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Chapter 4 Chance experiments（第1页）

Chapter4ceexperiments

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。

Thislumpsistsofagigaoms，alladepely。Fo:atsomeradecaysbyemittingaparticle。Wedonotkhiswillbe，butsiheatomsinthelumpde22years，thecethat　thispartidecayseriodis50%。Supposeithaserfiveyears:atthattime，itisjustoomintheresiduallumpof　210Pb，sotheceitdecayswithinafurther22yearsisagain50%。Andifithashehesameapplies，andsoon。