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Chapter4ceexperiments
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Thinkofawithes–buyi，bettingonahoingonablinddate，undergoireatmeheword distributiontospecifyallthepossibleoutes，alongwiththeirassociatedprobabilities.（WeslippedinthatritingaboutPoisson’sanalysisofhowmasen，givenalargenumberofopportunities.）
&ributiooanalysingtherangeofaceexperiment.Plaiobeclearaboutthefullextentofthepossibleoutes.Togivesensiblevaluesfortheirprobabilities，wemustspelloutourassumptions，aheyareappropriatefortheexperimeoie.
Discretedistributions
First，welookatceswherethepossibleoutesbewrittenasalist，eaehavingitsownprobability.Thephrasediscrete distributionapplieshere.
&raightforwardcaseiswhenwetthees，aheyshouldallbetakenasequallylikely.Thetermuniformdistributiohetotalprobabilityisspreaduniformlyovertheoutasareexpectedtofitthisbill–roulette，didsofcards，selegthewinningnumbersic.Accurateggeheappropriateanswer.
&heterm‘Bernoullitrials’tomeanasequendepeswithatprobabilityofSuccesseachtime.WithafixednumberofBernoullitrials，thereisasimpleformula，calledthebinomialdistribution，thatgivestherespectiveprobabilitiesofexactly0，1，2，...Successes.Thisformuladependsonlyontherials，andtheSuccessprobability.Asyhtheoutesiheirprobabilitiesinitiallyioamaximumvalue，thenfallawaytowardszero.（Poissondistributionsalsofollowthispattern.）
&abinomialdistributionforthenumberofSixesamohrowsofadie;orthenumberofswerswheguessesrandomlyamongfivechoicesateachofthirtyquestionsonamultiple.Butwedo whenaskinghowmanyClubsabridgeplayerhasamonghisthirteencards:althougheachseparatecardhasprobabilityoerofbeingaClub，successivecardsare，astheceofathecardisaffectedbyallpreviousoutes.
Alwaysreadthesmallprint.Abinomialdistributiohrees:afixedrials，eade，andwithatceofSuccess.
Inasequerials，whatistheceittakesexactlyfivegoestoachievethefirstSuccess?TheonlywaythishappensistobeginwithfourFails，thenhaveaSudsirialsareiheanswerultiplyiiveprobabilitiesoftheseoutestapleasinglysimpleexpression，theso-etric distribution.
Theprobabilitiesoftakily1，2，3，...trialsforthefirstSuccessdecreasesteadily.Eachtime，theprobabilityultiplyivaluebytheoreFail，somefixedvaluelessthanunity.Thus，whatevertheceofSuccess，thesilikelyrialstoachievethefirstSuccessisalwaysunity!
Makealeapoffaith，a，i，successiveballsformBernoullitrials.Abowler，whois‘Success’asmeaniakesawithinkoptimistically:wheobowl，thesilikelytimehewilltakehishthedelivery.versely，abatsmanwhotakesasimilarviewmustfatalisticallyacceptthatthemostlikelydurationofhisinningsisthathefacesjustoneball.（Evesmen，rethattheirsilikelytotalscoreisusuallyzero!）
4.Someoributions
Figure 4illustratessomeoftheoributions.Foreachpossiblevalue，theheightoftheverticalbargivesitsprobability，andthesumofalltheheightsisalways，ofity.
uousdistributions
Howmightweextendtheclassicalideasofprobabilitytodealwiththeexperimentofgarandompointonastigth80cm?Herethereisaofpossibleoutes，notjustalist.
‘Atraallindividualpoihesameprobability.Butifthatooexceedzero，then，bytakingsuffitlymanypoialprobabilitywouldexity，whichisimpossible.Eachseparatepointmusthaveprobabilityzero，andwegerusepictureslike Figure 4.Ratherthanassociateprobabilitieswithindividualpoioassociateprobabilitieswithsegments，orintervals.
Togiveequaltreatmentaloick，allsegmentshavingthesamelengthmusthavethesameprobability.Imaginegthestitoeightequalpieces:a‘random’pointmust，bydefinition，fallihthesameprobability，so，forexample，thesegmentfrom20usthaveprobability18.
Figure 5ashowshowtoprogthemantra‘Arearepresentsprobability’.Theheightofthehorizontallinelabelled hissothattheshadedareabeliy，represehatitis100%thattherandompointfallsstheintervalfrom0to80.Then Figure 5bshowshowtofindtheprobabilityoffallifrom32，bygthedingshadedarea.Plainly，thisis14.
Tofindtheprobabilitythataraedpointiswithin10cmofeithereick，orwithin10cmofthetre，wecoulduse Figure 5dappealtotheAdditionLaw.Therequiredprobabilityisthesumofthethreeshadedareas，namelyonehalf.
Figure 6illustratesasimilarpathforothersituatioetakesuousvalues，suchasthetimeuaaparticularstretotorillarguebelowthatthegeneralshapeofthecurveshownisreasohissituation，butthemainpointisthatthescaleissothatthetotalareaabovetheliime’，butbelinniE，isunity，asitis100%thatthetimetowaittakessomeivevalue.
5a.Theshadedareaisunity
5b.Theprobabilityoffalliween32and52is14
5c.Seetext
6.Auousdistribution
TheprobabilitythatthetimeisatleastB，buthanC，isthesizeoftheareashaded.Inasimilarfashion，wedtheprobabilitythatthetimetowaitfallsierval，andthen，usiionLawasabove，theceitfallsinmoreplexregions.
Acurvethatgeesprobabilitiesinthismanneriscalledaprobability density.Nowareaiscalculatedas‘leh’，ahofanylineiszero.Hehe‘area’ofeitherofthevertiesatAure 6iszero，soboththoseindividualpointshaveprobabilityzero，asbefore.ButthedensitycurveishigheratAthanatD，sovaluesnearAaremorelikelythanvaluesaglaheFigureiheregioivelylhprobability.Thetermuousdistributionisused.
Inallsuchexperiments，sindividualpointshaveprobabilityzero，webealittleslipshod:whetheranintervalihendpoints，justohem，orheprobabilitytheoutefallsinitisthesame.
Toqualifyasaprobabilitydensity，acurvemusthavetwoproperties:itottakeivevalues，aalareaumustbeunity.Thiseallcalsofprobabilitiesleadtosensiblevalues.
Manyprobabilitydensityfunsariseoftenenoughforthemtobegiveheexperimeingarandompointwithierval，thedensityfunwillbepletelyfiatoverthatinterval，asinFigure5:plainly，allsegmentsofthesamelengthdoihesameprobability.Agaiermuniformdistributionisused.
Supposeweareihetimetowaitforsomespet.Forexample， 210PbisaopeofLead，andtheclaim‘Itshalf-lifeis22years’appearsinphysieaningisthat，wheakealumpofthissubstanlyhalfofitisuer22years，theresthavihersubstahroughradioactiveemission.
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