision theory, and game theory, and features interesting examples of games of chance to illustrate the presented mathematical concepts

Ranging from simple games to strategic games, Probability, Decisions and Games features a variety of gaming and gambling examples to build a better understanding of quantitative reasoning. The authors establish fundamental concepts before moving on to more strategic games that illustrate how multiple concepts fit together. Organized into thirteen chapters, this book presents equal coverage on the general mathematical analysis concepts found in a wide variety of games as well as the specific theories and problems associated with well-known casino games. The authors cover zero-sum games, non-zero-sum games, and sequential games in order to introduce ideas such as mathematical expectation and variance, bias, combinatorial calculus, conditional probability and Bayes Theorem, Bernoulli trials, and the Binomial distribution.

This book presents interesting gaming examples to highlight the practical applications and methodologies behind the basic concepts of probability, statistics, decision theory, and game theory. The first two chapters of Probability, Decisions and Games: A Gentle Introduction using R feature an introductory discussion of utility and probability theory in finite and discrete spaces. Subsequent chapters utilize popular casino games to illustrate the practical probabilistic methodologies. Finally, the book concludes with discussions on game theory and strategic games.

  • Features introductory coverage of probability, statistics, decision theory, game theory, and mathematical analysis and has been class-tested at University of California, Santa Cruz for the past six years
  • Illustrates mathematical concepts through interesting and fun examples using five popular casino games: roulette, lotto, craps, blackjack, and poker
  • Provides a variety of gaming and gambling examples of well-known simple games, such as casino games, zero-sum games, non-zero-sum games, and sequential games, in order to help readers better understand quantitative reasoning
  • Features computer simulations using R throughout in order to illustrate complex concepts and help readers calculate the presented problems
  • Contains exercises throughout and approaches games and gambling at a level that is accessible for readers with minimal experience
  • Presents a unique approach by presenting simple games first to demonstrate individual concepts before moving on to more strategic games that illustrate how these concepts work together

Probability, Decisions and Games: A Gentle Introduction using R is a unique and helpful textbook for undergraduate courses on statistical reasoning, introduction to probability, statistical literacy, and quantitative reasoning for students from a variety of disciplines.

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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 Méré’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 Méré’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 Labouchère 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