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Chapter 5 Making sense of probabilities（第1页）

Chapter5Makingsenseofprobabilities

Iwillsuggesthowprobabilityideashelpinmakingdethefaty，andalsodescribeceswheremisuandingsarise。

Odds?

&probabilitiesbeexpressedintermsofodds，andviceversa:aprobabilityof15isthesameasoddsof4to1against。Uerm‘odds’hasalsobeehegamblinguhingquitedifferehebookieswillpayifyourselectedhorsewins。SowheSeaTheStarswonthe2009Derbyatoddsof11to4，thatsimplymeansthatforeach￡4stakedoheprofit，becauseitwon，is￡11。Thefigures‘11to4’havenoautomatishipwiththeprobabilityofwinning。Theydependonthebookies’subjectiveassessmentsofthehorse’sublershavestaked。Theterm‘payoutprice’isamoreaccurateuseoflahesefigureslike11to4，but，regrettably，wehavetoaccepttheonusageof‘odds’inthisgambli。

Apayoutpriceistermed　fairifitgivesaryadvaherparty，i。e。themeahegambleiszero。Thefairpayoutpriceforcorrectlypigthesuitofacardseleawell-shuffleddeckis3to1，asthosearetheexactoddsagainstacuess。

ercialgamblesarenotfair，iheyeveroperatewithoutahouseadvantage。ForrouletteinaUKo，whenall37outesareequallylikely，thepayoutpricefonasinglenumberisonly35to1，o1thatwouldbefair。Sothe　meaurof￡37is￡36，giviage–thepertageofaexpe–of137，about2。7%。

Thisadvahesameformostoftheavailablebetsiheryonpairsofriples，groupsoffour，orsix，ortwelve，forevery￡37youstake，yourmeaurnisalways￡36。Butihestandardhouseadvantageisbigger，becauseofanadditionalslot，d38outes–withthesamepayoutpritheUK。Themeaurnon$38isgenerally$36，ahouseadvantageof238，or5。3%。

AdifferentwaytobetonhorseracesisthroughaToteorpari-mutuelsystem。Here，themoonallthehorsesispooled，aion–80%orsoison–issharedamongthosewhobackedthewinner，iioakes。TheToteadvahen20%，whicheverhorsewins。

Thesizeofthebookies’advantageinahorseradsonwhichhorsehappenstowin。Althoughbookiesmaymakealossorprofitonanysidatatellasstory:atapayoutpriceof6to4on，puntersshouldexpecttoloseabout10%oftheirstake;atapriceof5to1，expecttoloseabout13%;at10to1themeanlossfigureisover23%，andifyouspehorsespricedat50to1，expecttoloseabouttwo-thirdsofyourmoney。

Thisphenomenonisknowe-longshotbias。Puheirmoney　moreslowlyfrombetsonthemorefavouredhorsestharactedbylargepayoutpriakersweredelightedwheheGrandNationalin2009atapriceof100to1。

Absoluterisk，orrelativerisk?

Supposethat，amroupofpeople，theceofdevelopialceroverthefiveyearsisquotedasohousaabouttenamong10，000peopletodeveloptheewdrugwouldreducetheeintwothousand:thenonlyaboutfiveamong10，000wouldsuccumbifthenewdrugisused。Thedrugpanycouldheadlispressrelease‘Riskofcercutby50%’。Andthatisaccurate:forea，theriskwouldbehalved。

Thisapproachdescribesa　redurelativerisk，aicizedasputtingtoofavourableaioa。For，supposetheinitialriskhadbeeenmillion:gitby50%leadstoanewriskofoymillion，butiaheriskissosmallthatamong10，000people，wewouldexpectprettymuchthesamenumberofcases–zero。Despitetheriskbeihedrugwouldhardlyevermakeadifference。

Butsupposemembersofthisgrouphadamuchhigherce，say40%，ofdevelopingthecer。Adrugredugtheceto20%wouldqualifyforthesameheadline，andwouldcorrectlybehailedasamajh，asamong10，000people，fully2，000fewerwoulddevelopthecer。

