商务统计学（第7版）英文课件3 Numerical Descriptive Measures、4 Basic Probability

NumericalDescriptiveMeasuresChapter3Inthischapter,youlearnto:

Describethepropertiesofcentraltendency,variation,andshapeinnumericaldataConstructandinterpretaboxplotComputedescriptivesummarymeasuresforapopulationCalculatethecovarianceandthecoefficientofcorrelationObjectivesSummaryDefinitionsThecentraltendencyistheextenttowhichthevaluesofanumericalvariablegrouparoundatypicalorcentralvalue.Thevariationistheamountofdispersionorscatteringawayfromacentralvaluethatthevaluesofanumericalvariableshow.Theshapeisthepatternofthedistributionofvaluesfromthelowestvaluetothehighestvalue.DCOVAMeasuresofCentralTendency:

TheMeanThearithmeticmean(oftenjustcalledthe“mean”)isthemostcommonmeasureofcentraltendencyForasampleofsizen:SamplesizeObservedvaluesTheithvaluePronouncedx-barDCOVAMeasuresofCentralTendency:

TheMean(con’t)ThemostcommonmeasureofcentraltendencyMean=sumofvaluesdividedbythenumberofvaluesAffectedbyextremevalues(outliers)11121314151617181920Mean=1311121314151617181920Mean=14DCOVAMeasuresofCentralTendency:

TheMedianInanorderedarray,themedianisthe“middle”number(50%above,50%below)

LesssensitivethanthemeantoextremevaluesMedian=13Median=131112131415161718192011121314151617181920DCOVAMeasuresofCentralTendency:

LocatingtheMedianThelocationofthemedianwhenthevaluesareinnumericalorder(smallesttolargest):Ifthenumberofvaluesisodd,themedianisthemiddlenumberIfthenumberofvaluesiseven,themedianistheaverageofthetwomiddlenumbers Notethatisnotthevalueofthemedian,onlythepositionofthemedianintherankeddataDCOVAMeasuresofCentralTendency:

TheModeValuethatoccursmostoftenNotaffectedbyextremevaluesUsedforeithernumericalorcategoricaldataTheremaybenomodeTheremaybeseveralmodes01234567891011121314

Mode=90123456NoModeDCOVAMeasuresofCentralTendency:

ReviewExampleHousePrices:

$2,000,000$500,000

$300,000

$100,000

$100,000Sum$3,000,000Mean:($3,000,000/5) =$600,000Median:middlevalueofrankeddata

=$300,000Mode:mostfrequentvalue

=$100,000DCOVAMeasuresofCentralTendency:

WhichMeasuretoChoose?Themeanisgenerallyused,unlessextremevalues(outliers)exist.Themedianisoftenused,sincethemedianisnotsensitivetoextremevalues.Forexample,medianhomepricesmaybereportedforaregion;itislesssensitivetooutliers.Insomesituationsitmakessensetoreportboththemeanandthemedian.DCOVAMeasuresofCentralTendency:

SummaryCentralTendencyArithmeticMeanMedianModeMiddlevalueintheorderedarrayMostfrequentlyobservedvalueDCOVASamecenter,differentvariationMeasuresofVariationMeasuresofvariationgiveinformationonthespreadorvariabilityordispersionofthedatavalues.

VariationStandardDeviationCoefficientofVariationRangeVarianceDCOVAMeasuresofVariation:

TheRangeSimplestmeasureofvariationDifferencebetweenthelargestandthesmallestvalues:Range=Xlargest–Xsmallest01234567891011121314Range=13-1=12Example:DCOVAMeasuresofVariation:

WhyTheRangeCanBeMisleadingDoesnotaccountforhowthedataaredistributedSensitivetooutliers789101112Range=12-7=5789101112Range=12-7=5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120Range=5-1=4Range=120-1=119DCOVAAverage(approximately)ofsquareddeviationsofvaluesfromthemeanSample

variance:MeasuresofVariation:

TheSampleVarianceWhere

=arithmeticmeann=samplesizeXi=ithvalueofthevariableXDCOVAMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthevarianceHasthesameunitsastheoriginaldataSample

standarddeviation:MeasuresofVariation:

TheSampleStandardDeviationDCOVAMeasuresofVariation:

TheStandardDeviationStepsforComputingStandardDeviation1. Computethedifferencebetweeneachvalueandthemean.2. Squareeachdifference.3. Addthesquareddifferences.4. Dividethistotalbyn-1togetthesamplevariance.5. Takethesquarerootofthesamplevariancetogetthesamplestandarddeviation.DCOVAMeasuresofVariation:

