Table of Contents

1. Cover
2. Title Page
3. Copyright
4. Dedication
5. Preface
   1. Why Gambling and Gaming?
   2. Using this Book
6. Acknowledgments
7. About the Companion Website
8. Chapter 1: An Introduction to Probability
   1. 1.1 What is Probability?
   2. 1.2 Odds and Probabilities
   3. 1.3 Equiprobable Outcome Spaces and De Méré's Problem
   4. 1.4 Probabilities for Compound Events
   5. 1.5 Exercises
9. Chapter 2: Expectations and Fair Values
   1. 2.1 Random Variables
   2. 2.2 Expected Values
   3. 2.3 Fair Value of a Bet
   4. 2.4 Comparing Wagers
   5. 2.5 Utility Functions and Rational Choice Theory
   6. 2.6 Limitations of Rational Choice Theory
   7. 2.7 Exercises
10. Chapter 3: Roulette
    1. 3.1 Rules and Bets
    2. 3.2 Combining Bets
    3. 3.3 Biased Wheels
    4. 3.4 Exercises
11. Chapter 4: Lotto and Combinatorial Numbers
    1. 4.1 Rules and Bets
    2. 4.2 Sharing Profits: De Méré's Second Problem
    3. 4.3 Exercises
12. Chapter 5: The Monty Hall Paradox and Conditional Probabilities
    1. 5.1 The Monty Hall Paradox
    2. 5.2 Conditional Probabilities
    3. 5.3 Independent Events
    4. 5.4 Bayes Theorem
    5. 5.5 Exercises
13. Chapter 6: Craps
    1. 6.1 Rules and Bets
    2. 6.2 Exercises
14. Chapter 7: Roulette Revisited
    1. 7.1 Gambling Systems
    2. 7.2 You are a Big Winner!
    3. 7.3 How Long will My Money Last?
    4. 7.4 Is This Wheel Biased?
    5. 7.5 Bernoulli Trials
    6. 7.6 Exercises
15. Chapter 8: Blackjack
    1. 8.1 Rules and Bets
    2. 8.2 Basic Strategy in Blackjack
    3. 8.3 A Gambling System that Works: Card Counting
    4. 8.4 Exercises
16. Chapter 9: Poker
    1. 9.1 Basic Rules
    2. 9.2 Variants of Poker
    3. 9.3 Additional Rules
    4. 9.4 Probabilities of Hands in Draw Poker
    5. 9.5 Probabilities of Hands in Texas Hold'em
    6. 9.6 Exercises
17. Chapter 10: Strategic Zero-Sum Games with Perfect Information
    1. 10.1 Games with Dominant Strategies
    2. 10.2 Solving Games with Dominant and Dominated Strategies
    3. 10.3 General Solutions for Two Person Zero-Sum Games
    4. 10.4 Exercises
18. Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
    1. 11.1 Finding Mixed-Strategy Equilibria
    2. 11.2 Mixed Strategy Equilibria in Sports
    3. 11.3 Bluffing as a Strategic Game with a Mixed-Strategy Equilibrium
    4. 11.4 Exercises
19. Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
    1. 12.1 The Prisoner's Dilemma
    2. 12.2 The Impact of Communication and Agreements
    3. 12.3 Which Equilibrium?
    4. 12.4 Asymmetric Games
    5. 12.5 Exercises
20. Chapter 13: Tic-Tac-Toe and Other Sequential Games of Perfect Information
    1. 13.1 The Centipede Game
    2. 13.2 Tic-Tac-Toe
    3. 13.3 The Game of Nim and the First- and Second-Mover Advantages
    4. 13.4 Can Sequential Games be Fun?
    5. 13.5 The Diplomacy Game
    6. 13.6 Exercises
21. Appendix A: A Brief Introduction to R
    1. A.1 Installing R
    2. A.2 Simple Arithmetic
    3. A.3 Variables
    4. A.4 Vectors
    5. A.5 Matrices
    6. A.6 Logical Objects and Operations
    7. A.7 Character Objects
    8. A.8 Plots
    9. A.9 Iterators
    10. A.10 Selection and Forking
    11. A.11 Other Things to Keep in Mind
22. Index
23. End User License Agreement

