商务与经济统计(原书第14版) 课件 第4章　概率

Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann©2020CengageLearning©2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.1Chapter4-IntroductiontoProbability 4.1-RandomExperiments,CountingRules,andAssigningProbabilities4.2-EventsandTheirProbabilities4.3-SomeBasicRelationshipsofProbability4.4-ConditionalProbability4.5-Bayes’Theorem2UncertaintiesManagersoftenbasetheirdecisionsonananalysisofuncertaintiessuchasthefollowing:Whatarethechancesthatthesaleswilldecreaseifweincreaseprices?Whatisthelikelihoodanewassemblymethodwillincreaseproductivity?Whataretheoddsthatanewinvestmentwillbeprofitable?3ProbabilityProbabilityisanumericalmeasureofthelikelihoodthataneventwilloccur.Probabilityvaluesarealwaysassignedonascalefrom0to1.Aprobabilitynearzeroindicatesaneventisquiteunlikelytooccur.Aprobabilitynearoneindicatesaneventisalmostcertaintooccur.4StatisticalExperimentsInstatistics,thenotionofanexperimentdifferssomewhatfromthatofanexperimentinthephysicalsciences.Instatisticalexperiments,probabilitydeterminesoutcomes.Eventhoughtheexperimentisrepeatedexactlythesameway,anentirelydifferentoutcomemayoccur.Forthisreason,statisticalexperimentsaresometimescalledrandomexperiments.5RandomExperimentandItsSampleSpace(1of2)ARandomexperimentisaprocessthatgenerateswell-definedexperimentaloutcomes.Thesamplespaceforanexperimentisthesetofallexperimentaloutcomes.Anexperimentaloutcomeisalsocalledasamplepoint.ExperimentExperimentalOutcomesTossacoinHead,tailInspectapartDefective,non-defectiveConductasalecallPurchase,nopurchaseRolladie1,2,3,4,5,6PlayafootballgameWin,loss,tie6RandomExperimentandItsSampleSpace(2of2)Example:BradleyInvestmentsBradleyhasinvestedintwostocks,MarkleyOilandCollinsMining.Bradleyhasdeterminedthatthepossibleoutcomesoftheseinvestmentsthreemonthsfromnowareasfollows:InvestmentGainorLossin3Months(in$1000s):MarkeyOilCollinsMining1085-20EMPTYCELL-20.EMPTYCELL7ACountingRuleforMultiple-StepExperimentsIfanexperimentconsistsofasequenceofkstepsinwhichtherearen1possibleresultsforthefirststep,n2possibleresultsforthesecondstep,andsoon,thenthetotalnumberofexperimentaloutcomesisgivenby(n1)(n2)...(nk).Ahelpfulgraphicalrepresentationofamultiple-stepexperimentisatreediagram.MarkleyOil: n1=4CollinsMining: n2=2Totalnumberofexperimentaloutcomes:(4)(2)=8.8TreeDiagram(1of2)Example:BradleyInvestments9CountingRuleforCombinationsNumberofCombinationsofNObjectsTakennataTimeAsecondusefulcountingruleenablesustocountthenumberofexperimentaloutcomeswhennobjectsaretobeselectedfromasetofNobjects.10CountingRuleforPermutationsNumberofPermutationsofNObjectsTakennataTimeAthirdusefulcountingruleenablesustocountthenumberofexperimentaloutcomeswhennobjectsaretobeselectedfromasetofNobjects,wheretheorderofselectionisimportant.11AssigningProbabilities(1of2)BasicRequirementsforAssigningProbabilities1. Theprobabilityassignedtoeachexperimentaloutcomemustbebetween0and1,inclusively.whereEiistheithexperimentaloutcomeandP(Ei)isitsprobability2. Thesumoftheprobabilitiesforallexperimentaloutcomesmustequal1.wherenisthenumberofexperimentaloutcomes.12AssigningProbabilities(2of2)ClassicalMethodAssigningprobabilitiesbasedontheassumptionofequallylikelyoutcomesRelativeFrequencyMethodAssigningprobabilitiesbasedonexperimentalorhistoricaldataSubjectiveMethodAssigningprobabilitybasedonjudgment

