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Chapter 1 Fundamentals（第2页）

Thefiveassertionsabovehavedifferentnatures，andwehavedifferentkindsofevidehem.Forthefirst，ealtosymmetrybetweeails.Forthesed，wehavethehistoryoftheOsce1929toguideus.Inboththesecases，thetruthorotherwiseofthestatementwillbeeknownwithihethirdiseithertrueorfalse，andcouldbeestablishednowbyathhtrawlofOlympicrecords.Thefourthisalsoeithertrueorfalse，butwewillneverknoeotrerunhistorytoasthetruthorotherwiseofthefifthclaim.

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’.