# Probability, Decisions and Games: A Gentle Introduction using R

## Books INTRODUCES THE FUNDAMENTALS OF PROBABILITY, STATISTICS, DECISION THEORY, AND GAME THEORY, AND FEATURES INTERESTING EXAMPLES OF GAMES OF CHANCE AND STRATEGY TO MOTIVATE AND ILLUSTRATE ABSTRACT MATHEMATICAL CONCEPTS

Covering both random and strategic games, Probability, Decisions and Games features a variety of gaming and gambling examples to build a better understanding of basic concepts of probability, statistics, decision theory, and game theory. The authors present fundamental concepts such as random variables, rational choice theory, mathematical expectation and variance, fair games, combinatorial calculus, conditional probability, Bayes Theorem, Bernoulli trials, zero-sum games and Nash equilibria, as well as their application in games such as Roulette, Craps, Lotto, Blackjack, Poker, Rock-Paper-Scissors, the Game of Chicken and Tic-Tac-Toe. Computer simulations, implemented using the popular R computing environment, are used to provide intuition on key concepts and verify complex calculations.

The book starts by introducing simple concepts that are carefully motivated by the same historical examples that drove their original development of the field of probability, and then applies those concepts to popular contemporary games. The first two chapters of Probability, Decisions and Games: A Gentle Introduction using R feature an introductory discussion of probability and rational choice theory in finite and discrete spaces that builds upon the simple games discussed in the famous correspondence between Blaise Pascal and Pierre de Fermat. Subsequent chapters utilize popular casino games such as Roulette and Blackjack to expand on these concepts illustrate modern applications of these methodologies. Finally, the book concludes with discussions on game theory using a number of strategic games. This book:

· Features introductory coverage of probability, statistics, decision theory and game theory, and has been class-tested at University of California, Santa Cruz for the past six years

· Illustrates basic concepts in probability through interesting and fun examples using a number of popular casino games: roulette, lotto, craps, blackjack, and poker

· Introduces key ideas in game theory using classic games such as Rock-Paper-Scissors, Chess, and Tic-Tac-Toe.

· Features computer simulations using R throughout in order to illustrate complex concepts and help readers verify complex calculations

· Contains exercises and approaches games and gambling at a level that is accessible for readers with minimal experience

· Adopts a unique approach by motivating complex concepts using first simple games and then moving on to more complex, well-known 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.

ABEL RODRÍGUEZ, 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.

BRUNO MENDES, PhD, is Lecturer in the Department of Applied Mathematics and Statistics at the University of California, Santa Cruz, CA, USA.INTRODUCES THE FUNDAMENTALS OF PROBABILITY, STATISTICS, DECISION THEORY, AND GAME
THEORY, AND FEATURES INTERESTING EXAMPLES OF GAMES OF CHANCE AND STRATEGY TO
MOTIVATE AND ILLUSTRATE ABSTRACT MATHEMATICAL CONCEPTS
Covering both random and strategic games, Probability, Decisions and Games features a variety
of gaming and gambling examples to build a better understanding of basic concepts of probability,
statistics, decision theory, and game theory. The authors present fundamental concepts such as
random variables, rational choice theory, mathematical expectation and variance, fair games,
combinatorial calculus, conditional probability, Bayes Theorem, Bernoulli trials, zero-sum games and
Nash equilibria, as well as their application in games such as Roulette, Craps, Lotto, Blackjack, Poker,
Rock-Paper-Scissors, the Game of Chicken and Tic-Tac-Toe. Computer simulations, implemented
using the popular R computing environment, are used to provide intuition on key concepts and verify
complex calculations.
The book starts by introducing simple concepts that are carefully motivated by the same historical
examples that drove their original development of the field of probability, and then applies those concepts
to popular contemporary games. The first two chapters of Probability, Decisions and Games: A Gentle
Introduction using R feature an introductory discussion of probability and rational choice theory in finite
and discrete spaces that builds upon the simple games discussed in the famous correspondence between
Blaise Pascal and Pierre de Fermat. Subsequent chapters utilize popular casino games such as Roulette
and Blackjack to expand on these concepts illustrate modern applications of these methodologies. Finally,
the book concludes with discussions on game theory using a number of strategic games. This book:
• Features introductory coverage of probability, statistics, decision theory and game theory, and has
been class-tested at University of California, Santa Cruz for the past six years
• Illustrates basic concepts in probability through interesting and fun examples using a number of
popular casino games: roulette, lotto, craps, blackjack, and poker
• Introduces key ideas in game theory using classic games such as Rock-Paper-Scissors, Chess,
and Tic-Tac-Toe.
• Features computer simulations using R throughout in order to illustrate complex concepts and
• Contains exercises and approaches games and gambling at a level that is accessible for readers
with minimal experience
• Adopts a unique approach by motivating complex concepts using first simple games and then
moving on to more complex, well-known 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.
ABEL RODRÍGUEZ, 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.

Preface xi

Acknowledgments xv

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

4.1.2 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

9.4 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

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