Games, Gambling, and Probability: An Introduction to Mathematics

Description

Many experiments have shown the human brain generally has very serious problems dealing with probability and chance. A greater understanding of probability can help develop the intuition necessary to approach risk with the ability to make more informed (and better) decisions. The first four chapters offer the standard content for an introductory probability course, albeit presented in a much different way and order. The chapters afterward include some discussion of different games, different "ideas" that relate to the law of large numbers, and many more mathematical topics not typically seen in such a book. The use of games is meant to make the book (and course) feel like fun! Since many of the early games discussed are casino games, the study of those games, along with an understanding of the material in later chapters, should remind you that gambling is a bad idea; you should think of placing bets in a casino as paying for entertainment. Winning can, obviously, be a fun reward, but should not ever be expected. Changes for the Second Edition: New chapter on Game Theory New chapter on Sports Mathematics The chapter on Blackjack, which was Chapter 4 in the first edition, appears later in the book. Reorganization has been done to improve the flow of topics and learning. New sections on Arkham Horror, Uno, and Scrabble have been added. Even more exercises were added! The goal for this textbook is to complement the inquiry-based learning movement. In my mind, concepts and ideas will stick with the reader more when they are motivated in an interesting way. Here, we use questions about various games (not just casino games) to motivate the mathematics, and I would say that the writing emphasizes a "just-in-time" mathematics approach. Topics are presented mathematically as questions about the games themselves are posed. Table of Contents Preface 1. Mathematics and Probability 2. Roulette and Craps: Expected Value 3. Counting: Poker Hands 4. More Dice: Counting and Combinations, and Statistics 5. Game Theory: Poker Bluffing and Other Games 6. Probability/Stochastic Matrices: Board Game Movement 7. Sports Mathematics: Probability Meets Athletics 8. Blackjack: Previous Methods Revisited 9. A Mix of Other Games 10. Betting Systems: Can You Beat the System? 11. Potpourri: Assorted Adventures in Probability Appendices Tables Answers and Selected Solutions Bibliography Biography Dr. David G. Taylor is a professor of mathematics and an associate dean for academic affairs at Roanoke College in southwest Virginia. He attended Lebanon Valley College for his B.S. in computer science and mathematics and went to the University of Virginia for his Ph.D. While his graduate school focus was on studying infinite dimensional Lie algebras, he started studying the mathematics of various games in order to have a more undergraduate-friendly research agenda. Work done with two Roanoke College students, Heather Cook and Jonathan Marino, appears in this book! Currently he owns over 100 different board games and enjoys using probability in his decision-making while playing most of those games. In his spare time, he enjoys reading, cooking, coding, playing his board games, and spending time with his six-year-old dog Lilly. Cover Half Title Series Page Title Page Copyright Page Dedication Contents List of Figures List of Tables Preface 1. Mathematics and Probability 1.1. Introduction 1.2. About Mathematics 1.3. Probability Language Probability Adding Probabilities Multiplying Probabilities Shortcuts The Monty Hall Problem 1.4. Candy (Yum)! 1.5. Exercises 2. Roulette and Craps: Expected Value 2.1. Roulette Odds Expected Winnings (Expected Value) 2.2. Summations 2.3. Craps Free Odds Other Wagers 2.4. Exercises 3. Counting: Poker Hands 3.1. Cards and Counting Poker Hands Combinations (Binomial Coe cients) 3.2. Seven Card Pokers 3.3. Texas Hold'Em 3.4. Exercises 4. More Dice: Counting and Combinations, and Statistics 4.1. Liar's Dice Binomial Distribution Spread Using Tables 4.2. Arkham Horror 4.3. Yahtzee Permutations Multinomial Distribution Mini-Yahtzee 4.4. Exercises 5. Game Theory: Poker Bluffing and Other Games 5.1. Bluffing Optimal Bluffing 5.2. Game Theory Basics Dominance and Saddle Points Mixed Strategies Larger Games: 2 x m and n x 2 Larger Games: 3 x 3 and Beyond 5.3. Non-Zero Sum Games Movement Diagrams Nash Equilibrium Explored Prudential Strategies Pareto Optimality Prisoner's Dilemma Revisited 5.4. Three-Player Game Theory 5.5. Exercises 6. Probability/Stochastic Matrices: Board Game Movement 6.1. Board Game Movement Probability Matrices Steady-States Yahtzee, Revisited Cyclic Boards 6.2. Pay Day (The Board Game) 6.3. Monopoly The Properties' Real Values The "Long Jail" Strategy Fixing the Model 6.4. Spread, Revisited 6.5. Exercises 7. Sports Mathematics: Probability Meets Athletics 7.1. Sports Betting Moneyline and Spread Betting Horse Racing 7.2. Game Theory and Sports 7.3. Probability Matrices and Sports 7.4. Winning a Tennis Tournament Game Win Probability Set Win Probability 7.5. Repeated Play: Best of Seven 7.6. Exercises 8. Blackjack: Previous Methods Revisited 8.1. Blackjack Gameplay Card Counting Decision Making, Probability Trees, and Basic Strategy 8.2. Blackjack Variants Rule Variations for Normal Blackjack Blackjack Switch 8.3. Exercises 9. A Mix of Other Games 9.1. The Lottery Pick 3 Mega Millions 9.2. Bingo Counting Cards Bingo Probabilities 9.3. Uno 9.4. Baccarat 9.5. Farkle 9.6. Scrabble 9.7. Backgammon Probabilities for Hitting a Piece The Doubling Cube 9.8. Memory 9.9. Zombie Dice 9.10. Exercises 10. Betting Systems: Can You Beat the System? 10.1. Betting Systems Martingale System (Loss) Streaks Anti-Martingale System The Kelly Criterion 10.2. Gambler's Ruin Minimizing the Probability of Ruin 10.3. Exercises 11. Potpourri: Assorted Adventures in Probability 11.1. True Randomness? 11.2. Three Dice "Craps" 11.3. Counting "Fibonacci" Coins "Circularly" 11.4. Compositions and Probabilities 11.5. Sicherman Dice 11.6. Traveling Salesmen 11.7. Random Walks and Generating Functions Random Walks Generating Functions 11.8. More Probability! Appendices A. Probabilities with Infinity B. St. Petersburg Paradox C. Binomial Distribution versus Normal Distribution D. Matrix Multiplication Review Tables Answers and Selected Solutions Bibliography Image Credits Index