Guide

1. Cover
2. Table of Contents
3. Preface
4. Begin Reading

List of Illustrations

1. Chapter 1: An Introduction to Probability
   1. Figure 1.1 Cumulative empirical frequency of heads (black line) in 5000 simulated flips of a fair coin. The gray horizontal line corresponds to the true probability .
   2. Figure 1.2 Venn diagram for the (a) union and (b) intersection of two events.
   3. Figure 1.3 Venn diagram for the addition rule.
2. Chapter 2: Expectations and Fair Values
   1. Figure 2.1 Running profits from a wager that costs $1 to join and pays nothing if a coin comes up tails and $1.50 if the coin comes up tails (solid line). The gray horizontal line corresponds to the expected profit.
   2. Figure 2.2 Running profits from Wagers 1 (continuous line) and 2 (dashed line).
   3. Figure 2.3 Running profits from Wagers 3 (continuous line) and 4 (dashed line).
3. Chapter 3: Roulette
   1. Figure 3.1 The wheel in the French/European (left) and American (right) roulette and respective areas of the roulette table where bets are placed.
   2. Figure 3.2 Running profits from a color (solid line) and straight-up (dashed line) bet.
   3. Figure 3.3 Empirical frequency of each pocket in 5000 spins of a biased wheel.
   4. Figure 3.4 Cumulative empirical frequency for a single pocket in an unbiased wheel.
4. Chapter 5: The Monty Hall Paradox and Conditional Probabilities
   1. Figure 5.1 Each branch in this tree represents a different decision and the s represent the probability of each door being selected to contain the prize.
   2. Figure 5.2 The tree structure now represents an extra level, representing the contestant decisions and the probability for each decision to be the one chosen.
   3. Figure 5.3 Decision tree for the point when Monty decides which door to open assuming the prize is behind door 3.
   4. Figure 5.4 Partitioning the event space.
   5. Figure 5.5 Tree representation of the outcomes of the game of urns under the optimal strategy that calls yellow balls as coming from Urn 3 and blue and red balls as coming from urn 2.
5. Chapter 6: Craps
   1. Figure 6.1 The layout of a craps table.
   2. Figure 6.2 Tree representation for the possible results of the game of craps. Outcomes that lead to the pass line bet winning are marked with W, while those that lead to a lose are marked L.
   3. Figure 6.3 Tree representation for the possible results of the game of craps with the probabilities for each of the come-out roll.
   4. Figure 6.4 Tree representation for the possible results of the game of craps with the probabilities for all scenarios.
6. Chapter 7: Roulette Revisited
   1. Figure 7.1 The solid line represents the running profits from a martingale doubling system with $1 initial wagers for an even bet in roulette. The dashed horizontal line indicates the zero-profit level.
   2. Figure 7.2 Running profits from a Labouchère system with an initial list of $50 entries of $10 for an even bet in roulette. Note that the simulation stops when the cumulative profit is ; the number of spins necessary to reach this number will vary from simulation to simulation.
   3. Figure 7.3 Running profits over 10,000 spins from a D'Alembert system with an initial bet of $5, change in bets of $1, minimum bet of $1 and maximum bet of $20 to an even roulette bet.
7. Chapter 8: Blackjack
   1. Figure 8.1 A 52-card French-style deck.
8. Chapter 9: Poker
   1. Figure 9.1 Examples of poker hands.
9. Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
   1. Figure 11.1 Graphical representation of decisions in a simplified version of poker.
10. Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