SampleStandardDeviation:

CalculationExampleSample

Data(Xi):1012141517181824n=8Mean=X=16Ameasureofthe“average”scatteraroundthemeanDCOVAMeasuresofVariation:

ComparingStandardDeviationsMean=15.5S=3.338

11121314151617181920211112131415161718192021DataBDataAMean=15.5S=0.9261112131415161718192021Mean=15.5S=4.567DataCDCOVAMeasuresofVariation:

ComparingStandardDeviationsSmallerstandarddeviationLargerstandarddeviationDCOVAMeasuresofVariation:

SummaryCharacteristicsThemorethedataarespreadout,thegreatertherange,variance,andstandarddeviation.Themorethedataareconcentrated,thesmallertherange,variance,andstandarddeviation.Ifthevaluesareallthesame(novariation),allthesemeasureswillbezero.Noneofthesemeasuresareevernegative.DCOVAMeasuresofVariation:

TheCoefficientofVariationMeasuresrelativevariationAlwaysinpercentage(%)ShowsvariationrelativetomeanCanbeusedtocomparethevariabilityoftwoormoresetsofdatameasuredindifferentunitsDCOVAMeasuresofVariation:

ComparingCoefficientsofVariationStockA:Averagepricelastyear=$50Standarddeviation=$5StockB:Averagepricelastyear=$100Standarddeviation=$5Bothstockshavethesamestandarddeviation,butstockBislessvariablerelativetoitspriceDCOVAMeasuresofVariation:

ComparingCoefficientsofVariation(con’t)StockA:Averagepricelastyear=$50Standarddeviation=$5StockC:Averagepricelastyear=$8Standarddeviation=$2StockChasamuchsmallerstandarddeviationbutamuchhighercoefficientofvariationDCOVALocatingExtremeOutliers:

Z-ScoreTocomputetheZ-scoreofadatavalue,subtractthemeananddividebythestandarddeviation.TheZ-scoreisthenumberofstandarddeviationsadatavalueisfromthemean.AdatavalueisconsideredanextremeoutlierifitsZ-scoreislessthan-3.0orgreaterthan+3.0.ThelargertheabsolutevalueoftheZ-score,thefartherthedatavalueisfromthemean.DCOVALocatingExtremeOutliers:

Z-ScorewhereXrepresentsthedatavalue Xisthesamplemean SisthesamplestandarddeviationDCOVALocatingExtremeOutliers:

Z-ScoreSupposethemeanmathSATscoreis490,withastandarddeviationof100.ComputetheZ-scoreforatestscoreof620.Ascoreof620is1.3standarddeviationsabovethemeanandwouldnotbeconsideredanoutlier.DCOVAShapeofaDistributionDescribeshowdataaredistributedTwousefulshaperelatedstatisticsare:SkewnessMeasurestheextenttowhichdatavaluesarenotsymmetricalKurtosisKurtosisaffectsthepeakednessofthecurveofthedistribution—thatis,howsharplythecurverisesapproachingthecenterofthedistributionDCOVAShapeofaDistribution(Skewness)MeasurestheextenttowhichdataisnotsymmetricalMean=Median

Mean<Median

Median<MeanRight-SkewedLeft-SkewedSymmetricDCOVASkewnessStatistic<0 0 >0ShapeofaDistribution--Kurtosismeasureshowsharplythecurverisesapproachingthecenterofthedistribution

SharperPeakThanBell-Shaped(Kurtosis>0)FlatterThanBell-Shaped(Kurtosis<0)Bell-Shaped(Kurtosis=0)DCOVAGeneralDescriptiveStatsUsingMicrosoftExcelFunctionsDCOVAGeneralDescriptiveStatsUsingMicrosoftExcelDataAnalysisToolSelectData.SelectDataAnalysis.SelectDescriptiveStatisticsandclickOK.DCOVAGeneralDescriptiveStatsUsingMicrosoftExcel4.Enterthecellrange.5.ChecktheSummaryStatisticsbox.6.ClickOKDCOVAExceloutputMicrosoftExceldescriptivestatisticsoutput,usingthehousepricedata:HousePrices:

$2,000,000500,000

300,000

100,000

100,000DCOVAMinitabOutputMinitabdescriptivestatisticsoutputusingthehousepricedata:HousePrices:

