完本神战

Chapter 1 Fundamentals（第2页）

Specificexampleslaterwillillustratehowsubjectiveprobabilitieshavebeenassessed。Asidefromthumeleastthreedistineralapproaeisasthe　fair　priceforabetthattheeventwilloccur。Butthisdoesnotworkforeverybody:somepeoplehavepriiohersareunwillingtoplateaighteverleadtoaloss。Ahosewhodofeelfortablewithbetting，theirfairpricemightdifferagastowhichsideofthebettheywereon。

Asedwaytoassessyreeofbeliefiheobjectiveapproach。Whichofferwouldyouprefer:toreceive￡5iftheeventoccurs，ortoreceive￡5ifyuessthecolour，RedorBlack，ofthetopawell-shuffleddeck?Ifyoupreferthelatter，yreeofbeliefiisbelow50%。

Supposethatistheparetheprospect￡5iftheeventitifyuessthe　suitofarandomlydrawncard。Thelattershouldoe，soyourprefereellyreeofbeliefisbelow25%，orisbetween25%and50%。

&eparisonsalongtheseliyouhomeinonasituationwhereyouotsayonwhichsideyourpreferencelies。Yreeofbeliefiheheobjectiveprobabilityofthediiohanuseadeckof52cards，withitsawkwardfraightthinkofanurng20，ormaybe100，identicalballswithwhichtospecifythealters。

Giveyourahappropriatepre。TennisplayersJohnIsnerandNiahutplayedtheloWimbledonhistoryin2010;viag，thecethattheywetheragaiyear（ithappened!）isprecisely2iterrouo‘alittleunder1%’。Butitwasabsurdof　StarTrek’sMrSpocktotellKirkthattheoddsagainsttheireseepisoderoximately7，824。7to1’。

Forathirdmethod，thinkofamodestsumofmososmallthatyouaretotallyi（say，onepenny），nethatpossessingitwouldmakeadramatistances（￡1milliontomostpegerfates）。Forme，￡10fitsthebill–callthis　unitamount。Nowsupposethat，somehoworother，thetruthorfalsityoftheeventwillberevealedtomorrow:andyouwillreceivethisunitamountifitistrue，butzeroifitisfalse。Butratherthanwaitfortomorrow，youcouldreceiveadefiion　pofthisunitamounttoday。（Gettiodayortomorrowmakesoyou。）

If　pistiny，youarelikelytorejecttheoffer，andwillprefertowait;ifitisclosetounity，youmaytthatdefi。Buttherewillbesomeievalueof　pwhereyouareiweentakingthisoffer，andwaitietoberevealed。This　pisyreeofbeliefaboutthisstatemeiion。

Ioffermyowiveahefiveassertiohinkofnosensiblereasonwhyonesideshouldbemoreorlesslikelytowihaher，sureis50%。LookingatOscarhistory，notonlyforactorsbutalsotheories，theawardhasoallybeeedinsuccessiveyears:perhapstherearemoredidatesthesedays，leadiosuggest3%，iaedf，butée，foil，sabre，ahasappearedinalltheSummerGamesfrom1896。SomenativeofOslomighthavewoIstronglydoubtit–myfigurehereisabout95%。PrejudifavouroftheWhiteRosety，ratherthaiveevideosuggest10%forthefourthclaim。Forthefifthsideriesie，andthinkingofaplausibledivisioesNaderreceived，guidemetowards20%。

Pauseawhile，andmakeyestionsforthesefiveclaims。Thebetteryouareatassessingprobabilitieswhenmattersareuhemorelikelyyouaretobehappywiththedeakeinlife。

Odds

&herweusetheclassicalapproach，orfrequencies，reesofbelief，theterm　oddsisoftenusedwhendesgprobabilities。

&saythattheoddsofobtainingaSixwithafairdieare‘fivetoo’–foreverytimewegetaSixihrows，weexpeottodosofivetimes。Ifanouteisexpeorelikelythannot，suchasthehigherrankedplayerwich，thateventissaidtobe　oddson。

