The Mathematics of Games and Gambling (Anneli Lax New Mathematical Library)

190 Pages·2006·1.921 MB·English

by  Packel, Edward W

The Mathematics of Games
and Gambling
Cover design by Elizabeth Holmes Clark
©2006 by
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 2006926250
Print ISBN 978-0-88385-646-8
Electronic ISBN 978-0-88385-943-8
Printed in the United States of America
The Mathematics of Games
and Gambling
by
Edward W. Packel
Lake Forest College
®
PublishedandDistributedby
TheMathematicalAssociationofAmerica
AnneliLaxNewMathematicalLibrary
CouncilonPublications
RogerNelsen,Chair
EdwardW.Packel,Editor
CarolS.Schumacher
DanielJ.Teague
DonnaL.Beers
MichaelE.Boardman
HaroldP.Boas
RichardK.Guy
ANNELILAXNEWMATHEMATICALLIBRARY
1. Numbers:RationalandIrrationalbyIvanNiven
2. WhatisCalculusAbout?byW.W.Sawyer
3. AnIntroductiontoInequalitiesbyE.F.BeckenbachandR.Bellman
4. GeometricInequalitiesbyN.D.Kazarinoff
5. The Contest Problem Book I Annual High School Mathematics Examinations
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6. TheLoreofLargeNumbersbyP.J.Davis
7. UsesofInfinitybyLeoZippin
8. GeometricTransformationsIbyI.M.Yaglom,translatedbyA.Shields
9. ContinuedFractionsbyCarlD.Olds
10. (cid:2) ReplacedbyNML-34
11. HungarianProblemBooksIandII,BasedontheEo¨tvo¨sCompetitions
12. 1894–1905and1906–1928,translatedbyE.Rapaport
13. EpisodesfromtheEarlyHistoryofMathematicsbyA.Aaboe
14. GroupsandTheirGraphsbyE.GrossmanandW.Magnus
15. TheMathematicsofChoicebyIvanNiven
16. FromPythagorastoEinsteinbyK.O.Friedrichs
17. The Contest Problem Book II AnnualHighSchoolMathematicsExaminations
1961–1965.CompiledandwithsolutionsbyCharlesT.Salkind
18. FirstConceptsofTopologybyW.G.ChinnandN.E.Steenrod
19. GeometryRevisitedbyH.S.M.CoxeterandS.L.Greitzer
20. InvitationtoNumberTheorybyOysteinOre
21. GeometricTransformationsIIbyI.M.Yaglom,translatedbyA.Shields
22. ElementaryCryptanalysis—AMathematicalApproachbyA.Sinkov
23. IngenuityinMathematicsbyRossHonsberger
24. GeometricTransformationsIIIbyI.M.Yaglom,translatedbyA.Shenitzer
25. TheContestProblemBookIIIAnnualHighSchoolMathematicsExaminations
1966–1972.CompiledandwithsolutionsbyC.T.SalkindandJ.M.Earl
26. MathematicalMethodsinSciencebyGeorgePo´lya
27. International Mathematical Olympiads—1959–1977. Compiled and with
solutionsbyS.L.Greitzer
28. TheMathematicsofGamesandGambling,SecondEditionbyEdwardW.
Packel
29. TheContestProblemBookIVAnnualHighSchoolMathematicsExaminations
1973–1982. Compiled and with solutions by R. A. Artino,A. M. Gaglione,
andN.Shell
30. TheRoleofMathematicsinSciencebyM.M.SchifferandL.Bowden
31. InternationalMathematicalOlympiads1978–1985andfortysupplementary
problems.CompiledandwithsolutionsbyMurrayS.Klamkin
32. RiddlesoftheSphinxbyMartinGardner
33. U.S.A.MathematicalOlympiads1972–1986.Compiledandwithsolutions
byMurrayS.Klamkin
34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J.
Wilson
35. ExploringMathematicswithYourComputerbyArthurEngel
36. GameTheoryandStrategybyPhilipD.Straffin,Jr.
37. Episodes in Nineteenth and Twenthieth Century Euclidean Geometry by
RossHonsberger
38. TheContestProblemBookVAmericanHighSchoolMathematicsExaminations
andAmericanInvitationalMathematicsExaminations1983–1988. Compiled and
augmentedbyGeorgeBerzsenyiandStephenB.Maurer
39. OverandOverAgainbyGengzheChangandThomasW.Sederberg
40. TheContestProblemBookVIAmericanHighSchoolMathematicsExaminations
1989–1994.CompiledandaugmentedbyLeoJ.Schneider
41. The Geometry of Numbers by C. D. Olds, Anneli Lax, and Giuliana P.
Davidoff
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translatedbyAndyLiu
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44. GeometricTransformationsIVbyI.M.Yaglom,translatedbyA.Shenitzer
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Contents
PrefacetotheFirstEdition ix
PrefacetotheSecondEdition xiii
1 ThePhenomenonofGambling 1
1.1 Aselectivehistory . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thegamblerinfactandfiction . . . . . . . . . . . . . . . 5
2 FiniteProbabilitiesandGreatExpectations 13
2.1 Theprobabilityconceptanditsorigins . . . . . . . . . . . 13