$2,000,000500,000

300,000

100,000

100,000DCOVADescriptiveStatistics:HousePriceTotalVariableCountMeanSEMeanStDevVarianceSumMinimumHousePrice56000003577718000006.40000E+113000000100000 NforVariableMedianMaximumRangeModeMode

SkewnessKurtosisHousePrice300000200000019000001000002 2.014.13QuartileMeasuresQuartilessplittherankeddatainto4segmentswithanequalnumberofvaluespersegment25%Thefirstquartile,Q1,isthevalueforwhich25%oftheobservationsaresmallerand75%arelargerQ2isthesameasthemedian(50%oftheobservationsaresmallerand50%arelarger)Only25%oftheobservationsaregreaterthanthethirdquartileQ1Q2Q325%25%25%DCOVAQuartileMeasures:

LocatingQuartilesFindaquartilebydeterminingthevalueintheappropriatepositionintherankeddata,whereFirstquartileposition:

Q1=(n+1)/4rankedvalue

Secondquartileposition:

Q2=(n+1)/2

rankedvalue

Thirdquartileposition:

Q3=3(n+1)/4rankedvalue

where

n

isthenumberofobservedvaluesDCOVAQuartileMeasures:

CalculationRulesWhencalculatingtherankedpositionusethefollowingrulesIftheresultisawholenumberthenitistherankedpositiontouseIftheresultisafractionalhalf(e.g.2.5,7.5,8.5,etc.)thenaveragethetwocorrespondingdatavalues.Iftheresultisnotawholenumberorafractionalhalfthenroundtheresulttothenearestintegertofindtherankedposition.DCOVA(n=9)Q1isinthe

(9+1)/4=2.5positionoftherankeddata

sousethevaluehalfwaybetweenthe2ndand3rdvalues,

soQ1=12.5QuartileMeasures:

LocatingQuartilesSampleDatainOrderedArray:111213161617182122

Q1andQ3aremeasuresofnon-centrallocationQ2=median,isameasureofcentraltendencyDCOVA(n=9)Q1isinthe

(9+1)/4=2.5positionoftherankeddata,

soQ1=(12+13)/2=12.5Q2isinthe

(9+1)/2=5thpositionoftherankeddata,

soQ2=median=16Q3isinthe

3(9+1)/4=7.5positionoftherankeddata,

soQ3=(18+21)/2=19.5QuartileMeasures

CalculatingTheQuartiles:ExampleSampleDatainOrderedArray:111213161617182122

Q1andQ3aremeasuresofnon-centrallocationQ2=median,isameasureofcentraltendencyDCOVAQuartileMeasures:

TheInterquartileRange(IQR)TheIQRisQ3–Q1andmeasuresthespreadinthemiddle50%ofthedataTheIQRisalsocalledthemidspreadbecauseitcoversthemiddle50%ofthedataTheIQRisameasureofvariabilitythatisnotinfluencedbyoutliersorextremevaluesMeasureslikeQ1,Q3,andIQRthatarenotinfluencedbyoutliersarecalledresistantmeasuresDCOVACalculatingTheInterquartileRangeMedian(Q2)XmaximumXminimumQ1Q3Example:25%25%25%25%1230455770Interquartilerange=57–30=27DCOVATheFiveNumberSummaryThefivenumbersthathelpdescribethecenter,spreadandshapeofdataare:XsmallestFirstQuartile(Q1)Median(Q2)ThirdQuartile(Q3)XlargestDCOVARelationshipsamongthefive-numbersummaryanddistributionshapeLeft-SkewedSymmetricRight-SkewedMedian–Xsmallest>Xlargest–MedianMedian–Xsmallest≈Xlargest–MedianMedian–Xsmallest<Xlargest–MedianQ1–Xsmallest>Xlargest–Q3Q1–Xsmallest≈Xlargest–Q3Q1–Xsmallest<Xlargest–Q3Median–Q1>Q3–MedianMedian–Q1≈Q3–MedianMedian–Q1<Q3–MedianDCOVAFiveNumberSummaryand

TheBoxplotTheBoxplot:AGraphicaldisplayofthedatabasedonthefive-numbersummary:Example:Xsmallest

--Q1

--Median--Q3

--

Xlargest25%ofdata25%25%25%ofdata ofdataofdata Xsmallest Q1 MedianQ3

XlargestDCOVAFiveNumberSummary:

