(1.33)-PowerPoint Ch 04商务统计课件

Chapter4

IntroductiontoProbabilityExperiments,CountingRules, andAssigningProbabilitiesEventsandTheirProbabilitySomeBasicRelationships ofProbabilityConditionalProbabilityBayes’TheoremUncertaintiesManagersoftenbasetheirdecisionsonananalysisofuncertaintiessuchasthefollowing:Whatarethechancesthatsaleswilldecreaseifweincreaseprices?Whatisthelikelihoodanewassemblymethodmethodwillincreaseproductivity?Whataretheoddsthatanewinvestmentwillbeprofitable?Probability

Probabilityisanumericalmeasureofthelikelihoodthataneventwilloccur.Probabilityvaluesarealwaysassignedonascalefrom0to1.Aprobabilitynearzeroindicatesaneventisquiteunlikelytooccur.Aprobabilitynearoneindicatesaneventisalmostcertaintooccur.ProbabilityasaNumericalMeasure

oftheLikelihoodofOccurrence01.5IncreasingLikelihoodofOccurrenceProbability:Theeventisveryunlikelytooccur.Theoccurrenceoftheeventisjustaslikelyasitisunlikely.Theeventisalmostcertaintooccur.StatisticalExperimentsInstatistics,thenotionofanexperimentdifferssomewhatfromthatofanexperimentinthephysicalsciences.Instatisticalexperiments,probabilitydeterminesoutcomes.Eventhoughtheexperimentisrepeatedinexactlythesameway,anentirelydifferentoutcomemayoccur.Forthisreason,statisticalexperimentsaresome-timescalledrandomexperiments.AnExperimentandItsSampleSpaceAnexperiment

isanyprocessthatgenerateswell-definedoutcomes.Thesamplespaceforanexperimentisthesetofallexperimentaloutcomes.Anexperimentaloutcomeisalsocalledasample

point.AnExperimentandItsSampleSpaceExperimentTossacoinInspectionapartConductasalescallRolladiePlayafootballgameExperimentOutcomesHead,tailDefective,non-defectivePurchase,nopurchase1,2,3,4,5,6Win,lose,tie Bradleyhasinvestedintwostocks,MarkleyOilandCollinsMining.Bradleyhasdeterminedthatthepossibleoutcomesoftheseinvestmentsthreemonthsfromnowareasfollows.InvestmentGainorLossin3Months(in$000)MarkleyOilCollinsMining1050-208-2Example:BradleyInvestmentsAnExperimentandItsSampleSpaceACountingRulefor

Multiple-StepExperimentsIfanexperimentconsistsofasequenceofkstepsinwhichtherearen1possibleresultsforthefirststep,

n2possibleresultsforthesecondstep,andsoon,thenthetotalnumberofexperimentaloutcomesisgivenby(n1)(n2)...(nk).Ahelpfulgraphicalrepresentationofamultiple-stepexperimentisatreediagram.

BradleyInvestmentscanbeviewedasatwo-stepexperiment.Itinvolvestwostocks,eachwithasetofexperimentaloutcomes.MarkleyOil: n1=4CollinsMining: n2=2TotalNumberofExperimentalOutcomes: n1n2=(4)(2)=8ACountingRulefor

Multiple-StepExperimentsExample:BradleyInvestmentsTreeDiagramGain5Gain8Gain8Gain10Gain8Gain8Lose20Lose2Lose2Lose2Lose2EvenMarkleyOil(Stage1)CollinsMining(Stage2)ExperimentalOutcomes(10,8) Gain$18,000(10,-2) Gain$8,000(5,8) Gain$13,000(5,-2) Gain$3,000(0,8) Gain$8,000(0,-2) Lose

$2,000(-20,8) Lose

$12,000(-20,-2) Lose

$22,000Example:BradleyInvestmentsAsecondusefulcountingruleenablesustocountthenumberofexperimentaloutcomeswhennobjectsaretobeselectedfromasetofNobjects.CountingRuleforCombinationsNumberofCombinationsofNObjectsTakennataTimewhere:N!=N(N

-1)(N

-2)...(2)(1)

n!=n(n

-1)(n

-2)...(2)(1) 0!=1NumberofPermutationsofNObjectsTakennataTimewhere:N!=N(N

-1)(N

-2)...(2)(1)

n!=n(n

-1)(n

-2)...(2)(1) 0!=1CountingRuleforPermutationsAthirdusefulcountingruleenablesustocountthenumberofexperimentaloutcomeswhennobjectsaretobeselectedfromasetofNobjects,wheretheorderofselectionisimportant.AssigningProbabilitiesBasicRequirementsforAssigningProbabilities1.Theprobabilityassignedtoeachexperimentaloutcomemustbebetween0and1,inclusively.0<

