商务统计 美谭英平等译

[美]AmirD.Aczel,JayavelSounderpandian著，谭英平等译商务统计Chapter1IntroductionandDescriptiveStatisticsUsingStatisticsPercentilesandQuartilesMeasuresofCentralTendencyMeasuresofVariabilityGroupedDataandtheHistogramSkewnessandKurtosisRelationsbetweentheMeanandStandardDeviationMethodsofDisplayingDataExploratoryDataAnalysisUsingtheComputerIntroductionandDescriptiveStatistics1Distinguishbetweenqualitativedataandquantitativedata.Describenominal,ordinal,interval,andratioscalesofmeasurements.Describethedifferencebetweenpopulationandsample.Calculateandinterpretpercentilesandquartiles.Explainmeasuresofcentraltendencyandhowtocomputethem.Createdifferenttypesofchartsthatdescribedatasets.UseExceltemplatestocomputevariousmeasuresandcreatecharts.

LEARNINGOBJECTIVES1Afterstudyingthischapter,youshouldbeableto:Statisticsisasciencethathelpsusmakebetterdecisionsinbusinessandeconomicsaswellasinotherfields.Statisticsteachesushowtosummarize,analyze,anddrawmeaningfulinferencesfromdatathatthenleadtoimprovedecisions.Thesedecisionsthatwemakehelpusimprovetherunning,forexample,adepartment,acompany,theentireeconomy,etc.WHATISSTATISTICS?1-1.UsingStatistics(TwoCategories)InferentialStatisticsPredictandforecastvaluesofpopulationparametersTesthypothesesaboutvaluesofpopulationparametersMakedecisionsDescriptiveStatisticsCollectOrganizeSummarizeDisplayAnalyzeQualitative-CategoricalorNominal:

Examplesare-ColorGenderNationalityQuantitative-MeasurableorCountable:

Examplesare-TemperaturesSalariesNumberofpointsscoredona100

pointexamTypesofData-TwoTypesNominalScale

-groupsorclassesGenderOrdinalScale

-ordermattersRanks(toptenvideos)IntervalScale

-differenceordistancematters–hasarbitraryzerovalue.Temperatures(0F,0C)RatioScale

-Ratiomatters–hasanaturalzerovalue.SalariesScalesofMeasurementA

populationconsistsofthesetofallmeasurementsforwhichtheinvestigatorisinterested.A

sample

isasubsetofthemeasurementsselectedfromthepopulation.A

census

isacompleteenumerationofeveryiteminapopulation.SamplesandPopulationsSampling

fromthepopulationisoftendone

randomly,suchthateverypossiblesampleofequalsize(n)willhaveanequalchanceofbeingselected.Asampleselectedinthiswayiscalledasimplerandomsampleorjustarandomsample.Arandomsampleallowschancetodetermineitselements.SimpleRandomSamplePopulation(N)Sample(n)SamplesandPopulationsCensusofapopulationmaybe:Impossible ImpracticalToocostlyWhySample?Givenanysetofnumericalobservations,orderthemaccordingtomagnitude.ThePth

percentile

intheorderedsetisthatvaluebelowwhichlieP%(Ppercent)oftheobservationsintheset.ThepositionofthePthpercentileisgivenby(n+1)P/100,wherenisthenumberofobservationsintheset.1-2PercentilesandQuartiles

Alargedepartmentstorecollectsdataonsalesmadebyeachofitssalespeople.Thenumberofsalesmadeonagivendaybyeachof

20

salespeopleisshownonthenextslide.Also,thedatahasbeensortedinmagnitude.

Example1-2Example1-2(Continued)-

SalesandSortedSales

Sales

SortedSales

9 6 6 9 12 10 10 12 13 13 15 14 16 14 14 15 14 16 16 16 17 16 16 17 24 17 21 18 22 18 18 19 19 20 18 21 20 22 17 24

Findthe50th,80th,andthe90th

percentilesofthisdataset.Tofindthe50th

percentile,determinethedatapointinposition(n+1)P/100=(20+1)(50/100)

=10.5.Thus,thepercentileislocatedatthe10.5th

position.The10thobservationis16,andthe11th

observationisalso16.The50thpercentilewillliehalfwaybetweenthe10th

and11th

values(whichareboth16inthiscase)andisthus16.

Example1-2(Continued)Percentiles

Tofindthe80thpercentile,determinethedatapointinposition(n+1)P/100=(20+1)(80/100)=16.8.Thus,thepercentileislocatedatthe16.8th

position.The16th

observationis19,andthe17th

observationisalso20.The80thpercentileisapointlying0.8ofthewayfrom19to20andisthus19.8.