ShapeofBoxplotsIfdataaresymmetricaroundthemedianthentheboxandcentrallinearecenteredbetweentheendpointsABoxplotcanbeshownineitheraverticalorhorizontalorientationXsmallestQ1MedianQ3

XlargestDCOVADistributionShapeand

TheBoxplotRight-SkewedLeft-SkewedSymmetricQ1Q2Q3Q1Q2Q3Q1Q2Q3DCOVABoxplotExampleBelowisaBoxplotforthefollowingdata:

022233455927Thedataarerightskewed,astheplotdepicts023527XsmallestQ1Q2/MedianQ3

XlargestDCOVANumericalDescriptiveMeasuresforaPopulationDescriptivestatisticsdiscussedpreviouslydescribedasample,notthepopulation.Summarymeasuresdescribingapopulation,calledparameters,aredenotedwithGreekletters.Importantpopulationparametersarethepopulationmean,variance,andstandarddeviation.DCOVANumericalDescriptiveMeasures

foraPopulation:ThemeanµThepopulationmeanisthesumofthevaluesinthepopulationdividedbythepopulationsize,Nμ=populationmeanN=populationsizeXi=ithvalueofthevariableXWhere

DCOVAAverageofsquareddeviationsofvaluesfromthemeanPopulation

variance:NumericalDescriptiveMeasuresForAPopulation:TheVarianceσ2Where

μ=populationmeanN=populationsizeXi=ithvalueofthevariableXDCOVANumericalDescriptiveMeasuresForAPopulation:TheStandardDeviationσMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthepopulationvarianceHasthesameunitsastheoriginaldataPopulation

standarddeviation:DCOVASamplestatisticsversuspopulationparametersMeasurePopulationParameterSampleStatisticMeanVarianceStandardDeviationDCOVATheempiricalruleapproximatesthevariationofdatainabell-shapeddistributionApproximately68%ofthedatainabellshapeddistributioniswithin1standarddeviationofthemeanorTheEmpiricalRule68%DCOVAApproximately95%ofthedatainabell-shapeddistributionlieswithintwostandarddeviationsofthemean,orµ±2σApproximately99.7%ofthedatainabell-shapeddistributionlieswithinthreestandarddeviationsofthemean,orµ±3σTheEmpiricalRule99.7%95%DCOVAUsingtheEmpiricalRuleSupposethatthevariableMathSATscoresisbell-shapedwithameanof500andastandarddeviationof90.Then,Approximately68%ofalltesttakersscoredbetween410and590,(500±90).Approximately95%ofalltesttakersscoredbetween320and680,(500±180).Approximately99.7%ofalltesttakersscoredbetween230and770,(500±270).DCOVARegardlessofhowthedataaredistributed,atleast(1-1/k2)x100%ofthevalueswillfallwithinkstandarddeviationsofthemean(fork>1)

Examples:

(1-1/22)x100%=75%…..............k=2(μ±2σ) (1-1/32)x100%=88.89%………..k=3(μ±3σ)ChebyshevRuleWithinAtleastDCOVAWeDiscussTwoMeasuresOfTheRelationshipBetweenTwoNumericalVariablesScatterplotsallowyoutovisuallyexaminetherelationshipbetweentwonumericalvariablesandnowwewilldiscusstwoquantitativemeasuresofsuchrelationships.TheCovarianceTheCoefficientofCorrelationTheCovarianceThecovariancemeasuresthestrengthofthelinearrelationshipbetweentwonumericalvariables(X&Y)Thesamplecovariance:OnlyconcernedwiththestrengthoftherelationshipNocausaleffectisimpliedDCOVACovariancebetweentwovariables:cov(X,Y)>0XandYtendtomoveinthesamedirectioncov(X,Y)<0XandYtendtomoveinoppositedirectionscov(X,Y)=0XandYareindependentThecovariancehasamajorflaw:ItisnotpossibletodeterminetherelativestrengthoftherelationshipfromthesizeofthecovarianceInterpretingCovarianceDCOVACoefficientofCorrelationMeasurestherelativestrengthofthelinearrelationshipbetweentwonumericalvariablesSamplecoefficientofcorrelation:

whereDCOVAFeaturesofthe

CoefficientofCorrelationThepopulationcoefficientofcorrelationisreferredasρ.Thesamplecoefficientofcorrelationisreferredtoasr.Eitherρorrhavethefollowingfeatures:UnitfreeRangebetween–1and1Thecloserto–1,thestrongerthenegativelinearrelationshipThecloserto1,thestrongerthepositivelinearrelationshipThecloserto0,theweakerthelinearrelationshipDCOVAScatterPlotsofSampleDatawithVariousCoefficientsofCorrelationYXYXYXYXr=-1r=-.6r=+.3r=+1YXr=0DCOVATheCoefficientofCorrelationUsingMicrosoftExcelFunctionDCOVATheCoefficientofCorrelationUsingMicrosoftExcelDataAnalysisToolSelectDataChooseDataAnalysisChooseCorrelation&ClickOKDCOVATheCoefficientofCorrelation

