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Chapter 5 Making sense of probabilities（第1页）

Chapter5Makingsenseofprobabilities

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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.