Example1-2(Continued)Percentiles

Tofindthe90thpercentile,determinethedatapointinposition(n+1)P/100=(20+1)(90/100)=18.9.Thus,thepercentileislocatedatthe18.9th

position.The18th

observationis21,andthe19th

observationisalso22.The90thpercentileisapointlying0.9ofthe

wayfrom21to22andisthus21.9.

Example1-2(Continued)Percentiles

Quartilesarethepercentagepointsthatbreakdowntheordereddatasetintoquarters.Thefirstquartileisthe25thpercentile.Itisthepointbelowwhichlie1/4ofthedata.Thesecondquartileisthe50thpercentile.Itisthepointbelowwhichlie1/2ofthedata.Thisisalsocalledthemedian.Thethirdquartileisthe75thpercentile.Itisthepointbelowwhichlie3/4ofthedata.

Quartiles–SpecialPercentiles

Thefirstquartile,Q1,(25thpercentile)is

oftencalledthelowerquartile.Thesecondquartile,Q2,(50th

percentile)isoftencalledthemedian

orthemiddlequartile.Thethirdquartile,Q3,(75thpercentile)

isoftencalledtheupperquartile.Theinterquartilerangeisthedifference

betweenthefirstandthethirdquartiles.

QuartilesandInterquartileRange

SortedSales Sales 96 6 9 12 10 10 12 13 13 15 14 16 14 14 15 14 16 16 16 17 16 16 17 24 17 21 18 22 18 18 19 19 20 18 21 20 22 17 24 FirstQuartileMedianThirdQuartile(n+1)P/100(20+1)25/100=5.25(20+1)50/100=10.5(20+1)75/100=15.7513+(.25)(1)=13.2516+(.5)(0)=1618+(.75)(1)=18.75QuartilesExample1-3:FindingQuartilesPosition(n+1)P/100QuartilesExample1-3:UsingtheTemplate(n+1)P/100QuartilesExample1-3(Continued):UsingtheTemplateThisisthelowerpartofthesametemplatefromthepreviousslide.MeasuresofVariabilityRangeInterquartilerangeVarianceStandardDeviationMeasuresofCentralTendencyMedianModeMeanOthersummarymeasures:SkewnessKurtosisSummaryMeasures:PopulationParametersSampleStatisticsMedianMiddlevaluewhensortedinorderofmagnitude50thpercentileModeMostfrequently-occurringvalueMeanAverage1-3MeasuresofCentralTendency

orLocationSales SortedSales

9 6 6 9 12 10 10 12 13 13 15 14 16 14 14 15 14 16 16 16 17 16 16 17 24 17 21 18 22 18 18 19 19 20 18 21 20 22 17 24 MedianMedian50thPercentile(20+1)50/100=10.516+(.5)(0)=16Themedianisthemiddlevalueofdatasortedinorderofmagnitude.Itisthe50thpercentile.Example–Median(DataisusedfromExample1-2)Seeslide#21forthetemplateoutput

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---------------------------------------------------------------6910121314151617181920212224Mode=16Themodeisthemostfrequentlyoccurringvalue.Itisthevaluewiththehighestfrequency.Example-Mode(DataisusedfromExample1-2)Seeslide#21forthetemplateoutputThemeanofasetofobservationsistheiraverage-thesumoftheobservedvaluesdividedbythenumberofobservations.PopulationMeanSampleMeanm==åxNiN1xxnin==å1ArithmeticMeanorAveragexxnin====å1317201585.Sales

96121013151614141617162421221819182017317Example–Mean(DataisusedfromExample1-2)Seeslide#21forthetemplateoutput

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---------------------------------------------------------------6910121314151617181920212224MedianandMode=16Mean=15.85Example-Mode(DataisusedfromExample1-2)Seeslide#21forthetemplateoutputRangeDifferencebetweenmaximumandminimumvaluesInterquartileRangeDifferencebetweenthirdandfirstquartile(Q3-Q1)VarianceAverage*ofthesquareddeviationsfromthemeanStandardDeviationSquarerootofthevarianceDefinitionsofpopulationvarianceandsamplevariancedifferslightly.1-4MeasuresofVariabilityorDispersion