UsingMicrosoftExcelInputdatarangeandselectappropriateoptionsClickOKtogetoutputDCOVAInterpretingtheCoefficientofCorrelation

UsingMicrosoftExcelr=.733Thereisarelativelystrongpositivelinearrelationshipbetweentestscore#1andtestscore#2.Studentswhoscoredhighonthefirsttesttendedtoscorehighonsecondtest.DCOVAPitfallsinNumerical

DescriptiveMeasuresDataanalysisisobjectiveShouldreportthesummarymeasuresthatbestdescribeandcommunicatetheimportantaspectsofthedatasetDatainterpretationissubjectiveShouldbedoneinfair,neutralandclearmannerDCOVAEthicalConsiderationsNumericaldescriptivemeasures:ShoulddocumentbothgoodandbadresultsShouldbepresentedinafair,objectiveandneutralmannerShouldnotuseinappropriatesummarymeasurestodistortfactsDCOVAInthischapterwehavediscussed:

Describingthepropertiesofcentraltendency,variation,andshapeinnumericaldataConstructingandinterpretingaboxplotComputingdescriptivesummarymeasuresforapopulationCalculatingthecovarianceandthecoefficientofcorrelationChapterSummaryBasicProbabilityChapter4ObjectivesTheobjectivesforthischapterare:

Tounderstandbasicprobabilityconcepts.TounderstandconditionalprobabilityTobeabletouseBayes’TheoremtoreviseprobabilitiesTolearnvariouscountingrulesBasicProbabilityConceptsProbability–thechancethatanuncertaineventwilloccur(alwaysbetween0and1)ImpossibleEvent–aneventthathasnochanceofoccurring(probability=0)CertainEvent–aneventthatissuretooccur(probability=1)AssessingProbabilityTherearethreeapproachestoassessingtheprobabilityofanuncertainevent:

1.apriori--basedonpriorknowledgeoftheprocess 2.empiricalprobability 3.subjectiveprobabilitybasedonacombinationofanindividual’spastexperience,personalopinion,andanalysisofaparticularsituationAssumingalloutcomesareequallylikelyprobabilityofoccurrenceprobabilityofoccurrenceExampleofaprioriprobabilityWhenrandomlyselectingadayfromtheyear2015whatistheprobabilitythedayisinJanuary?ExampleofempiricalprobabilityTakingStatsNotTakingStatsTotalMale84145229Female76134210Total160279439Findtheprobabilityofselectingamaletakingstatisticsfromthepopulationdescribedinthefollowingtable:ProbabilityofmaletakingstatsSubjectiveprobabilitySubjectiveprobabilitymaydifferfrompersontopersonAmediadevelopmentteamassignsa60%probabilityofsuccesstoitsnewadcampaign.Thechiefmediaofficerofthecompanyislessoptimisticandassignsa40%ofsuccesstothesamecampaignTheassignmentofasubjectiveprobabilityisbasedonaperson’sexperiences,opinions,andanalysisofaparticularsituationSubjectiveprobabilityisusefulinsituationswhenanempiricaloraprioriprobabilitycannotbecomputedEventsEachpossibleoutcomeofavariableisanevent.SimpleeventAneventdescribedbyasinglecharacteristice.g.,AdayinJanuaryfromalldaysin2015JointeventAneventdescribedbytwoormorecharacteristicse.g.AdayinJanuarythatisalsoaWednesdayfromalldaysin2015ComplementofaneventA(denotedA’)AlleventsthatarenotpartofeventAe.g.,Alldaysfrom2015thatarenotinJanuarySampleSpaceTheSampleSpaceisthecollectionofallpossibleeventse.g.All6facesofadie:e.g.All52cardsofabridgedeck: Organizing&VisualizingEventsVennDiagramForAllDaysIn2015SampleSpace(AllDaysIn2015)JanuaryDaysWednesdaysDaysThatAreInJanuaryandAreWednesdaysOrganizing&VisualizingEventsContingencyTables--ForAllDaysin2015DecisionTreesAllDaysIn2015NotJan.Jan.NotWed.Wed.Wed.NotWed.SampleSpaceTotalNumberOfSampleSpaceOutcomesNotWed.