P(Ei)<1foralliwhere: EiistheithexperimentaloutcomeandP(Ei)isitsprobabilityAssigningProbabilitiesBasicRequirementsforAssigningProbabilities2.Thesumoftheprobabilitiesforallexperimentaloutcomesmustequal1.P(E1)+P(E2)+...+P(En)=1where: nisthenumberofexperimentaloutcomesAssigningProbabilitiesClassicalMethodRelativeFrequencyMethodSubjectiveMethodAssigningprobabilitiesbasedontheassumptionofequallylikelyoutcomesAssigningprobabilitiesbasedonexperimentationorhistoricaldataAssigningprobabilitiesbasedonjudgmentClassicalMethodIfanexperimenthasnpossibleoutcomes,theclassicalmethodwouldassignaprobabilityof1/ntoeachoutcome.Experiment:RollingadieSampleSpace:S={1,2,3,4,5,6}Probabilities:Eachsamplepointhasa 1/6chanceofoccurring

Example:RollingaDieRelativeFrequencyMethodNumberofPolishersRentedNumberofDays012344618102LucasToolRentalwouldliketoassignprobabilitiestothenumberofcarpolishersitrentseachday.Officerecordsshowthefollowingfrequenciesofdailyrentalsforthelast40days.

Example:LucasToolRental Eachprobabilityassignmentisgivenbydividingthefrequency(numberofdays)bythetotalfrequency(totalnumberofdays).RelativeFrequencyMethod4/40ProbabilityNumberofPolishersRentedNumberofDays01234461810240.10.15.45.25.051.00

Example:LucasToolRentalSubjectiveMethodWheneconomicconditionsandacompany’scircumstanceschangerapidlyitmightbeinappropriatetoassignprobabilitiesbasedsolelyonhistoricaldata.Wecanuseanydataavailableaswellasourexperienceandintuition,butultimatelyaprobabilityvalueshouldexpressourdegreeofbeliefthattheexperimentaloutcomewilloccur.Thebestprobabilityestimatesoftenareobtainedbycombiningtheestimatesfromtheclassicalorrelativefrequencyapproachwiththesubjectiveestimate.SubjectiveMethodAnanalystmadethefollowingprobabilityestimates.Exper.OutcomeNetGainorLossProbability(10,8)(10,-2)(5,8)(5,-2)(0,8)(0,-2)(-20,8)(-20,-2)$18,000Gain$8,000Gain$13,000Gain$3,000Gain$8,000Gain$2,000Loss$12,000Loss$22,000Loss.20.08.16.26.10.12.02.06Example:BradleyInvestmentsAnevent

isacollectionofsamplepoints.Theprobabilityofanyeventisequaltothesumoftheprobabilitiesofthesamplepointsintheevent.Ifwecanidentifyallthesamplepointsofanexperimentandassignaprobabilitytoeach,wecancomputetheprobabilityofanevent.EventsandTheirProbabilitiesEventsandTheirProbabilitiesEventM=MarkleyOilProfitableM={(10,8),(10,-2),(5,8),(5,-2)}P(M)=P(10,8)+P(10,-2)+P(5,8)+P(5,-2)=.20+.08+.16+.26=.70Example:BradleyInvestmentsEventsandTheirProbabilitiesEventC=CollinsMiningProfitableC={(10,8),(5,8),(0,8),(-20,8)}P(C)=P(10,8)+P(5,8)+P(0,8)+P(-20,8)=.20+.16+.10+.02=.48Example:BradleyInvestmentsSomeBasicRelationshipsofProbability Therearesomebasicprobabilityrelationshipsthatcanbeusedtocomputetheprobabilityofaneventwithoutknowledgeofallthesamplepointprobabilities.ComplementofanEventIntersectionofTwoEventsMutuallyExclusiveEventsUnionofTwoEventsThecomplementofAisdenotedbyAc.ThecomplementofeventAisdefinedtobetheeventconsistingofallsamplepointsthatarenotinA.ComplementofanEventEventAAcSampleSpaceSVennDiagramTheunionofeventsAandBisdenotedbyA

B

TheunionofeventsAandBistheeventcontainingallsamplepointsthatareinAorBorboth.UnionofTwoEventsSampleSpaceSEventAEventBUnionofTwoEventsEventM=MarkleyOilProfitableEventC=CollinsMiningProfitableM