SortedSales Sales Rank9 6 16 9 212 10 310 12 413 13 515 14 616 14 714 15 814 16 916 16 1017 16 1116 17 1224 17 1321 18 1422 18 1518 19 1619 20 1718 21 1820 22 1917 24 20FirstQuartileThirdQuartileQ1=13+(.25)(1)=13.25Q3=18+(.75)(1)=18.75MinimumMaximumRange:Maximum-Minimum=

24-6=18InterquartileRange:Q3-Q1=

18.75-13.25=5.5Example-RangeandInterquartileRange(DataisusedfromExample1-2)Seeslide#21forthetemplateoutputVarianceandStandardDeviation()smss22121221=-=-===åå=å()xNxNNiNiNxiNPopulationVariance()()sxxnxxnnssininin2212122111=-å-=-å-====å()SampleVariance()

6 -9.85 97.0225 369 -6.85 46.9225 8110 -5.85 34.2225 10012 -3.85 14.8225 14413 -2.85 8.122516914 -1.85 3.4225 19614 -1.85 3.4225 19615 -0.85 0.7225 22516 0.15 0.0225 25616 0.15 0.0225 25616 0.15 0.0225 25617 1.15 1.3225 28917 1.15 1.3225 28918 2.15 4.6225 32418 2.15 4.6225 32419 3.15 9.9225 36120 4.15 17.2225 40021 5.15 26.5225 44122 6.15 37.8225 48424 8.15 66.4225 576317 0 378.55005403CalculationofSampleVariance(n+1)P/100QuartilesExample:SampleVarianceUsingtheTemplateNote:Thisisjustareplicationofslide#21.DividingdataintogroupsorclassesorintervalsGroupsshouldbe:MutuallyexclusiveNotoverlapping-everyobservationisassignedtoonlyonegroupExhaustiveEveryobservationisassignedtoagroupEqual-width

(ifpossible)Firstorlastgroupmaybeopen-ended1-5GroupDataandtheHistogramTablewithtwocolumnslisting:EachandeverygrouporclassorintervalofvaluesAssociatedfrequencyofeachgroupNumberofobservationsassignedtoeachgroupSumoffrequenciesisnumberofobservationsNforpopulationnforsampleClass

midpoint

isthemiddlevalueofagrouporclassorintervalRelativefrequency

isthepercentageoftotalobservationsineachclassSumofrelativefrequencies=1FrequencyDistribution x f(x) f(x)/nSpendingClass($) Frequency(numberofcustomers) RelativeFrequency0tolessthan100 30 0.163100tolessthan200 38 0.207200tolessthan300 50 0.272300tolessthan400 31 0.168400tolessthan500 22 0.120500tolessthan600 13 0.070 184 1.000

Exampleofrelativefrequency:30/184=0.163Sumofrelativefrequencies=1Example1-7:FrequencyDistribution x F(x) F(x)/nSpendingClass($) CumulativeFrequency CumulativeRelativeFrequency0tolessthan100 30 0.163100tolessthan200 68 0.370200tolessthan300 118 0.641300tolessthan400 149 0.810400tolessthan500 171 0.929500tolessthan600 184 1.000

Thecumulativefrequency

ofeachgroupisthesumofthefrequenciesofthatandallprecedinggroups.CumulativeFrequencyDistributionA

histogram

isachartmadeofbarsofdifferentheights.WidthsandlocationsofbarscorrespondtowidthsandlocationsofdatagroupingsHeightsofbarscorrespondtofrequenciesorrelativefrequenciesofdatagroupingsHistogramFrequencyHistogramHistogramExampleRelativeFrequencyHistogramHistogramExampleSkewnessMeasureofasymmetryofafrequencydistributionSkewedtoleftSymmetricorunskewedSkewedtorightKurtosisMeasureofflatnessorpeakednessofafrequencydistributionPlatykurtic(relativelyflat)Mesokurtic(normal)Leptokurtic(relativelypeaked)1-6SkewnessandKurtosisSkewedtoleftSkewnessSkewnessSymmetricSkewnessSkewedtorightKurtosisPlatykurtic-flatdistributionKurtosisMesokurtic-nottooflatandnottoopeakedKurtosisLeptokurtic

-peakeddistributionChebyshev’sTheoremAppliestoany

distribution,regardlessofshapePlaceslowerlimitsonthepercentagesofobservationswithinagivennumberofstandarddeviationsfromthemeanEmpiricalRuleAppliesonlytoroughlymound-shapedandsymmetricdistributionsSpecifiesapproximatepercentagesofobservationswithinagivennumberofstandarddeviationsfromthemean1-7RelationsbetweentheMeanandStandardDeviationAtleastoftheelementsofany

distributionliewithinkstandarddeviationsofthemean

Atleast

LiewithinStandarddeviationsofthemean234Chebyshev’sTheoremForroughlymound-shapedandsymmetricdistributions,approximately:EmpiricalRulePieChartsCategoriesrepresentedaspercentagesoftotalBarGraphsHeightsofrectanglesrepresentgroupfrequenciesFrequencyPolygons

