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

375 Pages·2009·2.22 MB·English

by  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
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2010 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20110725
International Standard Book Number-13: 978-1-4398-0165-9 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copy-
right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222
Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-
vides licenses and registration for a variety of users. For organizations that have been granted a pho-
tocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
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