Mathematics in Games, Sports, and Gambling : - The Games People Play

375 Pages·2009·2.225 MB·English

by  Gould, Ronald J

MATHEMATICS in
GAMES, SPORTS,
and GAMBLING
– The Games People Play

MATHEMATICS in
GAMES, SPORTS,
and GAMBLING
– The Games People Play
Ronald J. Gould
Emory University
Atlanta, Georgia, U.S.A.
CRC Press
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Version Date: 20110725
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Dedication
To Lyn,
For putting up with all of this writing and more.
v

Contents
List of Figures xi
List of Tables xv
Preface xvii
Author Biography xix
1 Basic Probability 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Of Dice and Men . . . . . . . . . . . . . . . . . . . . . 2
1.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 The Laws Which Govern Us . . . . . . . . . . . . . . . 12
1.4.1 Dependent Events . . . . . . . . . . . . . . . . . 15
1.5 Poker Hands Versus Batting Orders . . . . . . . . . . . 23
1.6 Let’s Play for Money! . . . . . . . . . . . . . . . . . . . 28
1.7 Is That Fair? . . . . . . . . . . . . . . . . . . . . . . . . 32
1.8 The Odds Are against Us . . . . . . . . . . . . . . . . . 39
1.9 Things Vary . . . . . . . . . . . . . . . . . . . . . . . . 41
1.10 Conditional Expectation . . . . . . . . . . . . . . . . . 45
2 The Game’s Afoot 49
2.1 Applications to Games . . . . . . . . . . . . . . . . . . 49
2.2 Counting and Probability in Poker Hands . . . . . . . . 49
2.3 Roulette . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Craps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.1 Street Craps . . . . . . . . . . . . . . . . . . . . 60
2.4.2 Casino Craps . . . . . . . . . . . . . . . . . . . . 61
2.4.3 Other Bets . . . . . . . . . . . . . . . . . . . . . 62
2.5 Let’s Make A Deal — The Monty Hall Problem . . . . 66
2.6 Carnival Games . . . . . . . . . . . . . . . . . . . . . . 69
2.7 Other Casino Games . . . . . . . . . . . . . . . . . . . 72
2.7.1 Caribbean Stud Poker . . . . . . . . . . . . . . . 72
vii
viii
2.7.2 Keno . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.7.3 Blackjack . . . . . . . . . . . . . . . . . . . . . . 76
2.8 Backgammon . . . . . . . . . . . . . . . . . . . . . . . . 79
2.8.1 Hitting Blots . . . . . . . . . . . . . . . . . . . . 81
2.8.2 Off the Bar . . . . . . . . . . . . . . . . . . . . . 81
2.8.3 Bearing Off . . . . . . . . . . . . . . . . . . . . . 83
2.8.4 Doubling . . . . . . . . . . . . . . . . . . . . . . 86
3 Repeated Play 93
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2 Binomial Coefficients . . . . . . . . . . . . . . . . . . . 94
3.3 The Binomial Distribution . . . . . . . . . . . . . . . . 103
3.4 The Poisson Distribution . . . . . . . . . . . . . . . . . 113
3.5 Streaks — Are They Real? . . . . . . . . . . . . . . . . 120
3.6 Betting Strategies . . . . . . . . . . . . . . . . . . . . . 124
3.7 The Gambler’s Ruin . . . . . . . . . . . . . . . . . . . . 129
4 Card Tricks and More 135
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 The Five-Card Trick . . . . . . . . . . . . . . . . . . . . 135
4.2.1 Adding a Joker to the Deck . . . . . . . . . . . . 139
4.2.2 More Variations of the Trick . . . . . . . . . . . 140
4.3 The Two-Deck Matching Game . . . . . . . . . . . . . 147
4.4 More Tricks . . . . . . . . . . . . . . . . . . . . . . . . 151
4.4.1 Friends Find Each Other . . . . . . . . . . . . . 152
4.4.2 The Small Arithmetic Trick . . . . . . . . . . . . 153
4.4.3 The Nine-Card Trick . . . . . . . . . . . . . . . . 154
4.5 The Paintball Wars . . . . . . . . . . . . . . . . . . . . 155
5 Dealing with Data 159
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2 Batting Averages and Simpson’s Paradox . . . . . . . . 160
5.3 NFL Passer Ratings . . . . . . . . . . . . . . . . . . . . 166
5.4 Viewing Data — Simple Graphs . . . . . . . . . . . . . 169
5.4.1 Time Plots and Regression Lines . . . . . . . . . 175
5.4.2 When To Find the Regression Line . . . . . . . . 181
5.5 Confidence in Our Estimates . . . . . . . . . . . . . . . 186
5.6 Measuring Differences in Performance . . . . . . . . . . 190
5.6.1 Coefficient of Variation . . . . . . . . . . . . . . 194
5.6.2 Relative Performance . . . . . . . . . . . . . . . 196
ix
6 Testing and Relationships 201
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 Suzuki Versus Pujols . . . . . . . . . . . . . . . . . . . 201
6.3 I’ll Decide If I Believe That . . . . . . . . . . . . . . . . 204
6.3.1 Errors . . . . . . . . . . . . . . . . . . . . . . . . 207
6.3.2 Summary of Hypothesis Testing . . . . . . . . . 207
6.3.3 One-Sided Tests . . . . . . . . . . . . . . . . . . 208
6.3.4 Small Sample Sizes . . . . . . . . . . . . . . . . . 209
6.4 Are the Old Adages True? . . . . . . . . . . . . . . . . 212
6.4.1 Home Field Advantage . . . . . . . . . . . . . . . 212
6.4.2 Lefty Versus Righty . . . . . . . . . . . . . . . . 215
6.5 How Good Are Certain Measurements? . . . . . . . . . 219
6.5.1 Batting Average and Runs Scored . . . . . . . . 222
6.6 Arguing over Outstanding Performances . . . . . . . . 224
6.7 A Last Look at Comparisons . . . . . . . . . . . . . . . 227
6.7.1 Small Sample Comparisons . . . . . . . . . . . . 227
7 Games and Puzzles 233
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 233
7.2 Number Arrays . . . . . . . . . . . . . . . . . . . . . . 233
7.2.1 Magic Squares . . . . . . . . . . . . . . . . . . . 233
7.2.2 Variations on Magic Squares . . . . . . . . . . . 239
7.2.3 Sudoku . . . . . . . . . . . . . . . . . . . . . . . 242
7.3 The Tower of Hanoi . . . . . . . . . . . . . . . . . . . . 245
7.3.1 Finding Solutions . . . . . . . . . . . . . . . . . . 248
7.3.2 Bicolored Tower of Hanoi . . . . . . . . . . . . . 251
7.3.3 The Derangement Tower of Hanoi . . . . . . . . 253
7.4 Instant Insanity . . . . . . . . . . . . . . . . . . . . . . 255
7.5 Lights Out . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.6 Peg Games . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.6.1 English Board . . . . . . . . . . . . . . . . . . . 267
7.6.2 Triangular Peg Solitaire . . . . . . . . . . . . . . 272
8 Combinatorial Games 277
8.1 Introduction to Combinatorial Games . . . . . . . . . . 277
8.2 Subtraction Games . . . . . . . . . . . . . . . . . . . . 278
8.3 Nim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
8.3.1 Poker Nim . . . . . . . . . . . . . . . . . . . . . 286
8.3.2 Moore’s Nim . . . . . . . . . . . . . . . . . . . . 287
8.3.3 Other Games . . . . . . . . . . . . . . . . . . . . 289