Probability, Decisions and Games

A Gentle Introduction using R

Abel Rodríguez, Bruno Mendes

• Publisher: Wiley
• Year: 2018
• ISBN:
  9781119302629
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

Chapter 1
An Introduction to Probability

The study of probability started in the seventeenth century when Antoine Gambaud (who called himself the “Chevalier” de Méré) reached out to the French mathematician Blaise Pascal for an explanation of his gambling loses. De Méré would commonly bet that he could get at least one ace when rolling 4 six-sided dice, and he regularly made money on this bet. When that game started to get old, he started betting on getting at least one double-one in 24 rolls of two dice. Suddenly, he was losing money!

De Méré was dumbfounded. He reasoned that two aces in two rolls are 1/6 as likely as one ace in one roll. To compensate for this lower probability, the two dice should be rolled six times. Finally, to achieve the probability of one ace in four rolls, the number of the rolls should be increased fourfold (to 24). Therefore, you would expect a couple of aces to turn up in 24 double rolls with the same frequency as an ace in four single rolls. As you will see in a minute, although the very first statement is correct, the rest of his argument is not!

1.1 What is Probability?

Let's start by establishing some common language. For our purposes, an experiment is any action whose outcome cannot necessarily be predicted with certainty; simple examples include the roll of a die and the card drawn from a well-shuffled deck. The outcome space of an experiment is the set of all possible outcomes associated with it; in the case of a die, it is the set

, while for the card drawn from a deck, the outcome space has 52 elements corresponding to all combinations of 13 numbers (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K) with four suits (hearts, diamonds, clubs, and spades):

A probability is a number between 0 and 1 that we attach to each element of the outcome space. Informally, that number simply describes the chance of that event happening. A probability of 1 means that the event will happen for sure, a probability of 0 means that we are talking about an impossible event, and numbers in between represent various degrees of certainty about the occurrence of the event. In the future, we will denote events using capital letters; for example,

while the probability associated with these events is denoted by

and

. By definition, the probability of at least one event in the outcome space happening is 1, and therefore the sum of the probabilities associated with each of the outcomes also has to be equal to 1. On the other hand, the probability of an event not happening is simply the complem...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Table of Contents
• Preface
• Acknowledgments
• About the Companion Website
• Chapter 1: An Introduction to Probability
• Chapter 2: Expectations and Fair Values
• Chapter 3: Roulette
• Chapter 4: Lotto and Combinatorial Numbers
• Chapter 5: The Monty Hall Paradox and Conditional Probabilities
• Chapter 6: Craps
• Chapter 7: Roulette Revisited
• Chapter 8: Blackjack
• Chapter 9: Poker
• Chapter 10: Strategic Zero-Sum Games with Perfect Information
• Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
• Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
• Chapter 13: Tic-Tac-Toe and Other Sequential Games of Perfect Information
• Appendix A: A Brief Introduction to R
• Index
• End User License Agreement