概率与统计(英文)2 probability

2probability2.1SampleSpaceandEvents2.2Axioms,Interpretations,andPropertiesofProbability2.3CountingTechniques2.4ConditionalProbability2.5IndependenceIntroductionThetermprobabilityreferstothestudyofrandomnessanduncertainty.Inanysituationinwhichoneofanumberofpossibleoutcomesmayoccur,thetheoryofprobabilityprovidesmethodsforquantifyingthechances,orlikelihoods,associatedwiththevariousoutcomes.Thelanguageofprobabilityisconstantlyusedinaninformalmannerinbothwrittenandspokencontexts.Inthischapter,weintroducesomeelementaryprobabilityconcepts,indicatehowprobabilitiescanbeinterpreted,andshowhowtherulesofprobabilitycanbeappliedtocomputetheprobabilitiesofmanyinterestingevents.Themethodologyofprobabilitywillthenpermitustoexpressinpreciselanguagesuchinformalstatementsasthosegivenabove.2.1SampleSpacesandEventsAnexperimentisanyactionorprocessthatgeneratesobservations.Forexamples,tossingacoinonceorseveraltimes,selectingacardorcardsfromadeck,weighingaloafofbread,ascertainingthecommutingtimefromhometoworkonparticularmorning,obtainingbloodtypesfromagroupofindividuals,etc.RandomexperimentSamplespaceandsamplepointDefinition:Thesamplespaceofanexperiment,denotedbyS,orΩ,isthesetofallpossibleoutcomesofthatexperiment,andsamplepointofthesamplespace,denotedbys,isaoutcomeoftheexperiment.Example2.1:Thesimplestexperimenttowhichprobabilityappliesisonewithtwopossibleoutcomes.Onesuchexperimentconsistsofexaminingasinglefusetoseewhetheritisdefective.ThesamplespaceforthisexperimentcanbeabbreviatedasS={N,D},whereNrepresentsnotdefective,Drepresentsdefective,andthebracesareusedtoenclosetheelementsofaset.Anothersuchexperimentwouldinvolvetossingathumbtackandnotingwhetheritlandedpointuporpointdown,withsamplespaceS={U,D},andyetanotherwouldconsistofobservingthesexofthenextchildbornatthelocalhospital,withS={M,F}.Example2.2:Ifweexaminethreefusesinsequenceandnotetheresultofeachexamnation,thenanoutcomefortheentireexperimentisanysequenceofN’sandD’soflength3,soS={NNN,NND,NDN,NDD,DNN,DND,DDN,DDD}.Example2.2:drivingtowork,acommuterpassesthroughasequenceofthreeintersectionswithtrafficlights.Ateachlight,sheeitherstops,s,orcontinues,c,thesamplespaceisthesetofallpossibleoutcomes:S={ccc,ccs,css,csc,sss,ssc,scc,scs}Wherecscforexampledenotestheoutcomesthatthecommutercontinuesthroughthefirstlight,stopsatthesecondlight,andcontinuesthroughthethirdlight.Example2.3Thenumberofjobsinaprintqueueofamainframecomputermaybemodeledasrandom.HerethesamplespacecanbetakenasS={0,1,2,3,….}Thatis,allthenonnegativeintegers.Inpracticethereisprobablyanupperlimit,N,onhowlargetheprintqueuecanbe,soinsteadthesamplespacemightbedefinedasS={0,1,2,3,…,N}Example2.3Twogasstationsarelocatedatacertainintersection.Eachonehassixgaspumps.Considertheexperimentinwhichthenumberofpumpsinuseataparticulartimeofdayisdeterminedforeachofthestations.Anexperimentaloutcomespecifieshowmanypumpsareinuseatthefirststationandhowmanyareinuseatthesecondone.The49outcomesinSaredisplayedintheaccompanyingtable. 0 1 2 3 4 5 6 0 (0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,7) 1 (1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,0) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)Example2.4Ifanewtype-Dflashlightbatteryhasavoltagethatisoutsidecertainlimits,thatbatteryischaracterizedasafailure(F);ifthebatteryhasavoltagewithintheprescribedlimits,itisasuccess(S).Supposeanexperimentconsistsoftestingeachbatteryasitcomesoffanassemblylineuntilwefirstobserveasuccess.Althoughitmaynotbeverylikely,apossibleoutcomeofthisexperimentisthatthefirst10(or100or1000or…)aref’sandthenextoneisanS.thesamplespaceisΩ={S,FS,FFS,FFFS,…..