13ClassicalMethodExample:RollingaDieIfanexperimenthasnpossibleoutcomes,theclassicalmethodwouldassignaprobabilityof1/ntoeachoutcome.Experiment:RollingadieSampleSpace:S={1,2,3,4,5,6}Probabilities:Eachsamplepointhasa1/6chanceofoccurring14RelativeFrequencyMethodExample:LucasToolRentalLucasToolRentalwouldliketoassignprobabilitiestothenumberofcarpolishersitrentseachday.Officerecordsshowthefollowingfrequenciesofdailyrentalforthelast40days.Eachprobabilityassignmentisgivenbydividingthefrequency(numberofdays)bythetotalfrequency(totalnumberofdays).NumberofPolishersRentedNumberofDaysProbability04.10=4/4016.15218.45310.2542.05sum401.0015SubjectiveMethod(1of2)Wheneconomicconditionsoracompany’scircumstanceschangerapidly,itmightbeinappropriatetoassignprobabilitiesbasedsolelyonhistoricaldata.Wecanuseanydataavailableaswellasourexperienceandintuition,butultimatelyaprobabilityvalueshouldexpressourdegreeofbeliefthattheexperimentaloutcomewilloccur.Thebestprobabilityestimatesoftenareobtainedbycombiningtheestimatesfromtheclassicalorrelativefrequencyapproachwiththesubjectiveestimate.16SubjectiveMethod(2of2)Example:BradleyInvestmentsAnanalystmadethefollowingprobabilityestimates.ExperimentalOutcomeNetGainorLossProbability(10,8)$18,000Gain0.20(10,–2)$8,000Gain0.08(5,8)$13,000Gain0.16(5,–2)$3,000Gain0.26(0,8)$8,000Gain0.10(0,–2)$2,000Loss0.12(–20,8)$12,000Loss0.02(–20,–2)$22,000Loss0.06EmptycellEmptycellSumequals1.0017EventsandTheirProbabilities(1of2)Aneventisacollectionofsamplepoints.Theprobabilityofanyeventisequaltothesumoftheprobabilitiesofthesamplepointsintheevent.Ifwecanidentifyallthesamplepointsofanexperimentandassignaprobabilitytoeach,wecancomputetheprobabilityofanevent.EventM=MarkleyOilisProfitableM={(10,8),(10,–2),(5,8),(5,–2)}P(M)=P(10,8)+P(10,–2)+P(5,8)+P(5,–2)=0.20+0.08+0.16+0.26=0.7018EventsandTheirProbabilities(2of2)Example:BradleyInvestmentsEventC=CollinsMiningisProfitableC={(10,8),(5,8),(0,8),(–20,8)}P(C)=P(10,8)+P(5,8)+P(0,8)+P(–20,8)=0.20+0.16+0.10+0.02=0.4819SomeBasicRelationshipsofProbabilityTherearesomebasicprobabilityrelationshipsthatcanbeusedtocomputetheprobabilityofaneventwithoutknowledgeofallthesamplepointprobabilities.ComplementofanEventUnionofTwoEventsIntersectionofTwoEventsMutuallyExclusiveEvents20ComplementofanEventThecomplementofeventAisdefinedtobetheeventconsistingofallsamplepointsthatarenotinA.ThecomplementofAisdenotedbyAC.21UnionofTwoEvents(1of2)

22UnionofTwoEvents(2of2)

23IntersectionofTwoEvents(1of2)TheintersectionofeventsAandBisthesetofallsamplepointsthatareinbothAandB24IntersectionofTwoEvents(2of2)Example:BradleyInvestments31AdditionLawTheadditionlawprovidesawaytocomputetheprobabilityofeventA,orB,orbothAandBoccurring.Thelawiswrittenas:26MutuallyExclusiveEventsTwoeventsaresaidtobemutuallyexclusiveiftheeventshavenosamplepointsincommonTwoeventsaremutuallyexclusiveif,whenoneeventoccurs,theothercannotoccur.27ConditionalProbability(1of2)

