Digital Dice: Computational Solutions to Practical Probability Problems

289 Pages·2013·10.688 MB·English

by  Paul J. Nahin

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Computational Solutions to
Practical Probability Problems
i
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ii
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Computational Solutions to
Practical Probability Problems
Paul J. Nahin
With a new preface by the author
PrincetonUniversityPress
PrincetonandOxford
iii
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Copyright(cid:1)c 2008byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:
PrincetonUniversityPress,
6OxfordStreet,
Woodstock,OxfordshireOX201TW
press.princeton.edu
CoverdesignbyKathleenLynch/BlackKatDesign.
IllustrationbyPepMontserrat/MarlenaAgency.
AllRightsReserved
Secondprinting,andfirstpaperbackprinting,withanewprefacebytheauthor,
2013
LibraryofCongressControlNumber2012954924
ClothISBN978-0-691-12698-2
PaperbackISBN978-0-691-15821-1
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinITCNewBaskerville
Printedonacid-freepaper.∞
PrintedintheUnitedStatesofAmerica
1098765432
iv
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Tothememoriesof
Victor Benedict Hassing (1916–1980)
who taught me mathematics at the “old’’ Brea-Olinda
Union High School, Brea, California (1954–58), and
who,whenheoccasionallywalkedtoschoolwithmein
themorning,wouldenthusiasticallytellmeofhislatest
mathematicalreading
and
Bess Louise (Lyman) Satterfield (1910–2001)
who, after a 1930s and ’40s “Golden Age’’ career in
Minnesota and Kentucky radio as a writer, editor, and
on-the-airperformer,taughtmehowtowriteaproper
sentence in my senior English class at Brea-Olinda
High.
v
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vi
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Comments on Probability and Monte Carlo
An asteroid or comet impact is the only natural disaster
that can wipe out human society....A large impact is an
improbable event that is absolutely guaranteed to occur.
Over the span of geological time, very large impacts have
happened countless times, and will occur countless more
times in ages to come. Yet in any given year, or in one
person’slifetime,thechanceofalargeimpactisvanishingly
small. The same, it should be noted, was also true when
the dinosaurs ruled Earth. Then, on one ordinary day,
probability arrived in the form of a comet, and their world
ended.
—Curt Pebbles, Asteroids: A History (Smithsonian Institution Press
2000), illustrating how even extremely-low-probability events
becomevirtuallycertaineventsifonejustwaitslongenough
MonteCarloistheunsophisticatedmathematician’sfriend.
Itdoesn’ttakeanymathematicaltrainingtounderstandand
use it.
—MITProfessorBillyE.Goetz,writingwithperhapsjustabittoo
muchenthusiasminManagementTechnology(January1960)
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viii CommentsonProbabilityandMonteCarlo
The truth is that random events can make or break us. It is
more comforting to believe in the power of hard work and
meritthantothinkprobabilityreignsnotonlyinthecasino
but in daily life.
—Richard Friedman, M.D., writing in the New York Times (April
26,2005)onmathematicsinmedicine
Analytical results may be hard to come by for these cases;
however, they canall be handledeasily by simulation.
—Alan Levine, writing in Mathematics Magazine in 1986 about a
classofprobabilityproblems(seeProblem13inthisbook)
The only relevant difference between the elementary arith-
metic on which the Court relies and the elementary proba-
bility theory [of the case in hand] is that calculation in the
latter can’t bedone on one’s fingers.
—Supreme Court Justice John Harlan, in a 1971 opinion
Whitcombv.Chevis
January23,2013 Time: 01:54pm prelims.tex
Contents
PrefacetothePaperbackEdition xiii
Introduction 1
TheProblems 35
1.TheClumsyDishwasherProblem 37
2.WillLilandBillMeetattheMaltShop? 38
3.AParallelParkingQuestion 40
4.ACuriousCoin-FlippingGame 42
5.TheGamow-SternElevatorPuzzle 45
6.Steve’sElevatorProblem 48
7.ThePipeSmoker’sDiscovery 51
8.AToiletPaperDilemma 53
9.TheForgetfulBurglarProblem 59
10.TheUmbrellaQuandary 61
11.TheCaseoftheMissingSenators 63
12.HowManyRunnersinaMarathon? 65