Thereisadeweenprobabilitiesandodds，acheasilybetweehinkingoffrequenhelp。Iftheprobabilityis20%，oroh，weexpecttheeventtooeooutoffive，sotheoddsare‘fourtoo’。Foraprobabilityof75%，weexpectittooccurthreetimesoutoddsof‘threetooneon’。Andiftheoddsarestatedassixtofiveagainst，thisiforeachfivetimestheevefailstodososixtimes，soitsprobabilityis511。

Youdoibers。Theprobabilitythatthetopawell-shuffleddeckiseitheraKingoraQueeakenas213。Thiscouldbequotedas‘eleventotwoagainst’or，equallyaccurately，‘Fivepoioo’。Usewhicheveryoulike。

Althoughthephrase‘theoddsareooone’iserfese。Itiaisexpectedtohappenjustasoftenasnot，soitsprobabilityisonehalf。Ihastraightface，wesay‘theoddsareevens’。

&oresolve

Therearenoimportasabouthoithprobabilities，butadherehreeapproacheswehavedescribedmaydeducetheirvaluesiectivehasitsuses。Iouaillappealtowhicheverviepropriate。

&iveapproachislimitedtogfinitelymanyoutes，alljudgedequallylikely。Butnoordieisperfectlysymmetridonwhatbasiswedismissitsimperfesasirrelevant?weeveweagreeonthenumberofpossibleoutple，supposewearetoldthatanurnstwoballs，eitherbothWhite，bothBlaeofeachcolour。Shuewehave　threeequallylikelycases，orthattherearereally　fourequallylikelycases，arisiheballswereied，iherasorBB?ThesedifferentoutlookswouldgivedifferehecethatbothballsareBlack。Orsupposeyoureacharoadjunwiththreepossibleexits，twoofthemleadihethirdta‘randomchoice’，istheforSeaportohird（ohree），oronehalf（owodestinations）?

&seekstodealwithcesthatarerepeatableieiditions。TheesbefihinkoftossingthesametilHeadsappearthreetimesinsu，orselegarandompointonastick。But，howevermuchcarewetake，theexperimentalsotbe　absolutelyidentidanylimitingvaluelybeestimated。Howshouldtheerroriebedescribed?gthattheprobabilityisatleast99%thattheerrorisunder2%requiresacircularargumeoknorobabilityis，iodefi!

Forquestionssuchastheprobabilitythatoryiher，orthecethataparticularhearttransplantissuccessful，thecesariseonly，aivesotbereducedtoafiofequallylikelycases。Theobjedfrequencyapproachesaresileers。Asubjectiveapproachisrequired。

Asubjectivistmusteherbeliefsaretwitheachother。Forexample，iionalLottery，amaeseleumbersfromthelist{1，2，3，…，49}，

andSusiemaybettotakeall14millionorsopossibleselesasequallylikely。Then，whenaskedwhichismorelikely，

（a）thatnonumberdrawnexceeds44，or

（b）thatthosedrawndonotiwoseumbers，

&eralittlethought，edownoheother。Butifsheselects　eitheroftheseeventsasmorelikelythaher，shewillbeguiltyofincy，asprshowsthattheyoexactlythesamenumberofways!Nothiiveapproachspecifieshowsucyshouldberesolved，merelythatitmustbe。

Becausewewishtothinkaboutprobabilitiesinceswiderthaherearefinitelymanyequallylikelychoidincesthatotberepeatedieakethesubjectiveapproachasthedefaultoption。Butwearelikelytoholdmorefirmlytoouropiheyarebackedupbyeitheraive，orbyafrequency，argument。

