Probability

PROBABILITYPage1of3PROBABILITYProbabilityisameasureofhowlikelyaneventistohappen.Itisusuallymeasuredasafractionbetweenzeroand1butmaysometimesberepresentedasaPercentagebetween0%and100%ortoagambleasbettingoddstoagivenratio.LetEbeasingleeventandP(E)theprobabilityofthateventoccurring.IfP(E)=0,theeventisIMPOSSIBLEIfP(E)=1theeventisABSOLUTELYCERTAIN.LAWSOFPROBABILITY1.ForanysingleeventE,0P(E)12.IfEisoneofanumberofNequallylikelyevents,n(E)isthenumberofwaysinwhichEcantakeplacethenP(E)=NEn)(3.IfEisaneventandEistheoppositeofthateventhappening(theeventNOTE)ThenP(E)+P(E)=1,(sinceeitherEorNotEcanhappenwithabsolutecertainty).4.ForanytwoeventsAandB,P(AorB)iswrittenasP(BA)P(AandB)iswrittenasP(BA)P(BA)=P(A)+P(B)–P(BA)5.ForanytwoeventsAandBP(BA)=P(A)+P(B)–P(BA)6.MUTUALLYEXCLUSIVEEVENTS.TwoeventsareMutuallyExclusive(ME)iftheycannotbothhappenatonce(simultaneously)AnytwoeventsareMutuallyExclusiveifP(BA)=07.IfeventsAandBaremutuallyexclusive(ME)PROBABILITYPage2of3P(BA)=P(A)+P(B)ifAandBareMutuallyExclusive(ME)8.EXHAUSTIVEEVENTS.TwoeventsAandBareExhaustiveifP(AorB)=P(BA)=19.CONDITIONALPROBABILTY.TheprobabilityofeventAhappeninggiventhateventBhashappenedisdenotedby:P(A/B)whereP(A/B)=)()(BPBAP,whereP(B)0/10.INDEPENDENTEVENTS.EventsAandBareindependentiftheoutcomeofoneeventhasnoeffectontheoutcomeoftheotherevent.Intermsofprobabilitylaws,AandBareINDEPENDENTifP(A/B)=P(A)i.edoesnotdependuponBoccurring.11.IfAandBareindependenteventsthenP(BA)=P(A)P(B).Proof:BydefinitionP(A/B)=)()(BPBAP,henceP(BA)=P(A/B)P(B)ButAandBareindependentandsoP(A/B)=P(A)henceP(BA)=P(A/B)P(B)=P(A)P(B)12.ForanyeventsAandBP(B/A)=)()()/(APBPBAPProof:WeknowthatforanyeventsAandB,P(A/B)=)()(BPBAP,HenceP(BA)=P(A/B)P(B)(1)ButitisalsotruebysymmetrythatP(B/A)=)()(APBAP(2)PROBABILITYPage3of3SubstitutingforP(BA)in(2)fromformula(1)yieldstheresult:P(B/A)=)()()/(APBPBAP