HeightoflinerepresentsfrequencyOgivesHeightoflinerepresentscumulativefrequencyTimePlotsRepresentsvaluesovertime1-8MethodsofDisplayingDataPieChartBarChart

RelativeFrequencyPolygonOgiveFrequencyPolygonandOgive504030201000.30.20.10.0RelativeFrequencySales504030201001.00.50.0CumulativeRelativeFrequencySales(Cumulativefrequencyorrelativefrequencygraph)OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ8.57.56.55.5MonthMillions

of

TonsMonthly

Steel

Production

TimePlotStem-and-LeafDisplaysQuick-and-dirtylistingofallobservationsConveyssomeofthesameinformationasahistogramBoxPlotsMedianLowerandupperquartilesMaximumandminimumTechniquestodeterminerelationshipsandtrends,identifyoutliersandinfluentialobservations,andquicklydescribeorsummarizedatasets.1-9ExploratoryDataAnalysis-EDA

1122355567

201112223467778993012457411257

50236

602Example1-8:Stem-and-LeafDisplayFigure1-17:TaskPerformanceTimesXX*oMedianQ1Q3InnerFenceInnerFenceOuterFenceOuterFenceInterquartileRangeSmallestdatapointnotbelowinnerfenceLargestdatapointnotexceedinginnerfenceSuspectedoutlierOutlierQ1-3(IQR)Q1-1.5(IQR)Q3+1.5(IQR)Q3+3(IQR)ElementsofaBoxPlotBoxPlotExample:BoxPlot1-10UsingtheComputer–TheTemplateOutputwithBasicStatisticsUsingtheComputer–TemplateOutputfortheHistogramFigure1-24UsingtheComputer–TemplateOutputforHistogramsforGroupedDataFigure1-25UsingtheComputer–TemplateOutputforFrequencyPolygons&theOgiveforGroupedDataFigure1-25UsingtheComputer–TemplateOutputforTwoFrequencyPolygonsforGroupedDataFigure1-26UsingtheComputer–PieChartTemplateOutputFigure1-27UsingtheComputer–BarChartTemplateOutputFigure1-28UsingtheComputer–BoxPlotTemplateOutputFigure1-29UsingtheComputer–BoxPlotTemplatetoCompareTwoDataSetsFigure1-30UsingtheComputer–TimePlotTemplate

Figure1-31UsingtheComputer–TimePlotComparisonTemplate

Figure1-32ScatterPlotsScatterPlotsareusedtoidentifyandreportanyunderlyingrelationshipsamongpairsofdatasets.Theplotconsistsofascatterofpoints,eachpointrepresentinganobservation.ScatterPlots

Scatterplotwithtrendline.Thistypeofrelationshipisknownasapositivecorrelation.Correlationwillbediscussedinlaterchapters.COMPLETE

BUSINESS

STATISTICSbyAMIRD.ACZEL&JAYAVELSOUNDERPANDIAN6thedition.Chapter2ProbabilityUsingStatisticsBasicDefinitions:Events,SampleSpace,andProbabilitiesBasicRulesforProbabilityConditionalProbabilityIndependenceofEventsCombinatorialConceptsTheLawofTotalProbabilityandBayes’TheoremJointProbabilityTableUsingtheComputerProbability2Defineprobability,samplespace,andevent.Distinguishbetweensubjectiveandobjectiveprobability.Describethecomplementofanevent,theintersection,andtheunionoftwoevents.Computeprobabilitiesofvarioustypesofevents.Explaintheconceptofconditionalprobabilityandhowtocomputeit.Describepermutationandcombinationandtheiruseincertainprobabilitycomputations.ExplainBayes’theoremanditsapplications.LEARNINGOBJECTIVESAfterstudyingthischapter,youshouldbeableto:22-1Probabilityis:AquantitativemeasureofuncertaintyAmeasureofthestrengthofbelief

intheoccurrenceofanuncertaineventAmeasureofthedegreeofchanceorlikelihoodofoccurrence

ofanuncertaineventMeasuredbyanumberbetween0and1(orbetween0%and100%)TypesofProbability