28ConditionalProbability(2of2)Example:BradleyInvestments29MultiplicationLawThemultiplicationlawprovidesawaytocomputetheprobabilityoftheintersectionoftwoevents.Thelawiswrittenas:30JointProbabilityTableJointprobabilitiesappearinthebodyofthetableMarginalprobabilitiesappearinthemarginsofthetableMarkleyOilCollinsminingProfitable(C)CollinsminingnotProfitable(CC)TotalProfitable(M).36.34.70NotProfitable(MC).12.18.30Total.48.521.0031IndependentEventsIftheprobabilityofeventAisnotchangedbytheexistenceofeventB,wewouldsaythateventsAandBandareindependent.TwoeventsAandBareindependentif:32MultiplicationLawforIndependentEventsThemultiplicationlawalsocanbeusedasatesttoseeiftwoeventsareindependent.Thelawiswrittenas:33MutualExclusivenessandIndependenceDonotconfusethenotionofmutuallyexclusiveeventswiththatofindependentevents.Twoeventswithnonzeroprobabilitycannotbothmutuallyexclusiveandindependent.Ifonemutuallyexclusiveeventisknowntooccur,theothercannotoccur;thus,theprobabilityoftheothereventoccurringisreducedtozero(andthereforedependent).Twoeventsthatarenotmutuallyexclusivemightornightnotbeindependent.34Bayes’Theorem(1of2)Oftenwebeginprobabilityanalysiswithinitialorpriorprobabilities.Then,fromasample,specialreport,oraproducttest,weobtainsomeadditionalinformation.Giventhisinformation,wecalculaterevisedorposteriorprobabilities.Bayes’theoremprovidesthemeansforrevisingthepriorprobabilities.35Bayes’Theorem(2of2)AproposedshoppingcenterwillprovidestrongcompetitionfordowntownbusinesseslikeL.S.Clothiers.Iftheshoppingcenterisbuilt,theownerofL.S.Clothiersfeelsitwouldbebesttorelocatetotheshoppingcenter.Theshoppingcentercannotbebuiltunlessazoningchangeisapprovedbythetowncouncil.Theplanningboardmustfirstmakearecommendation,fororagainstthezoningchange,tothecouncil.Let: A1=towncouncilapprovesthezoningchange A2=towncouncildisapprovesthechangeUsingsubjectivejudgment:36NewInformationTheplanningboardhasrecommendedagainstthezoningchange.LetBdenotetheeventofanegativerecommendationbytheplanningboard.GiventhatBhasoccurred,shouldL.S.Clothiersrevisetheprobabilitiesthatthetowncouncilwillapproveordisapprovethezoningchange?Pasthistorywiththeplanningboardandthetowncouncilindicatesthefollowing:Hence:37TreeDiagram(2of2)Example:L.S.Clothiers38Bayes’TheoremTofindtheposteriorprobabilitythateventAiwilloccurgiventhateventBhasoccurred,weapplyBayes’theorem.Bayes’theoremisapplicablewhentheeventsforwhichwewanttocomputeposteriorprobabilitiesaremutuallyexclusiveandtheirunionistheentiresamplespace.39PosteriorProbabilities(1of2)Example:L.S.ClothiersGiventheplanningboard’srecommendationnottoapprovethezoningchange,werevisethepriorprobabilitiesasfollows:40PosteriorProbabilities(2of2)Theplanningboard’srecommendationisgoodnewsforL.S.Clothiers.Theposteriorprobabilityofthetowncouncilapprovingthezoningchangeis0.34comparedtoapriorprobabilityof0.70.41Bayes’Theorem:TabularApproach(1of6)Step1: Preparethefollowingthreecolumns:Column1-Themutuallyexclusiveeventsforwhichposteriorprobabilitiesaredesired.Column2-Thepriorprobabilitiesfortheevents.Column3-Theconditionalprobabilitiesofthenewinformationgiveneachevent.42Bayes’Theorem:TabularApproach(2of6)Example:L.S.Clothiers,Step1

(1)(2)(3)(4)(5)EventsPriorProbabilitiesConditionalProbability..AiP(Ai)P(B|Ai)..A1.7.2..A2.3.9...1.0...43Bayes’Theorem:TabularApproach(3of6)Step2:PreparethefourthcolumnColumn4:ComputethejointprobabilitiesforeacheventandthenewinformationBbyusingthemultiplicationlaw.Multiplytheprobabilitiesincolumn2bythecorrespondingconditionalprobabilitiesincolumn3.Thatis,(1)(2)(3)(4)(5)EventsPriorProbabilitiesConditionalProbabilityJoint

Probabil