&ions

Usingthe‘ballsinabag’viewpoint，theprobabilityofsomeeveheproportionofRedballsinthebag。SoavalueofzeroolyiftherearenoRedballs，inwhichcasetheeventwillneverhappen。Similarly，aprobabilityofuoeveryballbeingRed，soheretheeventoccurseverytime。Thesevalues，zeroanduheoclusivelyprbyexperimeheevesprobabilityotbezero，ifitfailstohappen，itsprobabilityotbeunity。Andthisistrueforthefrequency，orsubjectiveapproachesalso。Sosupposetheprobabilityhassomeievalue，say34。

&disposeofonefiniatterhowwellaroulettewheelhasbeeisphysicallyimpossiblethatalltheshave　exactlythesamece。Whattheorequiresisthatthecesareoughtotheidealthatitisihatanynumbercouldbepioreorlesslikelythananother。Similarremarksapplytodis，oreheprobabilityis34’willmeanthattheprobabilityisoughto34forallpracticalpurposes。Otheredantmightsmuglytellyouthatheknorobabilityisnot34，withoutfearoftradi。

Iofrepeatableexperiments，whatdoweexpetheclaim:‘TheprobabilityofaRedballis34’?Emphatically，wedoifweductthisexperimentfourtimes（replagtheballdrawnoneacho），weshalldrareciselythreeofthem。Itispossiblethatfourrepetitionsthrowupall，oreventhatRedhappeime。Butseriesofrepetitions，wedoexpecttheoverallfrequeobecloseto34。

Therearenoblackhatstitutesalongseriesofexperiments，nortohowcloseto34isacceptable。IfIobtainedRedonly20timesi40repetitions，Iwouldhaveverystrongdoubtsaboutaclaimthattheprobabilitywas34;butthosedoubtswouldbelargelyassuagediftheionsgave28Reds。Believingordisbelievingthisbeaprovisionalpositionforquitesometime。Assumialaihroughout，useallthedatacollectedtoreachade–shortrunsmislead。

Iwilluidelines，andjustifythemlater。Takethewemakeoions，andthesupposedprobabilityissomemiddlingvalue，nearonehalf。putethedifferehisfigureaualfrequendata:ifthisdifferenceexceeds0。1，Iwouldhavesomedoubtsaboutthedifitexceeded0。15，Iwouldhavestrongdoubts。Withathousaiohanahuclreemehosenumbersby0。03and0。05。Ifthesupposedprobabilityisclosertozeroorunity，say10%or90%，Iwouldalsorequirebetteragreement。Itbemucheasiertobethebasisofrepeatedexperiments，thataparticularprobabilityis　notsomeallegedvalue。

Whataboutasubjectiveassessment，suchasthattheprobabilityofraintomorrowis60%?Weotrecreatetoday’sweathershuimes，andcheckhowoftenitrains。This‘experimeedonly。Butwemighttesttheclaimbylookingattheprocessthatledtoitbeingmade。Forecastersusemodelsofatterheirs，ahefigureoersis31。067%，theysensiblyures。Youhear‘Theisabout30%’。Sonowyoucollectdatafordifferentdays，andlookattheempirianyofthe83dayslastyearwhentheutat30%diditactuallyrain?Solongasthatproportionwasreasonablycloseto30%，yourbeliefihodisreinforced，soagthefiguregivenfor‘tomorrow’isarationalresponse。

Probabilityisthekeytomakingdedersofuy。Ifyouhohattheprobabilityofapartitorstatementisunity，youshouldactasthoughitwilldefinitelyodifyourhotheprobabilityiszero，actasthoughitotoccur。

Ifyouthinktheprobabilityissomevaluebetweenzeroaasthoughyouexpectittooccurthatproportioime。Forexample，ifyementisthattheprobabilityis60%，imagiyouwillfacethissituatioimes，insixtyofwhich（butyouhavenoideawhichsixty）thiseventen，andfortytimesitwillnot。Swallowhard，anddeyoura，takingintoatthisbalance。Hadyoujudgedtheprobabilitytobe80%，sothatyouexpecttheeventtohappenrathermoreoften，youraightwellbedifferent。

AsBishopJosephButlerwroteinhis1736Analion，‘Tous，probabilityistheveryguidetolife’。