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Chapter 9 Curiosities and dilemmas（第1页）

Chapter9Curiositiesanddilemmas

Atthebeginningofthisbook，Isomeaspectsofprobabilityappear，atfirstsight，todefyonsense。Exampleshaveariseoryhasunfolded。Herearesomeotherceswhereintuitionbemisleading，but，withsuffitcare，theseapparenttradibeexplaiofprobabilityiswhollyfreefromrealparadoxes。Butalthoughideasofprobabilityhelpusmakesensibledeayalsofihinkingabouttheprobabilitiesofeventsmightleadtounfortabledilemmas。

Parrondo’sparadox

GrahamGreeakesAllisafibasedonafalsepremise:thatthereissomeclevermathematibisoewheeltogivetheplayeranadvaherthanthehouse。Orary:mathematicshasprovedthat，whenallindividualbetsfavourthehouse，nobinatiournmattersroundandfavourtheplayer。Sorry，folks。

JuanParrondohasshownthatyouhavetobeveryprehowyouformulateageneralclaimthat，whesfavouroisimpossibletobihattheothersidehasanadvantage。Idescribehereamodifiofhisidea，duetoDeanAstumian，whodescribedasimplegameplayedontheboardwithfiveslots，shownihisisname。Itwasstructedmerelytomakethispoint。）

11。TheboardforAstumian’sgame

Youneedsomewayarawilloe:perhapsabagwith99WhiteballsandoneBlackball，oraspiisequallylikelytoetorestosonehuobeginthegame，plaomarked‘Start’。Everymovewilltakethetoke，orht，andyouwiokenreabeforeithitsLose。

&wobasicsetsofrules，callthemAhAndy，fromStartyoualwaysmovetoLeft，andfrht，youalwaysmovetoWi，youusethespiogivea1%ovingtoLose，anda99%ovingbacktoStart。WithBert，thespinarttogivea99%ht，anda1%gtht，

youalwaysreturntoStart，whilefromLeft，itisthesameasinAndy–thespinnergivesa1%ovingtoLose，a99%cetoStart。

Analysisofthesegamesissimple。InAndy，thereisnoprovisioht;youshufflearouailrandomcetakesyoufromLefttoLose。I，youusuallyshufflebetweenStartandRight，withoalvisitstoLeft。Eventually，ohesesojour，randomcetakesyoutoLose。IheceWiniszero。

Forthenewgame，Chris，youalsoneedafair。Atea，tossthis:ifitshowsHeads，usetherulesofAndy，ifitshowsTails，usetherulesofBert。

ItturnsoutthatyChrisexceeds98%!Itiseasytoseewhyyfavourite:ifeveryougettoLeft，youarelylikelytoreturyofStart。FromStart，youplayBerthalfthetime，withits99%ceofgettingtht;andinRight，youplayAndyhalfthetime，iablywinning。

FollowiherA，you　mustlose:flipbetweerandom，andyouwiime!Framiheoremthatexcludesexampleslikethis，butsthatGreerestsonshakyground，requiresverypreguage!

2+2=4，or2+2=6?

SupposewecarryoutBernoullitrialswithafair，i。e。eachtoss，ily，isequallylikelytobeHeadsorTails。AtypiewillbeHHTHTTTHT。。。。ThemeaossesuntilHeadsappearsistwo;butwhatisthemeaossesu，orHH?

&iveanswerisfour，asweexpecttosforthefirstsymbol，thehrowsforthesedthemeahrowsuisindeedfour，butthisisnotthecaseforHH。Toseethatpattern，themeahrowsissix!

Thereasonforthediffere，togetHT，itiscorrecttuethatweexpecttotaketwettheH，theogettheTthatpletesthepattern。AndTwoplusTwoequalsFour。ButforHH，afterwehavethefirstH，thehrowwillbeThalfthetime，abeginagain–allthrowsuptothatpointwillhavebeehealgebraleadiaheAppendix。

&weenHandT，eachisequallylikelytoappearfirst;whataboutbetweenHHandHT?Again，eachisequallylikelytoarisebeforetheother，siwaitforthefirstHead，ahehrowdetermiheanswer。However，betweenHHaeristhreetimesaslikelytoappearfirst!Thereasonissimple:thesequencewillbeginwithHHohetime，butuhishappens，itisiHappearsfirst。（Thinkaboutit。）

Thegame　Peeisbasedontheaboveideas。YouinviteyouroppoapossibletripleslikeHHT，orTHT，etc。，thatmightothreesecutivethrowsofafair。Youselectadiffereralpersoherepeatedly，andthewihepersoripleisseenfirst。Despitetheappareyofallowingyouroppoohavefirstpick，thisgamefavoursyou–ifyouknowwhaty。Whatevershechooses，youselectatriplethatearbeforehersatleast23ofthetime!ThewinniheAppendix。

Givemeaclue…