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UnderstandingProbability
Chanceeventsarecommonplaceinourdailylives.Everydaywefacesituationswhere
theresultisuncertain,and,perhapswithoutrealizingit,weguessaboutthelikelihood
ofoneoutcomeoranother.Fortunately,masteringtheconceptsofprobabilitycancast
newlightonsituationswhererandomnessandchanceappeartorule.
InthisfullyrevisedsecondeditionofUnderstandingProbability,thereadercan
learnabouttheworldofprobabilityinanappealingway.Theauthordemystifiesthe
lawoflargenumbers,bettingsystems,randomwalks,thebootstrap,rareevents,the
centrallimittheorem,theBayesianapproach,andmore.
Thissecondeditionhaswidercoverage,moreexplanationsandexamplesand
exercises,andanewchapterintroducingMarkovchains,makingitagreatchoicefora
firstprobabilitycourse.Butitseasy-goingstylemakesitjustasvaluableifyouwantto
learnaboutthesubjectonyourown,andhighschoolalgebraisreallyallthe
mathematicalbackgroundyouneed.
Henk Tijms
isProfessorofOperationsResearchattheVrijeUniversityin
Amsterdam.Theauthorofseveraltextbooks,includingAFirstCourseinStochastic
Models,heisintensivelyactiveinthepopularizationofappliedmathematicsand
probabilityinDutchhighschools.Hehasalsowrittennumerouspapersonapplied
probabilityandstochasticoptimizationforinternationaljournals,includingApplied
ProbabilityandProbabilityintheEngineeringandInformationalSciences.
Understanding Probability
Chance Rules in Everyday Life
SecondEdition
HENK TIJMS
VrijeUniversity
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521701723
© H. Tijms 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-34626-2 eBook (Adobe Reader)
ISBN-10 0-511-34626-3 eBook (Adobe Reader)
ISBN-13 978-0-521-70172-3 paperback
ISBN-10 0-521-70172-4 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
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guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface pageix
Introduction 1
PARTONE: PROBABILITYINACTION 9
1 Probabilityquestions 11
2 Thelawoflargenumbersandsimulation 17
2.1 Thelawoflargenumbersforprobabilities 18
2.2 Basicprobabilityconcepts 27
2.3 Expectedvalueandthelawoflargenumbers 32
2.4 Thedrunkard’swalk 37
2.5 TheSt.Petersburgparadox 39
2.6 Rouletteandthelawoflargenumbers 41
2.7 TheKellybettingsystem 44
2.8 Random-numbergenerator 50
2.9 Simulatingfromprobabilitydistributions 55
2.10 Problems 64
3 Probabilitiesineverydaylife 73
3.1 Thebirthdayproblem 74
3.2 Thecouponcollector’sproblem 79
3.3 Craps 82
3.4 Gamblingsystemsforroulette 86
3.5 The1970draftlottery 89
3.6 Bootstrapmethod 93
3.7 Problems 95
v
vi Contents
4 Rareeventsandlotteries 103
4.1 Thebinomialdistribution 104
4.2 ThePoissondistribution 108
4.3 Thehypergeometricdistribution 125
4.4 Problems 134
5 Probabilityandstatistics 141
5.1 Thenormalcurve 143
5.2 Theconceptofstandarddeviation 151
5.3 Thesquare-rootlaw 159
5.4 Thecentrallimittheorem 160
5.5 Graphicalillustrationofthecentrallimit
theorem 164
5.6 Statisticalapplications 166
5.7 Confidenceintervalsforsimulations 170
5.8 Thecentrallimittheoremandrandomwalks 177
5.9 FalsifieddataandBenford’slaw 191
5.10 Thenormaldistributionstrikesagain 196
5.11 Statisticsandprobabilitytheory 197
5.12 Problems 200
6 ChancetreesandBayes’rule 206
6.1 TheMontyHalldilemma 207
6.2 Thetestparadox 212
6.3 Problems 217
PARTTWO: ESSENTIALSOFPROBABILITY 221
7 Foundationsofprobabilitytheory 223
7.1 Probabilisticfoundations 223
7.2 Compoundchanceexperiments 231
7.3 Somebasicrules 235
8 ConditionalprobabilityandBayes 243
8.1 Conditionalprobability 243
8.2 Bayes’ruleinoddsform 251
8.3 Bayesianstatistics 256
9 Basicrulesfordiscreterandomvariables 263
9.1 Randomvariables 263
Contents vii
9.2 Expectedvalue 264
9.3 Expectedvalueofsumsofrandomvariables 268
9.4 Substitutionruleandvariance 270
9.5 Independenceofrandomvariables 275
9.6 Specialdiscretedistributions 279
10 Continuousrandomvariables 284
10.1 Conceptofprobabilitydensity 285
10.2 Importantprobabilitydensities 296
10.3 Transformationofrandomvariables 308
10.4 Failureratefunction 310
11 Jointlydistributedrandomvariables 313
11.1 Jointprobabilitydensities 313
11.2 Marginalprobabilitydensities 319
11.3 Transformationofrandomvariables 323
11.4 Covarianceandcorrelationcoefficient 327
12 Multivariatenormaldistribution 331
12.1 Bivariatenormaldistribution 331
12.2 Multivariatenormaldistribution 339
12.3 Multidimensionalcentrallimittheorem 342
12.4 Thechi-squaretest 348
13 Conditionaldistributions 352
13.1 Conditionalprobabilitydensities 352
13.2 Lawofconditionalprobabilities 356
13.3 Lawofconditionalexpectations 361
14 Generatingfunctions 367
14.1 Generatingfunctions 367
14.2 Moment-generatingfunctions 374
15 Markovchains 385
15.1 Markovmodel 386
15.2 TransientanalysisofMarkovchains 394
15.3 AbsorbingMarkovchains 398
15.4 Long-runanalysisofMarkovchains 404
viii Contents
Appendix Countingmethodsandex 415
Recommendedreading 421
Answerstoodd-numberedproblems 422
Bibliography 437
Index 439