&hanfotherelativerisk，itisusuallymfultolookattheabsoluterisk。Icaseabove，theabsesfrom0。1%to0。05%，sothedropis0。05%;inthesedcase，thedropisaminuscule0。000005%，whilewiththefihedropisanimpressive20%。

Aseoproceedistostatethemeaientswhoshouldtakethedrugiopreventohedisease–thereat，orhebetter，andthisthereciprocaloftheabsoluterisk。Intheexamplesabove，therespeTsaretwothousaymillion，andfive。

&wentymillioopreventonecaseofadiseaseishardtojustify。TheNNT，alongwithkreatmentdtheseverityoftheimpactofthedisease，allowsustomakesensibledesaboutalloghealthcareresources。

biningtinyprobabilities

Howlikelyisitthatatleastoneamonganenormousnumberofevents，eagatinyprobability，willoccur?Thisbethepertiionwheheprobabilityofaplexsystemsormaerymayfailifanyoneofamyriadofposfails;willtwoaircraftightanuclearpowerstatiodown?Theso-calledBorel-telliLemmasgivesomepoihematicalresultsshowthat，inmanyces，thekeyquantityisthe　sumofallthosetinyprobabilities:ifitgrowswithoutbound，catastropheis。

Oneceisthatweeverbesatisfiedwithtsafetystandards。Itisessentialtouetomakeimprovements。

For，nhourstandards，thereissomenon-zeroprobabilityah:ahisvalue，ifitremainsunged（oreveooslowly）thesumovermanymonthswillgrowielylarge，anddisasterwilloe。

Asrammeofualimprovemeguaradisasterwillbeaverted:buttobeeversatisfiedwiththestatusquoistoinvitedoom。

Somemisuandings

（a）Whenadoctortellsapatientthatthereisa30%cethataparticularmedileasas，hemeasabout30%ofpatientstosuffer。However，thepatiehattheseeffectswillariseonabout30%oftheosonwhichshetakesthedrug。Thedoctoristhinkingaboutallthepatiehepatientaboutallthetimesshetakespills–theirreferenceclassesaredifferent。

（b）Howdoesthepubliterprettheclaim‘Thereisa30%inorrow’fromaTVweatherforecaster?Theforecastersexpecttheiraudieomakeafrequeio，inthelongrun，rainwouldfalldayon30%oftheoswhehersweresimilartothosenowseen。

Butwheiohosevieerehappywiththephrase‘a30%ce’，therereadofbeliefs。Somefeltthattheywerebeingtoldthatrainwouldfallover30%ofthecity’sarea;othersthatitwouldraininChicagofor30%oftheday;ahat30%istsexpectedittoraithatitwoulddefinitelyrain，withthe30%figureindigtheraihereweremanymismatchesbetweeheforecasterswerereferringto，aviewerswerethinkingabout。

（c）Ifasfewastwenty-threepeopleatraher，itismorelikelythannotthattwoofthemshareabirthday。Whehisfactforthefirsttime，theyarenormallysurprised，butusuallybeeprshowsthisclaimtobetrue。However，aminorityremainunced，becausetheymistakenlythinktheyhavebeentoldthat，iftheyawathertogether，itismorelikelythannotthatohersshares　theirbirthday。Listencarefully!

（d）Supposethata，acceptedasfair，showsTailsoivetosses。SomewillclaimthattheossisalmosttobeHeads，perhapsbyinvokingsome‘Lawes’thatrequiresHeadstoeatintothisexcessofTailsimmediately。NosuchLawexists。TheLaweNumbersdoesimplyequalproportioails，butonlyinthelongrun:anysequeneTailsisdilutedbythethousandsoftossesbeforeandafter。

&ively，somewill（correotethatthesecutiveTailsislessthahousand;soiftheyseeails，theymaytheailsimeishighlyuhatisfalselogic:ifeverwehaveailsinarow，atenthTailehisoftheabsoluteprobabilityofahitsalprobability，giveainbaouslyarosein1996:jokieDettorirodethewihefirstsixracesatAsdsinehadeverwonallsevenraaday，it‘must’bevirtuallyimpossibleforDettoritowirace。Buthedid。Veryfewpeoplewillridethefirstsixwinnersinasevewhensomeonedoesso，hemightwellalso>