    1. Figure 12.1 Expected utilities for Ileena (solid line) and Hans (dashed line) in the game of chicken as function of the probability that Hans will swerve with probability if we assume that Ileena swerves with probability .
    2. Figure 12.2 Expected utilities for Ileena (solid line) and Hans (dashed line) in the game of chicken as function of the probability that Ileena will swerve with probability if we assume that Hans swerves with probability .
    3. Figure 12.3 Expected utilities for Ileena (solid line) and Hans (dashed line) in the game of chicken as function of the probability that Hans will swerve with probability if we assume that Ileena always swerves.
11. Chapter 13: Tic-Tac-Toe and Other Sequential Games of Perfect Information
    1. Figure 13.1 Extensive-form representation of the centipede game.
    2. Figure 13.2 Reduced extensive-form representation of the centipede game after solving for Carissa's optimal decision during the third round of play.
    3. Figure 13.3 Reduced extensive-form representation of the centipede game after solving for Sahar's optimal decision during the second round of play and Carissa's optimal decision during the third round of play.
    4. Figure 13.4 A game of tic-tac-toe where the player represented by X plays first and the player represented by O wins the game. The boards should be read left to right and then top to bottom.
    5. Figure 13.5 A small subsection of the extensive-form representation of tic-tac-toe.
    6. Figure 13.6 Examples of boards in which the player using the X mark created a fork for themselves, a situation that should be avoided by their opponent. In the left figure, player 1 (who is using the X) claimed the top left corner in their first move, then player 2 claimed the top right corner, player 1 responded by claiming the bottom right corner, which forces player 2 to claim the center square (in order to block a win), and player 1 claims the bottom left corner too. At this point, player 1 has created a fork since they can win by placing a mark on either of the cells marked with an F. Similarly, in the right Figure player 1 claimed the top left corner, player 2 responded by claiming the bottom edge square, then player 1 took the center square, which forced player 2 to take the bottom right corner to block a win. After that, if player 1 places their mark on the bottom left corner they would have created a fork.
    7. Figure 13.7 Extended-form representation of the game of Nim with four initial pieces.
    8. Figure 13.8 Pruned tree for a game of Nim with four initial pieces after the optimal strategy at the third round has been elucidated.
    9. Figure 13.9 Pruned tree for a game of Nim with four initial pieces after the optimal strategy at the second round has been elucidated.
    10. Figure 13.10 The diplomacy game in extensive form.
    11. Figure 13.11 First branches pruned in the diplomacy game.
    12. Figure 13.12 Pruned tree associated with the diplomacy game.
12. Appendix A: A Brief Introduction to R
    1. Figure A.1 The R interactive command console in a Mac OS X computer. The symbol > is a prompt for users to provide instructions; these will be executed immediately after the user presses the RETURN key.
    2. Figure A.2 A representation of a vector x of length 6 as a series of containers, each one of them corresponding to a different number.
    3. Figure A.3 An example of a scatterplot in R]An example of a scatterplot in R.
    4. Figure A.4 An example of a line plot in R.
    5. Figure A.5 Adding multiple plots and reference lines to a single graph.
    6. Figure A.6 Example of a barplot in R.