ObjectiveorClassicalProbabilitybasedonequally-likelyeventsbasedonlong-runrelativefrequencyofeventsnotbasedonpersonalbeliefsisthesameforallobservers(objective)examples:tossacoin,throwadie,pickacardTypesofProbability(Continued)SubjectiveProbabilitybasedonpersonalbeliefs,experiences,prejudices,intuition-personaljudgmentdifferentforallobservers(subjective)examples:SuperBowl,elections,newproductintroduction,snowfallSet-acollectionofelementsorobjectsofinterestEmptyset(denotedby)

asetcontainingnoelementsUniversalset(denotedbyS)asetcontainingallpossibleelementsComplement(Not).ThecomplementofAisasetcontainingallelementsofSnotinA2-2BasicDefinitionsComplementofaSetASVennDiagramillustratingtheComplementofaneventIntersection

(And)asetcontainingallelementsinbothAandBUnion

(Or)asetcontainingallelementsinAorBorbothBasicDefinitions(Continued)Sets:AIntersectingwithBABSSets:AUnionBABSMutuallyexclusiveordisjointsetssetshavingnoelementsincommon,havingnointersection,whoseintersectionistheemptysetPartitionacollectionofmutuallyexclusivesetswhichtogetherincludeallpossibleelements,whoseunionistheuniversalsetBasicDefinitions(Continued)MutuallyExclusiveorDisjointSetsABSSetshavenothingincommonSets:PartitionA1A2A3A4A5SProcessthatleadstooneofseveralpossibleoutcomes

*,e.g.:CointossHeads,TailsThrowdie1,2,3,4,5,6PickacardAH,KH,QH,...IntroduceanewproductEachtrialofanexperimenthasasingleobservedoutcome.Thepreciseoutcomeofarandomexperimentisunknownbeforeatrial.

*Alsocalledabasicoutcome,elementaryevent,orsimpleeventExperimentSampleSpaceorEventSetSetofallpossibleoutcomes(universalset)foragivenexperimentE.g.:Rollaregularsix-sideddieS={1,2,3,4,5,6}EventCollectionofoutcomeshavingacommoncharacteristicE.g.:Evennumber

A={2,4,6}

EventAoccursifanoutcomeinthesetAoccursProbabilityofaneventSumoftheprobabilitiesoftheoutcomesofwhichitconsistsP(A)=P(2)+P(4)+P(6)Events:DefinitionForexample:ThrowadieSixpossibleoutcomes{1,2,3,4,5,6}Ifeachisequally-likely,theprobabilityofeachis1/6=0.1667=16.67%

Probabilityofeachequally-likelyoutcomeis1dividedbythenumberofpossibleoutcomesEventA(evennumber)P(A)=P(2)+P(4)+P(6)=1/6+1/6+1/6=1/2foreinAEqually-likelyProbabilities

(HypotheticalorIdealExperiments)PickaCard:SampleSpaceEvent‘Ace’UnionofEvents‘Heart’and‘Ace’Event‘Heart’Theintersectionoftheevents‘Heart’and‘Ace’comprisesthesinglepointcircledtwice:theaceofheartsHeartsDiamondsClubsSpadesAAAAKKKKQQQQJJJJ1010101099998888777766665555444433332222RangeofValuesforP(A):Complements

-Probabilityof

not

AIntersection

-ProbabilityofbothA

and

BMutuallyexclusiveevents(AandC):2-3BasicRulesforProbabilityUnion

-ProbabilityofAor

Borboth(ruleofunions)Mutuallyexclusiveevents:IfAandBaremutuallyexclusive,thenBasicRulesforProbability(Continued)Sets:P(AUnionB)ABSConditionalProbability

-ProbabilityofAgivenBIndependentevents:2-4ConditionalProbabilityRulesofconditionalprobability:IfeventsAandDarestatisticallyindependent:sosoConditionalProbability(continued)AT&TIBMTotalTelecommunication401050Computers203050Total6040100CountsAT&TIBMTotalTelecommunication.40.10.50Computers.20.30.50Total.60.401.00ProbabilitiesProbabilitythataprojectisundertakenbyIBMgivenitisatelecommunicationsproject:ContingencyTable-Example2-2ConditionsforthestatisticalindependenceofeventsAandB:2-5IndependenceofEventsEventsTelevision(T)andBillboard(B)areassumedtobeindependent.IndependenceofEvents–