}Example2.4Earthquakesexhibitedveryerraticbehavior,whichissometimesmodeledasrandom.Forexample,thelengthoftimebetweensuccessiveearthquakesinaparticularregionthataregreaterinmagnitudethanagiventhresholdmayberegardedasanexperiment.HereΩisthesetofallnonnegativenumbers:Ω=RandomeventDefinition:Anevent,areusuallydenotedbyitalicuppercaseletters,isanycollection(subset)ofoutcomescontainedinthesamplespaceS.Aneventissaidtobesimpleifitconsistsofexactlyoneoutcomeandcompoundifitconsistsofmorethanoneoutcome.Remark:Whenanexperimentisperformed,aparticulareventAissaidtooccuriftheresultingexperimentaloutcomeiscontainedinA.Ingeneral,exactlyonesimpleeventwilloccur,butmanycompoundeventswilloccursimultaneously.TheeventthatthecommuterstopsatthefirstlightisthesubsetofSdenotedbyA={sss,ssc,scc,scs}Example2.5:Consideranexperimentinwhicheachofthreeautomobilestakingaparticularfreewayexitturnsleft(L)orright(R)attheendoftheexitramp.TheeightpossibleoutcomesthatcomprisethesamplespaceareLLL,RLL,LRL,LLR,LRR,RLR,RRL,andRRR.Thus,thereareeightsimpleevents,amongwhichareE1={LLL}andE5={LRR}.SomecompoundeventsincludeA={RLL,LRL,LLR}=theeventthatexactlyoneofthethreecarsturnsrightB={LLL,RLL,LRL,LLR}=theeventthatatmostoneofthecarsturnsrightC={LLL,RRR}=theeventthatallthreecarsturninthesamedirectionSupposethatwhentheexperimentisperformed,theoutcomeisLLL.ThenthesimpleeventE1hasoccurredandsoalsohavetheeventsBandC(butnotA).Example2.6Whenthenumberofpumpsinuseateachoftwosix-pumpgasstationsisobserved,thereare49possibleoutcomesA={(0,0),(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}=theeventthatthenumberofpumpsinuseisthesameforbothstationsB={(0,4),(1,3),(2,2),(3,1),(4,0)}=theeventthatthetotalnumberofpumpsinuseisfourC={(0,0),(0,1),(1,0),(1,1)}=theeventthatatmostonepumpisinuseateachstationExample2.7Thesamplespaceforthebatteryexaminationexperimentcontainsaninfinitenumberofoutcomes,sothereareaninfinitenumberofsimpleevents.CompoundeventsincludeA={S,FS,FFS}=theeventthatatmostthreebatteriesareexaminedE={FS,FFFS,FFFFFS,….}=theeventthatanevennumberofbatteriesareexamined.ExerciseP57.2.Supposethatvehiclestakingaparticularfreewayexitcanturnright(R),turnleft(L),orgostraight(S).Considerobservingthedirectionforeachofthreesuccessivevehicles. ListalloutcomesintheeventAthatallthreevehiclesgointhesamedirection. ListalloutcomesintheeventBthatallthreevehiclestakedifferentdirections. ListalloutcomesintheeventCthatexactlytwoofthethreevehiclesturnright. ListalloutcomesintheeventDthatexactlytwovehiclesgointhesamedirection.Thealgebraofsettheorycarriesoverdirectlyintoprobabilitytheory.Theunionoftwoevents,AandB,istheeventCthateitherAoccursorBoccursorbothoccur:C=A∪B.Forexample,ifAistheeventthatcommuterstopsatthefirstlightandifBistheeventthatshestopsatthethirdlight,Union(∪)ThenCistheeventthathestopsatthefirstlightorstopsatthethirdlightandconsistsoftheoutcomesthatareinAorinBorinboth:SomerelationsfromsettheoryAneventisnothingbutaset,sothatrelationshipsandresultsfromelementarysettheorycanbeusedtostudyevents.Thefollowingconceptsfromsettheorywillbeusedtoconstructneweventsfromgivenevents.Definition:WhenAandBhavenooutcomesincommon,theyaresaidtobemutuallyexclusiveordisjointevents.Example2.10Asmallcityhasthreeautomobiledealerships:aGMdealersellingChevrolets,Pontiacs,andBuicks;aForddealersellingFordsandMercurys;andaChryslerdealersellingPlymouthsandChryslers.Ifanexperimentconsistsofobservingthebrandofthenextcarsold,thentheeventsA=(Chevrole,Pontiac,Buick}andB={Ford,Mercury}aremutuallyexclusivebecausethenextcarsoldcannotbebothaGMproductandaFordproduct.VenndiagramsExerciseP572,4ExerciseSupposeA,B,Carerandomevents.PleasedescribethefollowingeventsbyA,B,C,andtheirsetrelations.(1)OnlyApresent(2)onlyAnotpresent(3)Thethreeeventsareoccuratthesametime(4)Noneofthethreeispresent(5)Onlyoneofthethreeeventsispresent(6)Twoofthethreeeventsarepresent(7)Atleastoneispresent(8)Atleasttwoarepresent(9)Atmosttwoarepresent(10)AtmostoneispresentExerciseP57.2.Supposethatvehiclestakingaparticularfreewayexitcanturnright(R),turnleft(L),orgostraight(S).Considerobservingthedirectionforeachofthreesuccessivevehicles. ListalloutcomesintheeventAthatallthreevehiclesgointhesamedirection. ListalloutcomesintheeventBthatallthreevehiclestakedifferentdirections. ListalloutcomesintheeventCthatexactlytwoofthethreevehiclesturnright. ListalloutcomesintheeventDthatexactlytwovehiclesgointhesamedirection. ListoutcomesinExerciseP57.4.Eachofasampleoffourhomemortgagesisclassifiedasfixedrate(F)orvariablerate(V). Whatarethe16outcomesinS? Whichoutcomesareintheeventthatexactlythreeoftheselectedmortgagesarefixedrate? Whichoutcomesareintheeventthatallfourmortgagesareofthesametype? Whichoutcomesareintheeventthatatmostoneofthefourisavariables-ratemortgage? Whatistheunionoftheeventsinpart(c)and(d),andwhatistheintersectionofthesetwoevents? Whataretheunionandintersectionofthetwoeventsinparts(b)andpart(c)?2.2Axioms,interpretationsandpropertiesofprobabilityGivenanexperimentandasamplespaceS,theobjectiveofprobabilityistoassigntoeacheventAanumberP(A),calledtheprobabilityoftheeventA,whichwillgiveaprecisemeasureofthechancethatAwilloccur.Toensurethattheprobabilityassignmentswillbeconsistentwithourintuitivenotionsofprobability,allassignmentsshouldsatisfythefollowingaxioms:Thefirsttwoaxiomsareratherobvious.SinceSconsistsofallpossibleoutcomes.Thesecondaxiomsimplystatesthataprobabilityisnonnegative.ThethirdaxiomstatesthatifAandBaredisjointthatishavenooutcomesincommonthenandalsothatthispropertyextendstolimits.Forexampletheprobabilitythattheprintqueuecontainseitheroneorthreejobsisequaltotheprobabilitythatitcontainsoneplustheprobabilitythatitcontainsthree.Example2.12ConsidertheexperimentinExample2.4,inwhichbatteriescomingofanassemblylinearetestedonebyoneuntilonehavingavoltagewithinprescribedlimitsisfound.ThesimpleeventsareE1={S},E2={FS},E3=[FFS},E4={FFFS},….Supposetheprobabilityofanyparticularbatterybeingsatisfactoryis0.99.ThenThefollowingpropertiesofprobabilitymeasuresareconsequencesoftheaxioms.Property1thispropertyfollowssinceAandaredisjointwithandthus,bythefirstandthirdaxioms,inwordsthispropertysaysthattheprobabilitythataneventdoesnotoccurequalsoneminustheprobabilitythatitdoesoccur.Example2.13Considerasystemoffiveidenticalcomponentsconnectedinseries,asillustratedinthefollowingfigureDenoteacomponentthatfailsbyFandonethatdoesn’tfailbyS(forsuccess).Supposethattheprobabilityofonecomponentdoesn’tfailis0.9.LetAbetheeventthatthesystemfails.thispropertyfollowsfromproperty1sinceInwordsthissaysthattheprobabilitythatthereisnooutcomeatalliszero.thispropertyfollowssinceBcanbeexpressedastheunionoftwodisjointsets:Thenfromthethirdaxiom,AndthusThispropertystatesthatifBoccurswheneverAoccurs,then.Forexampleifwheneveritrains(A)itiscloudy(B),thentheprobabilitythatitrainsislessthanorequaltotheprobabilitythatitiscloudy.Property4Additionlawtoseethis,wedecomposeintothreedisjointsubsets,asshowninfigure2.1:FIGURE2.1Venndiagramillustratingtheadditionlaw.Example2.14Inacertainresidentialsuburb,60%ofallhousehokdssubscribetothemetropolitannewspaperpublishedinanearbycity,80%subscribetothelocalafternoonpaper,and50%ofallhouseholdssubscribetobothpapers.Ifahouseholdisselectedatrandom,whatistheprobabilitythatitsubscribesto(1)atleastoneofthetwonewspapersandwhat(2)exactlyoneofthetwonewspapers?Solution:WithA={subscribestothemetropolitanpaper}B={subscribestothelocalpaper}ThenP(A)=0.6,P(B)=0.8,andSo(1)P(subscribestoatleastoneofthetwonewspapers)=(2)P(exactlyone)=Theprobabilityofaunionofmorethantwoeventscanbecomputedanalogously.ForthreeeventsA,B,andC,theresultisExample:Twofightersshootaplane,theprobabilityofonefighterhittheplaneis0.4,theotheris0.6,andbothatthesametimehittheplaneprobabilityis0.3.Determinetheprobabilitythatatleastonefighterhittheplane?Example:Inarecentstudyonteenagedrugandalcoholuse,researchersfoundthatoneintenteenagersadmittedtousingmarijuanaatleastonceamonth,oneinfiveadmittedtodrinkingalcoholatleastonceaweek,andoneinfouradmittedtosmokingcigarettesonadailybasis.Theresearchersalsofoundthat10%ofrespondentsadmittedtobothsmokingcigarettesdailyanddrinkingalcoholatleastonceaweek;5%ofrespondentsadmittedtobothsmokingcigarettesdailyandusingmarijuanaatleastonceamonth;3%ofrespondentsadmittedtobothdrinkingalcoholatleastonceaweekandusingmarijuanaatleastonceamonth;and1%ofrespondentsadmittedtodoingallthree.Assumingthattheteenagerisaslikelyastheothersinthestudytosmoke,drink,ortakedrugs,whatistheprobabilitythattheteenagerengagesinatleastoneofthesethreepoorbehaviors?Whatistheprobabilitythattheteenagerdoesnotengageinanyofthesebehaviors?Solution:WesetM={Teenageradmitstousingmarijunanatleastonceamonth}A={Teenageradmitstoconsumingalcoholatleastonceaweek}C={Teenageradmitstosmokingcigareteesonadailybasis}So,P(M)=0.1,P(A)=0.2,P(C)=0.25ThenExample:Considerarecentstudyconductedbythepersonnelmanagerofamajorcomputersoftwarecompany.Itwasfoundthat30%oftheemployeeswholeftthefirmwithintwoyearsdidsoprimarilybecausetheyweredissatisfiedwiththeirsalary,20%leftbecausetheyweredissatisfiedwiththeirworkassignments,and12%oftheformeremployeesindicateddissatisfactionwithboththeirsalaryandtheirworkassignment.Whatistheprobabilitythatanemployeewholeaveswithintwoyearsdoessobecauseofdissatisfactionwithsalary,dissatisfactionwiththeworkassignment,orboth?Solution:S={leavesbecauseofdissatisfactionwithsalary}W={leavesbecauseofdissatisfactionwithworkassignment}ExampleifP(A)=P(B)=P(C)=1/4,P(AB)=P(AC)=0,P(BC)=3/16,Determinetheprobabilityofevent={noneofA,B,Coccurs}ExampleSupposeP(A)=0.4,P(B)=0.3,thenEquallylikelyoutcomesInmanyexperimentsconsistingofNoutcomes,itisreasonabletoassignequalprobabilitiestoallNsimpleevents.Withp=P(Ei)foreveryi,Thatis,ifthereareNpossibleoutcomes,thentheprobabilityassignedtoeachis1/N.NowconsideraneventA,withN(A)denotingthenumberofoutcomescontainedinA.ThenExample2.6:Whentwodicearerolledseparately,thereareN=36outcomes.Ifboththedicearefair,all36outcomesareequallylikely,soP(Ei)=1/36.Otherquestion:A={thesumoftwodiceis7},determineP(A)?Therearetwoanswer,whichiscorrect?Answer1:thesamplespaceΩ={2,3,4,5,6,7,8,9,10,11,12},soP(A)=1/11Answer2:thesamplespaceisΩ={(1,1),(1,2),(1,3),(1,4)…..}SoP(A)=6/36=1/6ExerciseP65121314172.3CountingtechniquesWhenthevariousoutcomesofanexperimentareequallylikely,thenthetaskofcomputingprobabilitiesreducestocounting.Inparticular,ifNisthenumberofoutcomesinasamplespaceandN(A)isthenumberofoutcomescontainedinaneventA,thenIfalistoftheoutcomesisavailableoreasytoconstructandNissmall,thenthenumeratoranddenominatorofEquation(2.1)canbeobtainedwithoutthebenefitofanygeneralcountingprinciples.Example2.17:ablackurncontains5redand6greenballsandawhiteurncontains3redand4greenballs.Youareallowedtochooseanurnandthenchooseaballatrandomfromtheurn.Ifyouchoosearedball,yougetaprize.Whichurnshouldyouchoosetodrawfrom?Ifyoudrawfromtheblackurn,theprobabilityofchoosingaredballis0.455.Ifyouchoosetodrawfromthewhiteurn,theprobabilityofchoosingaredballis0.429,soyoushouldchoosetodrawfromtheblackurn.Simpson’sparadox.Nowconsideranothergameinwhichasecondblackurnhas6redand3greenballsandasecondwhiteurnhas9redand5greenballs.Ifyoudrawfromtheblackurn,theprobabilityofredballis6/9.whereasifyouchoosetodrawfromthewhiteurn,theprobabilityis9/14.so,againyoushouldchoosetodrawfromtheblackurn.Inthefinalgame,thecontentsofthesecondblackurnareaddedtothefirstblackurnandthecontentsofthesecondwhiteurnareaddedtothefirstwhiteurn.againyoucanchoosewhichurnyoucandrawfrom.whichshouldyouchoose?Intuitionsayschoosetheblackurn,butlet’scalculatetheprobabilities.Theblackurnnowcontains11redand9greenballs,sotheprobabilityofdrawingaredballfromitis11/20=0.55.Thewhiteurnnowcontains12redand9greenballs,sotheprobabilityofdrawingaredballfromitis12/21=0.571.so,youshouldchoosethewhiteurn.ThiscounterintuitiveresultisanexampleofSimpson’sparadox.TheproductrulefororderedpairsIfoneexperimenthasmoutcomesandanotherexperimenthasnoutcomes,thentherearem×npossibleoutcomesforthetwoexperiments.Example2.18:Afamilyhasjustmovedtoanewcityandrequirestheservicesofbothanobstetricianandapediatrician.Therearetwoeasilyaccessiblemedicalclinics,eachhavingtwoobstetriciansandthreepediatricians.Thefamilywillobtainmaximumhealthinsurancebenefitsbyjoiningaclinicandselectingbothdoctorsfromthatclinic.Inhowmanywayscanthisbedone?DenotetheobstetriciansbyO1,O2,O3,andO4andthepediatriciansbyP1,…,P6.Thenwewishthenumberofpairs(Oi,Pj)forwhichOiandPjareassociatedwiththesameclinic.Becausetherearefourobstetricians,n1=4,andforeachtherearethreechoicesofpediatrician,son2=3.ApplyingtheproductrulegivesN=n1n2=12possiblechoices.TreediagramsAmoregeneralproductruleExample2.9:an8-bitbinarywordisasequenceof8digits,ofwhicheachmaybeeither0or1.Howmanydifferent8-bitwordsarethere?Therearetwochoicesforfirstbit,twoforthesecond,etc,andthusthereare2×2×2×2×2×2×2×2=28=256suchwords.PermutationsDefinition:Foranypositiveintegerm,m!isread“mfactorial”andisdefinedbysoCombinationsExample2.22Abridgehandconsistsofany13cardsselectedfroma52-carddeckwithoutregardtoorder.LetA={thehandconsistsentirelyofspadesandclubswithbothsuitsrepresented}B={thehandconsistsofexactlytwosuits}WhatistheprobabilityofeventAandeventB?Solution:Example2.23Arentalcarservicefacilityhas10foreigncarsand15domesticcarswaitingtobeservicedonaparticularSaturdaymorning.BecausetherearesofewmechanicsworkingonSaturday,only6canbeserviced.Ifthe6arechosenatrandom,whatistheprobabilitythat3ofthecarsselectedaredomesticandtheother3areforeign?Whatistheprobabilitythatatleast3domesticcarsareselected?Solution:ExampleConsidertheexperimentofselectingfivecardsfromadeckof52cards.LetA={thefivecardsnumberdifferenteachother}B={twocardshavethesamenumber,andotherthreecardsalsohavehomologynumber}C={twocardshavethesamenumber,theotherthreecardshavedifferentnumber}D={fivecardshavefourkindsofdesigns}ExerciseP7438:Aboxinacertainsupplyroomcontainsfour40-wlightbulbs,five60-wbulbs,andsix75-wbulbs.Supposethatthreebulbsarerandomlyselected. Whatistheprobabilitythatexactlytwooftheselectedbulbsarerated75-w?b.Whatistheprobabilitythatallthreeoftheselectedbulbshavethesamerating?c.Whatistheprobabilitythatonebulbofeachtypeisselected?d.Supposenowthatbulbsaretobeselectedonebyoneuntila75-wbulbisfound.Whatistheprobabilitythatitisnecessarytoexamineatleastsixbulbs?ExerciseP7539Fifteentelephoneshavejustbeenreceivedatanauthorizedservicecenter.Fiveofthesetelephonesarecellular,fivearecordless,andtheotherfivearecordedphones.Supposethatthesecomponentsarerandomlyallocatedthenumbers1,2,..,15toestablishtheorderinwhichtheywillbeserviced.a.Whatistheprobabilitythatallthecordlessphonesareamongthefirsttentobeserviced?b.Whatistheprobabilitythatafterservicingtenofthesephones,phonesofonlytwoofthethreetypesremaintobeserviced?c.Whatistheprobabilitythattwophonesofeachtypeareamongthefirstsixserviced?ExerciseP7542Threemarriedcoupleshavepurchasedtheaterticketsandareseatedinarowconsistingofjustsixseats.Iftheytaketheirseatsinacompletelyrandomfashion(randomorder),whatistheprobabilitythatJimandPaula(husbandandwife)sitinthetwoseatsonthefarleft?WhatistheprobabilitythatJimandPaulaendupsittingnexttooneanother?Whatistheprobabilitythatatleastoneofthewivesendsupsittingnexttoherhusband?Example:Therearemboysandngirls,theystandrandomlyonahorizontalline,LetA={girlsstandtogether}B={girlsstandtogether,andboysstandtogether,too}C={thereisatleastoneboybetweenanytwogirls}Solution:ExerciseP7433342.4ConditionalprobabilityInthissection,weexaminehowtheinformation“aneventBhasoccurred”affectstheprobabilityassignedtoA.forexample,Amightrefertoanindividualhavingaparticulardiseaseinthepresenceofcertainsymptoms.Ifabloodtestisperformedontheindividualandtheresultisnegative(B=negativebloodtest),thentheprobabilityofhavingthediseasewillchange(itshoulddecrease,butnotusuallytozero,sincebloodtestsarenotinfallible).WewillusethenotationP(A|B)torepresenttheconditionalprobabilityofAgiventhattheeventBhasoccurred.WhatisP(A|B)=? Condition B Line A 2 6 1 9Unawareofthisinformation,thesalesmanagerrandomlyselects1ofthese18componentsforademonstration.PriortothedemonstrationHowever,ifthechosencomponentturnsouttobedefective,thentheeventBhasoccurred,sothecomponentmusthavebeen1ofthe3intheBcolumnofthetable.Sincethese3componentsareequallylikelyamongthemselvesafterBhasoccurred,Thedefinitionofconditionalprobability