C=MarkleyOilProfitable

orCollinsMiningProfitable(orboth)M

C={(10,8),(10,-2),(5,8),(5,-2),(0,8),(-20,8)}P(M

C)=

P(10,8)+P(10,-2)+P(5,8)+P(5,-2) +P(0,8)+P(-20,8)=.20+.08+.16+.26+.10+.02=.82Example:BradleyInvestmentsTheintersectionofeventsAandBisdenotedbyA

TheintersectionofeventsAandBisthesetofallsamplepointsthatareinbothAandB.SampleSpaceSEventAEventBIntersectionofTwoEventsIntersectionofAandBIntersectionofTwoEventsEventM=MarkleyOilProfitableEventC=CollinsMiningProfitableM

C=MarkleyOilProfitable

andCollinsMiningProfitableM

C={(10,8),(5,8)}P(M

C)=

P(10,8)+P(5,8)=.20+.16=.36Example:BradleyInvestmentsTheadditionlawprovidesawaytocomputetheprobabilityofeventA,orB,orbothAandBoccurring.AdditionLawThelawiswrittenas:P(A

B)=P(A)+P(B)-

P(A

B

EventM=MarkleyOilProfitableEventC=CollinsMiningProfitableM

C=MarkleyOilProfitable

orCollinsMiningProfitableWeknow:P(M)=.70,P(C)=.48,P(M

C)=.36Thus:P(M

C)=P(M)+P(C)-

P(M

C)=.70+.48-.36=.82AdditionLaw(Thisresultisthesameasthatobtainedearlierusingthedefinitionoftheprobabilityofanevent.)Example:BradleyInvestmentsMutuallyExclusiveEventsTwoeventsaresaidtobemutuallyexclusiveiftheeventshavenosamplepointsincommon.Twoeventsaremutuallyexclusiveif,whenoneeventoccurs,theothercannotoccur.SampleSpaceSEventAEventBMutuallyExclusiveEventsIfeventsAandBaremutuallyexclusive,P(A

B

=0.Theadditionlawformutuallyexclusiveeventsis:P(A

B)=P(A)+P(B)Thereisnoneedtoinclude“-

P(A

B

”Theprobabilityofaneventgiventhatanothereventhasoccurrediscalledaconditionalprobability.Aconditionalprobabilityiscomputedasfollows:TheconditionalprobabilityofAgivenBisdenotedbyP(A|B).ConditionalProbabilityEventM=MarkleyOilProfitableEventC=CollinsMiningProfitableWeknow:P(M

C)=.36,P(M)=.70

Thus:ConditionalProbability=CollinsMiningProfitable

givenMarkleyOilProfitableExample:BradleyInvestmentsMultiplicationLawThemultiplicationlawprovidesawaytocomputetheprobabilityoftheintersectionoftwoevents.Thelawiswrittenas:P(A

B)=P(B)P(A|B)EventM=MarkleyOilProfitableEventC=CollinsMiningProfitableWeknow:P(M)=.70,P(C|M)=.5143MultiplicationLawM

C=MarkleyOilProfitable

andCollinsMiningProfitableThus:P(M

C)=P(M)P(M|C)=(.70)(.5143)=.36(Thisresultisthesameasthatobtainedearlierusingthedefinitionoftheprobabilityofanevent.)Example:BradleyInvestmentsJointProbabilityTableCollinsMiningProfitable(C)NotProfitable(Cc)MarkleyOilProfitable(M)

NotProfitable(Mc)Total.48.52Total.70.301.00.36.34.12.18JointProbabilities(appearinthebodyofthetable)MarginalProbabilities(appearinthemarginsofthetable)IndependentEventsIftheprobabilityofeventAisnotchangedbytheexistenceofeventB,wewouldsaythateventsAandBareindependent.TwoeventsAandBareindependentif:P(A|B)=P(A)P(B|A)=P(B)orThemultiplicationlawalsocanbeusedasatesttoseeiftwoeventsareindependent.Thelawiswrittenas:P(A

B)=P(A)P(B)MultiplicationLaw

forIndependentEventsEventM=MarkleyOilProfitableEventC=CollinsMiningProfitableWeknow:P(M

C)=.36,P(M)=.70,P(C)=.48

But:P(M)P(C)=(.70)(.48)=.34,not.36AreeventsMandCindependent?Does

P(M

C)=P(M)P(C)?