2.2 Dice,cards,andprobabilities . . . . . . . . . . . . . . . . 15
2.3 Roulette,probabilityandodds . . . . . . . . . . . . . . . 17
2.4 Compoundprobabilities:Therulesofthegame . . . . . . 20
2.5 Mathematicalexpectationanditsapplication. . . . . . . . 22
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 BackgammonandOtherDiceDiversions 29
3.1 Backgammonoversimplified . . . . . . . . . . . . . . . . 29
3.2 Rollingspotsandhittingblots . . . . . . . . . . . . . . . 32
3.3 Enteringandbearingoff . . . . . . . . . . . . . . . . . . 34
3.4 Thedoublingcube . . . . . . . . . . . . . . . . . . . . . 36
3.5 Craps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Chuck-a-Luck . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Permutations,Combinations,andApplications 51
4.1 Carefulcounting:Isorderimportant?. . . . . . . . . . . . 51
4.2 Factorialsandothernotation . . . . . . . . . . . . . . . . 53
vii
viii Contents
4.3 Probabilitiesinpoker . . . . . . . . . . . . . . . . . . . . 55
4.4 Bettinginpoker:Asimplemodel. . . . . . . . . . . . . . 59
4.5 Distributionsinbridge . . . . . . . . . . . . . . . . . . . 67
4.6 Kenotypegames . . . . . . . . . . . . . . . . . . . . . . 71
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 PlayitAgainSam:TheBinomialDistribution 79
5.1 Gamesandrepeatedtrials . . . . . . . . . . . . . . . . . . 79
5.2 Thebinomialdistribution . . . . . . . . . . . . . . . . . . 79
5.3 Beatingtheoddsandthe“law”ofaverages . . . . . . . . 83
5.4 Bettingsystems . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Abriefblackjackbreakthrough . . . . . . . . . . . . . . . 93
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 ElementaryGameTheory 99
6.1 Whatisgametheory? . . . . . . . . . . . . . . . . . . . . 99
6.2 Gamesinextensiveform . . . . . . . . . . . . . . . . . . 100
6.3 Two-persongamesinnormalform . . . . . . . . . . . . . 105
6.4 Zero-sumgames. . . . . . . . . . . . . . . . . . . . . . . 107
6.5 Nonzero-sumgames,Nashequilibriaandthe
prisoners’dilemma . . . . . . . . . . . . . . . . . . . . . 113
6.6 Simplen-persongames . . . . . . . . . . . . . . . . . . . 118
6.7 Powerindices . . . . . . . . . . . . . . . . . . . . . . . . 120
6.8 Gamescomputersplay . . . . . . . . . . . . . . . . . . . 123
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 OddsandEnds 135
7.1 Themathematicsofbluffingandthe
TexasHoldeminvasion . . . . . . . . . . . . . . . . . . . 135
7.2 Offtotheraces . . . . . . . . . . . . . . . . . . . . . . . 141
7.3 Lotteriesandyourexpectation . . . . . . . . . . . . . . . 147
7.4 Thegambler’sruin . . . . . . . . . . . . . . . . . . . . . 158
Answers/HintsforSelectedExercises 165
Bibliography 167
Index 171
AbouttheAuthor 175
Preface to the First Edition
The purpose of this book is to introduce and develop some of the impor-
tant and beautiful elementary mathematics needed for rational analysis of
variousgamblingand gameactivities.While theonlyformalmathematics
background assumed is high school algebra, some enthusiasm for and
facilitywithquantitativereasoningwillalsoservethereaderwell.Thebook
will,Ihope,beofinterestto:
1. Brighthighschoolstudentswithagoodmathematicsbackgroundandan
(oftenrelated)interestingamesofchance.
2. Studentsinelementaryprobabilitytheorycourseswhomightappreciate
an informal supplementary text focusing on applications to gambling
andgames.
3. Individualswithsomebackgroundinmathematicswhoareinterestedin
some common and uncommon elementary game-oriented applications
andtheiranalysis.
4. That subset of the numerate gambling and game-playing public who
wouldliketoexaminethemathematicsbehindgamestheymightenjoy
andwhowouldliketoseemathematicaljustificationforwhatconstitutes
“good”(rational)playinsuchgames.
One guiding principle of the book is that no mathematics is introduced
without specific examples and applications to motivate the theory. The
mathematicsdevelopedrangesfromthepredictableconceptsofprobability,
expectation, and binomial coefficients to some less well-known ideas of
elementarygametheory.Awidevarietyofstandardgamesareconsidered
ix