完本神战

Chapter 8 Other applications（第1页）

Chapter8Otherapplis

&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。

Buthowmightthepunterdeducehisdegreesofbeliefinthedifferentoutay2009，statistiDavidSpiegelhaltertookupthegeonBBoreorLessbyanalysiobeplayedinthePremierLeaguetwodayslater。Fame，heestimatedthenumberofgoalseachteammightsaverage，takingatitsthinattadtheiroppos’defensivecapabilities。Forexample，astroeam（Arseedtoals，aiy。