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Chapter8Otherapplis
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&ionsofprobabilityawayfromdibling，andvariousaspeaturalsbeoverlooked.Inthischapter，Ihavepieofitsappearanlaw，soce，sport，andeiphasizeitsubiquity.Theeisthatdeakewilldependontheprobabilitiesofvariousoutes，sowehatleadtoreasonablyreliableestimatesofthosedifferentprobabilities.
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Alth，o-knowhe20thtury，hadamathematicsdegree，fewlawyersfeelfortablewithprobability.Thisoughttobeastonishing，asphrasesrelatiareusedfreelyincivilcases，suchaslibel，tosay‘onthebalanceofprobabilities’clearlyputsthedividi50%.Butinalcases，whereajuryisaskedtolyiftheyare‘sure’ofGuilt，thereisnosusonafigure.Somepeoplewouldwishtovictiftheywere8uilt，otherswoulduse95%her.Theseareplaiiveprobabilities.Andalthoughthesamephraseisusedwhatevertheoffenelyalowerthresholdofproofforarelativelymihisakeithardertoassmurderersthanfaredodgers.
SupposeahattheDNAoftheaccusedmatAfoundatae，andthattheatthelatterandapersorandomisoneinseveralmillion.JurorsmayhavetwodistihisstatemeisthattheymaythinkthatitisequivalenttosayingthatthecetheeDNAisNOTthatofthedefendantisoneinseveralmilliohattheymaytreatallsuyfiguresasequivalehoughoenmilliondiffersfromoneinabillionbyafactorofahundred.
&errorhasbeeheProsecutor’sFallacy’.Starkly，itisequatingtheoaDNAmatch，totheatIhisislogise:theceivenafairroulettewheel，isnotthesameasthecethatthewheelisfair，giventhatzerooccurred.ThistrapbeavoidedbygivingthejuryaeofhowmamatchtheeDNA.ulationofaround60millioillihtbethirtyorso;ifitis1in20millihtbeaboutthree;itisuherearemorethanhalfadozen.Butdohephrase‘selectedatrandom’:themorecloserelativesthealhas，themorematcheswewouldexpect，thelessstrongthisevidetheaccused.
ThesederrorbestbeavoidedbyrememberinghowBayes’Rulemeasurestheusefulnessofanypieceofevidehisevideed，youhavesomeideaoftheoddsthattheaccusedisGuilty.IftheevideimesmorelikelyuhahentheoddsofGuiltgetmultipliedbytehatisthreetimesmorelikelyunderIhaheoddsbyafactorofthree，andsoon.WithDofteheceoftheevidengGuilt，is100%，whichmakestheimpactoftheevideheoddsofGuiltshouldbemultipliedbywhateverthe‘severalmilliouallyis.
Randomizedresponse
&eacherwishestoasortionofhissessmokeabis.Directquestionsareuoproducetruthfulaateownasrandomized responseisavailable.Themaitheteacherregtheaknowwhatquestionisactuallybeihatabisusersareabletoahidentified.
Thewords‘Ismokeabis’arewrittenoneachof80d‘Idonotsmokeabis’onanother20.Eachcardisplaidentivelope，these100envelopesaremixedthhlyihestudentsshouldseethisoperationbeitheyknowthatthebagsbothversioion，andiions.
&sora，readsthequestiontoherself，andsimplysayseither‘Agree’ree’.Shethecardbatheeurothebag，ahebagupreadyforthe.
Supposethatohirdoftherespree’.Becausethestudentsarepigtheerandom，‘Agree’isthehoresponsefrom80%ofusers，and20%ofnon-users.Afewlinesofalgebrashowthatthisistwith29ofthestudentsbeiheheadteacherhashisanswer，noindividualstudeified.
&ively，replaokeabis’withaquestionoedsubject，forwhichtheproportionof‘Agree’answersisknorevioussurveyhasestablishedthathalfthestudentsoet，andthereisoli-owningwithabissmokiementon20cardscouldbe‘Ioet’.Thehirdoftherespree’，weestimatethat724ofthestudentsareusers.
&ionsgiviesareintheAppendix.
&aintyastowhichquestionisbeiimeleadstosomeimprethefie.Theproportiohattheseionshouldbeashighaspossible，butlowenenuineabisuserstobelievethatgivianswerwillnothaverepers.Puttiivequestiononasmanyas95%ofthepostcardswouldnotwork.
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TheingAgeopromotesportasahealthyactivity，byidentifyiakeperformangdrugs，andexgthemfrompetitions.Butwhatevermethodsareused，aingprogrammeisliabletotwoopposingtypesoferror:gthataeisusingdrugswhe，andpassingaeastheyareadruguser.
Uhtheakiypeoferrorofteoiheceoftheother.Forexample，oheratiooftestosteroerohebodynormallyproducesthesesubstanfairlyequalamounts，butthoseathleteswhoseektocheatbyiestosteronewillhaveahighTEratio.Athleteswhoseratioisabovesomespecifiedamount，saysixtoone，willbebaheTEratiovariesnaturally:itenstrualcycle，itwillincreaseifyoucatchflu.SetthecriticalTEratiotoohighandncheatwillfailit;setittoolow，andmaeswillbewronglyaccused.
Supposethecethataparticulartestmakesamistakeis1%.Thatmeansthatiftheathleteisihecetheyfailis1%，iftheyareusersthecetheypassisalso1%.Samfailsthetest:whatisthecesheisi?
Putlikethat，thetemptationtosay‘1%’l–thistestgetsthingswroimeinahundred，soifitsaysshehasfailed，thatwillbewroimeithistemptation.Theonlyvalidansweris‘Wedonotknow.Itcouldbeanyfigure.Weokithepopulation.’
For，supposethatproportionis1%orso.Thenamong10，000athletesweexpect1d9，900iestijustcheattopass，leaving99whofail.But1%ofthe9，900is，i.e.ahletes，willalsofailthetest.Amongthosewhofail， halfareiheceSamisiwouldbe50%.
Iftheproportiosdiffersfrom1%，thisges.Ifitishigher，thecethatSamisiwillbeless，butifitislower，herocewillbeevehelowertheproporticheats，thelesssatisfactoryisthistest，despiteitsapparentlyimpressiveperformance.
Thissamelogicapplieswhenwesiderhowtodetetialterroristsatairports.Whateversgdevicesareused，theyotbeperfect，butsupposethattheprobabilityarealterroristevadesthesechey，110，000，whiletheipersonisledawayforierrogationisaminuscule1100，000.Howlikelyisitthatsomeoisguilty?
&aionwithouthavingsomeideaoftheproportionofwould-bepassengerswhoareterrorists.Tryoneinamillilyhigh，givehrowhaymillionpassehefiguresassureusthat，evenwithfiftypotes，itislylikelythatallwillbedetected.
Unfortunately，fivehundredipassengerswillalsobedetained!Amooppedbythissystem，fewerthaerrorists.Andiftherearefewerthas，theeonewhoisstoppedisiyiseveiohodsmusthavemuchbetterperfuresiftheyaretobeuseful.
Footballresults（1）
&isofsoccermateratesasubstaheUK.Allsortsofexotibemade–ohefirstthrow-iheshirtnumberswornbyallthegoal-sanyredandyellowcardswillbe?ourishedduringthegame–butmostiisonwhichofthethreeresults，Homeaywin，willoccur.Arationalpunterwillassesstherespectiveprobabilitiesoftheseresults，andhisdewhethertobet，andhowmuch，willrestontheseassessmehepayoutpricesofferedbythebookies.
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