Basic Gambling Mathematics: The Numbers Behind the Neon

278 Pages·2013·2.4 MB·English

by  Mark Bollman

Mathematics
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S BASIC GAMBLING
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“Basic Gambling Mathematics: The Numbers Behind the Neon is a must- C
have book for anyone interested in gambling mathematics. It not only covers all
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of the most popular casino games (blackjack, craps, roulette, keno), but is also MATHEMATICS
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a treasure trove of fascinating variants and novel games from casinos all over
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the world. Professors, students, and general gambling enthusiasts will all find
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something to capture their attention.” The Numbers
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—Mike Ferrara, Associate Professor and Graduate Program Director, Department
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of Mathematical and Statistical Sciences, University of Colorado Denver N
G Behind the Neon
This book explains the mathematics involved in analyzing games of chance,

including casino games, horse racing, and lotteries. The book helps you M
understand the mathematical reasons why some gambling games are better for A
the player than others. It is also suitable as a textbook for an introductory course T
on probability. H
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Along with discussing the mathematics of well-known casino games, the author
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examines game variations that have been proposed or used in actual casinos.
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Numerous examples illustrate the mathematical ideas in a range of casino games Mark Bollman
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while end-of-chapter exercises go beyond routine calculations to give you hands-
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on experience with casino-related computations.
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The book begins with a brief historical introduction and mathematical preliminaries
before developing the essential results and applications of elementary probability,
including the important idea of mathematical expectation. The author then
addresses probability questions arising from a variety of games, including roulette,
craps, baccarat, blackjack, Caribbean stud poker, Royal Roulette, and sic bo. The
final chapter explores the mathematics behind “get rich quick” schemes, such as
the martingale and the Iron Cross, and shows how simple mathematics uncovers
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the flaws in these systems.
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BASIC GAMBLING
MATHEMATICS
The Numbers
Behind the Neon
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BASIC GAMBLING
MATHEMATICS
The Numbers
Behind the Neon
Mark Bollman
Albion College
Albion, Michigan, USA
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CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2014 by Taylor & Francis Group, LLC
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Version Date: 20140418
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For Laura,
who took a gamble with me in Las Vegas that’s still paying off.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
Preface ix
Acknowledgments xi
Chapter 1 (cid:4) Introduction 1
1.1 HISTORICAL BACKGROUND 1
1.2 MATHEMATICAL BACKGROUND 3
1.3 WHAT DOES IT MEAN TO BE RANDOM? 7
1.4 EXERCISES 9
Chapter 2 (cid:4) Fundamental Ideas 11
2.1 DEFINITIONS 11
2.2 AXIOMS OF PROBABILITY 12
2.3 ELEMENTARY COUNTING ARGUMENTS 18
2.4 ADVANCED COUNTING ARGUMENTS 25
2.4.1 Fundamental Counting Principle 26
2.4.2 Where Order Matters: Permutations 29
2.4.3 Where Order Doesn’t Matter: Combinations 32
2.5 ODDS 45
2.6 EXERCISES 48
Chapter 3 (cid:4) Compound Events 51
3.1 ADDITION RULES 51
3.2 MULTIPLICATION RULES AND CONDITIONAL PROB-
ABILITY 52
3.3 EXERCISES 72
Chapter 4 (cid:4) Probability Distributions and Expectation 75
4.1 RANDOM VARIABLES 75
4.2 EXPECTED VALUE 79
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viii Basic Gambling Mathematics: The Numbers Behind the Neon
4.3 BINOMIAL DISTRIBUTION 121
4.4 EXERCISES 134
Chapter 5 (cid:4) Modified Casino Games 139
5.1 ROULETTE 139
5.2 DICE GAMES 145
5.3 CARD GAMES 155
5.4 CASINO PROMOTIONS 171
5.5 EXERCISES 176
Chapter 6 (cid:4) Blackjack: The Mathematical Exception 181
6.1 RULES OF BLACKJACK 182
6.2 MATHEMATICS OF BLACKJACK 189
6.3 BASIC STRATEGY 195
6.4 CARD COUNTING 203
6.5 EXERCISES 217
Chapter 7 (cid:4) Betting Strategies: Why They Don’t Work 221
7.1 ROULETTE STRATEGIES 221
7.2 CRAPS STRATEGIES 227
7.3 SLOT MACHINE STRATEGIES 229
7.4 BLACKJACK STRATEGIES 231
7.5 AND ONE THAT DOES: LOTTERY STRATEGIES 233
7.6 HOW TO DOUBLE YOUR MONEY 236
7.7 EXERCISES 240
Appendix A: House Advantages 245
Appendix B: Mathematical Induction 247
Appendix C: Internet Resources 251
Answers to Odd-Numbered Exercises 253
Bibliography 257
Index 265
Preface
Thisbookgrewoutofseveralyearsofteachingaboutgamblinginavariety
of contexts at Albion College beginning in 2002. For several years, I taught a
first-year seminar called “Chance,” which I came to describe as “probability
and statistics for the educated citizen” as distinguished from a formula-heavy
approach to elementary statistics. I also focused more on probability than
statistics in Chance. Part of probability is gambling, of course, and so over
theyears,thecourseevolvedtoincludemorecasinoexamplesinclass,whether
by simulation or actual in-class game play. The course included a field trip to
the Soaring Eagle Casino in Mount Pleasant, Michigan, late in the semester
afterallofthestudentshadturned18.Thisprovidedthestudentswithafine
opportunity to combine theory with practice and see for themselves how the
laws of probability worked, in a way that no classroom activity could mimic.
Later on, I expanded the gambling material into a course called Great Is-
sues in Humanities: Perspectives on Gambling, in Albion’s honors program.
The course combined mathematics from Chance (for mathematics, in the
words of one of my colleagues, is the first of the humanities) with other read-
ings from literature, philosophy, and history to provide a well-rounded view
of a subject that is not becoming less important in America.
Throughout my years teaching about gambling, I struggled to find a good
probability textbook that covered the topics germane to my course without
a lot of material that was not related to gambling. Basic Gambling Mathe-
matics is my effort to distill the mathematics involved in gambling, and only
that mathematics, into one place. While the final product started out as that
textbook, over the course of an intensive summer spent writing (with a goal
of 1000 words a day), it evolved to include more general information on the
mathematics that I have found so fascinating during my years of teaching.
The text necessarily contains a large number of examples, illustrating the
mathematical ideas in a range of casino games. The end of an example is
indicated with a (cid:4) symbol.
The exercises provided here are included for students, of course, and for
those casual readers who would like to try their hand at some casino-related
computations. Most of them either present other examples of the ideas in the
chapter or ask for fairly straightforward verification of computations men-
tioned in the main text. Answers to all of the odd-numbered exercises are
provided,andcompletesolutionstoallexercisesmaybefoundintheseparate
solutions manual.
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