ProblemSet21.Eachtimeashopperpurchasesatubeoftoothpaste,hechooseeitherbrandAorbrandB.Supposethatforeachpurchaseafterthefirst,theprobabilityis13thathewillchoosethesamebrandthathechoseonhisprecedingpurchaseandtheprobabilityis23thathewillswitchbrands.IfheisequallylikelytochooseeitherbrandAorbrandBonhisfirstpurchase,whatistheprobabilitythatbothhisfirstandsecondpurchaseswillbebrandAandbothhisthirdandfourthpurchaseswillbebrandB?2.IntheWorldSeriesofbaseball,twoteamsAandBplayasequenceofgamesagainsteachother,andthefirstteamthatwinsatotaloffourgamesbecomesthewinneroftheWorldseries.IftheprobabilitythatteamAwillwinanyparticulargameagainstteamBis1/3,whatistheprobabilitythatteamAwillwintheWorldseries?3.SupposethatA1,A2,...,Akareasequenceofkindependentevents.LetB1,B2,...,Bkbeanothersequenceofkeventssuchthatforeachvalueofj(j=1,2,...,k),eitherBj=AjorBj=ACj.ProvethatB1,B2,...,Bkarealsoindependentevents.Hint:UseaninductionargumentbasedonthenumberofeventsBjforwhichBj=ACj.4.Supposethatwhenamachineisadjustedproperly,50percentoftheitemsproducedbyitareofhighqualityandtheother50percentareofmediumquality.Suppose,however,thatthemachineisimproperlyadjustedduring10percentofthetimeandthat,undertheseconditions25percentoftheitemsproducedbyitareofhighqualityand75percentareofmediumquality.11)supposethatfiveitemsproducedbythemachineatacertaintimeareselectedatrandomandinspected.Iffouroftheseitemsareofhighqualityandoneitemisofmediumquality.Whatistheprobabilitythatthemachinewasadjustedproperlyatthattime?2)Supposethatoneadditionalitem,whichwasproducedbythemachineatthesametimeastheotherfiveitems,isselectedandfoundtobeofmediumquality.Whatisthenewposteriorprobabilitythatthemachinewasadjustedproperly?2