Example2-5Theprobabilityoftheunionofseveralindependenteventsis1minustheproductofprobabilitiesoftheircomplements:Example2-7:Theprobabilityoftheintersectionofseveralindependenteventsistheproductoftheirseparateindividualprobabilities:ProductRulesforIndependentEventsConsiderapairofsix-sideddice.Therearesixpossibleoutcomesfromthrowingthefirstdie{1,2,3,4,5,6}andsixpossibleoutcomesfromthrowingtheseconddie{1,2,3,4,5,6}.Altogether,thereare6*6=36possibleoutcomesfromthrowingthetwodice.Ingeneral,ifthereareneventsandtheeventicanhappeninNipossibleways,thenthenumberofwaysinwhichthesequenceofneventsmayoccurisN1N2...Nn.Pick5cardsfromadeckof52-withreplacement52*52*52*52*52=525380,204,032differentpossibleoutcomesPick5cardsfromadeckof52-withoutreplacement52*51*50*49*48=311,875,200differentpossibleoutcomes2-6CombinatorialConcepts.....Ordertheletters:A,B,andCABCBCABACACBCBA...........ABCACBBACBCACABCBAMoreonCombinatorialConcepts

(TreeDiagram)Howmanywayscanyouorderthe3lettersA,B,andC?Thereare3choicesforthefirstletter,2forthesecond,and1forthelast,sothereare3*2*1=6possiblewaystoorderthethreelettersA,B,andC.Howmanywaysaretheretoorderthe6lettersA,B,C,D,E,andF?(6*5*4*3*2*1=720)Factorial:Foranypositiveintegern,wedefinenfactorial

as:n(n-1)(n-2)...(1).Wedenotenfactorialasn!.Thenumbern!isthenumberofwaysinwhichnobjectscanbeordered.Bydefinition1!=1and0!=1.FactorialPermutationsarethepossibleorderedselectionsofrobjectsoutofatotalofnobjects.ThenumberofpermutationsofnobjectstakenratatimeisdenotedbynPr,whereWhatifwechoseonly3outofthe6lettersA,B,C,D,E,andF?Thereare6waystochoosethefirstletter,5waystochoosethesecondletter,and4waystochoosethethirdletter(leaving3lettersunchosen).Thatmakes6*5*4=120possibleorderingsorpermutations.Permutations(Orderisimportant)Combinations

arethepossibleselectionsofritemsfromagroupofnitemsregardlessoftheorderofselection.Thenumberofcombinationsisdenotedandisreadasnchooser.AnalternativenotationisnCr.Wedefinethenumberofcombinationsofroutofnelementsas:Supposethatwhenwepick3lettersoutofthe6lettersA,B,C,D,E,andFwechoseBCD,orBDC,orCBD,orCDB,orDBC,orDCB.(Thesearethe6(3!)permutationsororderingsofthe3lettersB,C,andD.)Buttheseareorderingsofthesamecombinationof3letters.Howmanycombinationsof6differentletters,taking3atatime,arethere?Combinations(OrderisnotImportant)Example:TemplateforCalculatingPermutations&CombinationsIntermsofconditionalprobabilities:Moregenerally(whereBimakeupapartition):2-7TheLawofTotalProbabilityandBayes’TheoremThelawoftotalprobability:EventU:StockmarketwillgoupinthenextyearEventW:EconomywilldowellinthenextyearTheLawofTotalProbability-

Example2-9Bayes’theoremenablesyou,knowingjustalittlemorethantheprobabilityofAgivenB,tofindtheprobabilityofBgivenA.Basedonthedefinitionofconditionalprobabilityandthelawoftotalprobability.ApplyingthelawoftotalprobabilitytothedenominatorApplyingthedefinitionofconditionalprobabilitythroughoutBayes’TheoremAmedicaltestforararedisease(affecting0.1%ofthepopulation[])isimperfect:Whenadministeredtoanillperson,thetestwillindicatesowithprobability0.92[]TheeventisafalsenegativeWhenadministeredtoapersonwhoisnotill,thetestwillerroneouslygiveapositiveresult(falsepositive)withprobability0.04[]Theeventisafalsepositive..Bayes’Theorem-Example2-10Example2-10(continued)PriorProbabilitiesConditionalProbabilitiesJointProbabilitiesExample2-10(TreeDiagram)GivenapartitionofeventsB1,B2,...,Bn:ApplyingthelawoftotalprobabilitytothedenominatorApplyingthedefinitionofconditionalprobabilitythroughoutBayes’TheoremExtendedAneconomistbelievesthatduringperiodsofhigheconomicgrowth,theU.S.dollarappreciateswithprobability0.70;inperio