List of Tables

1. Chapter 1: An Introduction to Probability
   1. Table 1.1 Two different ways to think about the outcome space associated with rolling two dice
2. Chapter 2: Expectations and Fair Values
   1. Table 2.1 Winnings for the different lotteries in Allais paradox
   2. Table 2.2 Winnings for 11% of the time for the different lotteries in Allais paradox
3. Chapter 3: Roulette
   1. Table 3.1 Inside bets for the American wheel
   2. Table 3.2 Outside bets for the American wheel
   3. Table 3.3 Outcomes of a combined bet of $2 on red and $1 on the second dozen
4. Chapter 4: Lotto and Combinatorial Numbers
   1. Table 4.1 List of possible groups of 3 out of 6 numbers, if the order of the numbers is not important
5. Chapter 5: The Monty Hall Paradox and Conditional Probabilities
   1. Table 5.1 Probabilities of winning if the contestant in the Monty problem switches doors
   2. Table 5.2 Studying the relationship between smoking and lung cancer
6. Chapter 6: Craps
   1. Table 6.1 Names associated with different combinations of dice in craps
   2. Table 6.2 All possible equiprobable outcomes associated with two dice being rolled
   3. Table 6.3 Sum of points associated with the roll of two dice
7. Chapter 7: Roulette Revisited
   1. Table 7.1 Accumulated losses from playing a martingale doubling system with an initial bet of $1 and an initial bankroll of $1000
   2. Table 7.2 Probability that you play exactly rounds before you lose your first dollar for between 1 and 6
8. Chapter 8: Blackjack
   1. Table 8.1 Probability of different hands assuming that the house stays on all 17s and that the game is being played with a large number of decks
   2. Table 8.2 Probability of different hands assuming that the house stays on all 17s, conditional on the face-up card
   3. Table 8.3 Optimal splitting strategy
   4. Table 8.4 Probability of different hands assuming that the house stays on all 17s and that the game is being played with a single deck where all Aces, 2s, 3s, 4s, 5s, and 6s have been removed
   5. Table 8.5 Probability of different hands assuming that the house stays on all 17s, conditional on the face-up card
9. Chapter 9: Poker
   1. Table 9.1 List of poker hands
   2. Table 9.2 List of opponent's poker hands that can beat our two-pair
10. Chapter 10: Strategic Zero-Sum Games with Perfect Information
    1. Table 10.1 Profits in the game between Pevier and Errian
    2. Table 10.2 Poll results for Matt versus Ling (first scenario)
    3. Table 10.3 Best responses for Matt (first scenario)
    4. Table 10.4 Best responses for Ling (first scenario)
    5. Table 10.5 Poll results for Matt versus Ling (second scenario)
    6. Table 10.6 Best responses for Matt (second scenario)
    7. Table 10.7 Best responses for Ling (second scenario)
    8. Table 10.8 Poll results for Matt versus Ling (third scenario)
    9. Table 10.9 Best responses for Ling (third scenario)
    10. Table 10.10 Best responses for Matt (third scenario)
    11. Table 10.11 Reduced Table for poll results for Matt versus Ling
    12. Table 10.12 A game without dominant or dominated strategies
    13. Table 10.13 Best responses for Player 1 in our game without dominant or dominated strategies
    14. Table 10.14 Best responses for Player 2 in our game without dominant or dominated strategies
    15. Table 10.15 Example of a game with multiple equilibria
11. Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
    1. Table 11.1 Player's profit in rock–paper–scissors
    2. Table 11.2 Best responses for Jiahao in the game of rock–paper–scissors
    3. Table 11.3 Utility associated with different actions that Jiahao can take if he assumes that Antonio selects rock with probability , paper with probability and scissors with probability
    4. Table 11.4 Utilities associated with different penalty kick decisions
    5. Table 11.5 Utility associated with different actions taken by the kicker if he assumes that goal keeper selects left with probability , center with probability , and right with probability
    6. Table 11.6 Expected profits in the simplified poker
    7. Table 11.7 Best responses for you in the simplified poker game
    8. Table 11.8 Best responses for Alya in the simplified poker game
    9. Table 11.9 Expected profits in the simplified poker game after eliminating dominated strategies
    10. Table 11.10 Expected profits associated with different actions you take if you assume that Alya will select with probability and with probability
12. Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
    1. Table 12.1 Payoffs for the prisoner's dilemma
    2. Table 12.2 Best responses for Prisoner 2 in the prisoner's dilemma game
    3. Table 12.3 Communication game in normal form
    4. Table 12.4 Best responses for Anastasiya in the communication game
    5. Table 12.5 Best responses for Anil in the communication game
    6. Table 12.6 Expected utility for Anil in the communication game
    7. Table 12.7 The game of chicken
    8. Table 12.8 Best responses for Ileena in the game of chicken
    9. Table 12.9 Expected utility for Ileena in the game of chicken
    10. Table 12.10 A fictional game of swords in Star Wars
    11. Table 12.11 Best responses for Ki-Adi in the sword game
    12. Table 12.12 Best responses for Asajj in the sword game
    13. Table 12.13 Expected utility for Ki-Adi in the asymmetric sword game
    14. Table 12.14 Expected utility for Asajj in the asymmetric sword game