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Chapter 8 Other applications（第1页）

Chapter8Otherapplis

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&ionsofprobabilityawayfromdibling，andvariousaspeaturalsbeoverlooked.Inthischapter，Ihavepieofitsappearanlaw，soce，sport，andeiphasizeitsubiquity.Theeisthatdeakewilldependontheprobabilitiesofvariousoutes，sowehatleadtoreasonablyreliableestimatesofthosedifferentprobabilities.

&ers

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.