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Chapter9Curiositiesanddilemmas
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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.
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