一道英文概率题Considerthefollowingversionofthegameofcraps：Theplayerrollstwodice.Ifthesumonthefirstrollis7or11,theplayerwinsthegameimmediately.Ifthesumonthefirstrollis2,3or12,theplayerlosesthegameimmed
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更新时间：2024-04-20 11:47:50
问题描述：
一道英文概率题
Considerthefollowingversionofthegameofcraps：Theplayerrollstwodice.Ifthesumonthefirstrollis7or11,theplayerwinsthegameimmediately.Ifthesumonthefirstrollis2,3or12,theplayerlosesthegameimmediately.Howeverifthesumonthefirstrollis45689or10,thenthetwodicearerolledagainandagainuntilthesumiseither7or11ortheoriginalvalue.Iftheoriginalvalueisobtainedasecondtimebeforeeither7or11isobtained,thentheplayerwins.Ifeither7or11isobtainedbeforetheoriginalvalueisobtainedasecondtime,thentheplayerloses.Determinetheprobabilitythattheplayerwillwinthisgame.
李云钢回答：
23456789101112
1/362/363/364/365/366/365/364/363/362/361/36
7or118/36=2/9winonfirstroll
23or124/36=1/9loseon1stroll
else24/36=2/3continuefromfirstroll
pofobtainingoriginalvaluefromsecondroll=1/36
pofcontuinuningthegamefromsecondrollandafter=(2/3-1/36)=23/36
(2/9)+(2/3)(1/36)+(2/3)*(23/36)*(1/36)+(2/3)*(23/36)^2(1/36)+.
=2/9+(2/3)(1/36)(1+23/36+(23/36)^2+(23/36)^3.)
=2/9+(2/3)(1/36)/(1-23/36)
=2/9+(2/3)(1/36)(36/13)
=2/9+2/39
=26/117+6/117
=32/117
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