Hence:MandCarenotindependent.Example:BradleyInvestmentsMultiplicationLaw

forIndependentEventsDonotconfusethenotionofmutuallyexclusiveeventswiththatofindependentevents.Twoeventswithnonzeroprobabilitiescannotbebothmutuallyexclusiveandindependent.Ifonemutuallyexclusiveeventisknowntooccur,theothercannotoccur.;thus,theprobabilityoftheothereventoccurringisreducedtozero(andtheyarethereforedependent).MutualExclusivenessandIndependenceTwoeventsthatarenotmutuallyexclusive,mightormightnotbeindependent.Bayes’TheoremNewInformationApplicationofBayes’TheoremPosteriorProbabilitiesPriorProbabilitiesOftenwebeginprobabilityanalysiswithinitialor

priorprobabilities.Then,fromasample,specialreport,oraproducttestweobtainsomeadditionalinformation.Giventhisinformation,wecalculaterevisedor

posteriorprobabilities.

Bayes’theoremprovidesthemeansforrevisingthepriorprobabilities.AproposedshoppingcenterwillprovidestrongcompetitionfordowntownbusinesseslikeL.S.Clothiers.Iftheshoppingcenterisbuilt,theownerofL.S.Clothiersfeelsitwouldbebesttorelocatetotheshoppingcenter.Bayes’Theorem

Example:L.S.ClothiersTheshoppingcentercannotbebuiltunlessazoningchangeisapprovedbythetowncouncil.Theplanningboardmustfirstmakearecommendation,fororagainstthezoningchange,tothecouncil.Let:PriorProbabilitiesA1=towncouncilapprovesthezoningchangeA2=towncouncildisapprovesthechangeP(A1)=.7,P(A2)=.3Usingsubjectivejudgment:

Example:L.S.ClothiersTheplanningboardhasrecommendedagainstthezoningchange.LetBdenotetheeventofanegativerecommendationbytheplanningboard.NewInformation

Example:L.S.ClothiersGiventhatBhasoccurred,shouldL.S.Clothiersrevisetheprobabilitiesthatthetowncouncilwillapproveordisapprovethezoningchange?Pasthistorywiththeplanningboardandthetowncouncilindicatesthefollowing:ConditionalProbabilitiesP(B|A1)=.2P(B|A2)=.9P(BC|A1)=.8P(BC|A2)=.1Hence:

Example:L.S.ClothiersP(Bc|A1)=.8P(A1)=.7P(A2)=.3P(B|A2)=.9P(Bc|A2)=.1P(B|A1)=.2P(A1

B)=.14P(A2

B)=.27P(A2

Bc)=.03P(A1

Bc)=.56TownCouncilPlanningBoardExperimentalOutcomesTreeDiagram

Example:L.S.ClothiersBayes’TheoremTofindtheposteriorprobabilitythateventAiwilloccurgiventhateventBhasoccurred,weapply

Bayes’theorem.Bayes’theoremisapplicablewhentheeventsforwhichwewanttocomputeposteriorprobabilitiesaremutuallyexclusiveandtheirunionistheentiresamplespace.Giventheplanningboard’srecommendationnottoapprovethezoningchange,werevisethepriorprobabilitiesasfollows:PosteriorProbabilities=.34

Example:L.S.ClothiersTheplanningboard’srecommendationisgoodnewsforL.S.Clothiers.Theposteriorprobabilityofthetowncouncilapprovingthezoningchangeis.34comparedtoapriorprobabilityof.70.

Example:L.S.ClothiersPosteriorProbabilitiesBayes’Theorem:TabularApproach

Example:L.S.Clothiers

Column1

-Themutuallyexclusiveeventsforwhichposteriorprobabilitiesaredesired.

Column2

-Thepriorprobabilitiesfortheevents.

Column3

-Theconditionalprobabilitiesofthenewinformationgiveneachevent.Preparethefollowingthreecolumns:Step1

(1)(2)(3)(4)(5)EventsAiPriorProbabilitiesP(Ai)ConditionalProbabilitiesP(B|Ai)A1A2.7.31.0.2.9

Example:L.S.ClothiersBayes’Theorem:TabularApproachStep1

Bayes’Theorem:TabularApproachColumn4

ComputethejointprobabilitiesforeacheventandthenewinformationBbyusingthemultiplicationlaw.Preparethefourthcolumn: Multiplythepriorprobabilitiesincolumn2bythecorrespondingconditionalprobabilitiesincolumn3.Thatis,P(AiIB)=P(Ai)P(B|Ai).