27286313

Wed.44852Total31334365

Jan.NotJan.Total42748286(continued)Definition:SimpleProbabilitySimpleProbabilityreferstotheprobabilityofasimpleevent.ex.P(Jan.)ex.P(Wed.)P(Jan.)=31/365P(Wed.)=52/365NotWed.

27286313

Wed.44852Total31334365

Jan.NotJan.TotalDefinition:JointProbabilityJointProbabilityreferstotheprobabilityofanoccurrenceoftwoormoreevents(jointevent).ex.P(Jan.andWed.)ex.P(NotJan.andNotWed.)P(Jan.andWed.)=4/365P(NotJan.andNotWed.)=286/365NotWed.

27286313

Wed.44852Total31334365

Jan.NotJan.TotalMutuallyexclusiveeventsEventsthatcannotoccursimultaneouslyExample:Randomlychoosingadayfrom2015A=dayinJanuary;B=dayinFebruaryEventsAandBaremutuallyexclusiveMutuallyExclusiveEventsCollectivelyExhaustiveEventsCollectivelyexhaustiveeventsOneoftheeventsmustoccurThesetofeventscoverstheentiresamplespaceExample:Randomlychooseadayfrom2015

A=Weekday;B=Weekend; C=January;D=Spring;EventsA,B,CandDarecollectivelyexhaustive(butnotmutuallyexclusive–aweekdaycanbeinJanuaryorinSpring)EventsAandBarecollectivelyexhaustiveandalsomutuallyexclusiveComputingJointand

MarginalProbabilitiesTheprobabilityofajointevent,AandB:Computingamarginal(orsimple)probability:WhereB1,B2,…,BkarekmutuallyexclusiveandcollectivelyexhaustiveeventsJointProbabilityExampleP(Jan.andWed.)NotWed.

27286313

Wed.44852Total31334365

Jan.NotJan.TotalMarginalProbabilityExampleP(Wed.)NotWed.

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Wed.44852Total31334365

Jan.NotJan.Total

P(A1andB2)P(A1)TotalEventMarginal&JointProbabilitiesInAContingencyTableP(A2andB1)P(A1andB1)EventTotal1JointProbabilitiesMarginal(Simple)ProbabilitiesA1A2B1B2P(B1)P(B2)P(A2andB2)P(A2)ProbabilitySummarySoFarProbabilityisthenumericalmeasureofthelikelihoodthataneventwilloccurTheprobabilityofanyeventmustbebetween0and1,inclusivelyThesumoftheprobabilitiesofallmutuallyexclusiveandcollectivelyexhaustiveeventsis1CertainImpossible0.5100≤P(A)≤1ForanyeventAIfA,B,andCaremutuallyexclusiveandcollectivelyexhaustiveGeneralAdditionRuleP(AorB)=P(A)+P(B)-P(AandB)GeneralAdditionRule:IfAandBaremutuallyexclusive,thenP(AandB)=0,sotherulecanbesimplified:P(AorB)=P(A)+P(B)FormutuallyexclusiveeventsAandBGeneralAdditionRuleExampleP(Jan.orWed.)=P(Jan.)+P(Wed.)-P(Jan.

andWed.)=31/365+52/365-4/365=79/365Don’tcountthefourWednesdaysinJanuarytwice!NotWed.

27286313

Wed.44852Total31334365

Jan.NotJan.TotalComputingConditionalProbabilitiesAconditionalprobabilityistheprobabilityofoneevent,giventhatanothereventhasoccurred:WhereP(AandB)=jointprobabilityofAandB P(A)=marginalorsimpleprobabilityofA P(B)=marginalorsimpleprobabilityofBTheconditionalprobabilityofAgiventhatBhasoccurredTheconditionalprobabilityofBgiventhatAhasoccurredWhatistheprobabilitythatacarhasaGPS,giventhatithasAC? i.e.,wewanttofindP(GPS|AC)ConditionalProbabilityExampleOfthecarsonausedcarlot,70%haveairconditioning(AC)and40%haveaGPS.20%ofthecarshaveboth.ConditionalProbabilityExampleNoGPSGPSTotalAC0.20.50.7NoAC0.20.10.3Total0.40.61.