完本神战

Chapter 1 Fundamentals（第1页）

Chapter1Fuals

Thescopeofprobability

Probabilityistheformalizatioudyofthenotioy。Theeffectsofblindceareapparenteverywhere。Biologically，weareallarandommixtureofthegenesofourparents。Catastrophes，likeoilspills，voloeruptions，tsunamis，orearthquakes，assuinglotteryprizes，randomlyanddramatigepeoples’lives。

Manypeoplehaveagoodiandingofprobability。Butthisuandinggoastraywheialideaaboutthelikelihoodofsomething，butthe，whoserelevaarehereareiorious‘trickquestions’，aboutbirthdays，orfamilieswithtwo，ortelevisiongameshowswiththreechoices，thatseemtohavebeeopersuadeyouthatthesubjectdefiesodoesnot。Solongasaioionsareflushedout，aof，sensibleaprobabilitydoesrequireclearthoughtprocesses。

Thedevelopmentofitsideasahodshasbeeslicability。TheD-DayinvasionofNormaaheadinJune1944oheprobabilityoffavourableweatherwasdeemedsuffitlyhigh。Eheherlandsmusttakeatofthecesofseverefloodswhehedykesthatprotecttheirtryfromthesea。Isareatmehahodstoeienttosurviveforfiveyears?Howmuchyoupaytoinsureyourlife，car，house，orpossessioheearlygmade。Mostdeake–whattostudyatschool，whotoselectasalifepartolive，whichcareertofollow–aremadeuioy。AsPierre-Simoein1814:

&importaionsihemostpart，onlyproblemsinprobability。

&hephrase‘theprobabilityis…’appears，someassumptions（thatmayilyhavebeenomitted）arebeihoseassumptionsareulerelianceshouldbeplatheclaim。Ihopethat，inthisbook，theseassumptioherimplicitlyorexplicitly。Beforerobabilitystatemeerpreted，wewilldescribedifferentwaysinwhichtheymayarise。

&iveview

The　classical，or　objective，viewofprobabilityisthatoftenusedduringgamesofce，suchasrollingdice，roulettewheels。Thereissomelistofoutes:theherfromsiderationsofsymmetry，orbecauseweogoodreasohemtooccurratherthaakethemallasequallylikely。Sowejusttthees，ahemallthesameprobability。Thentheprobabilityofaheexperimehe　proportionofoutesthatfavourit。

Forexample，whenaisthrowhefourpossibleHeadTailoutesareHH，HT，TH，TT。Witha　fair，HorTwillbeequallylikelyeachtime，sohosefouroutoreorlesslikelythanahers，eachshouldhaveprobability14。ThreeofthemHeadsatleastoheprobabilityoftheeventthatHeadsappearsatallis34。

Thereare1，326waysofdealingahandoftwocards。（Takemywordforit。）Ifthedeckhasbeenwellshuffled，wetakeallthesehandsasequallylikely。And64ofthemsistofaen-card’（i。e。Ten，Ja，），sowecludethattheprobabilityofbeisud–‘Blackjack’–is641326，justunder5%。

Sofarasprobabilitysiderationsareed，boththeseexamplescouldbereformulatedintermsofgoneballfromabagofidenticalballs。Thefirstbagwouldhavefourballs，threeofthemRed，thesed1，326balls，64ofthemRed。Indeed，everyexampleinthisobjectiveapproachtoprobabilityisessentiallyidentieproblemaboutseleeballfromsomebagorurn（whichperhapsexplaihoraofsuchexerstudebooks）。

Iemphasizethatitisotthenumberofpossibleoutes，andhowmanyofthemfavourtheeveheremustalsobereasonforaobemoreorlesslikelythaher。Otherwise，youcouldfallirapofbelievingthatyouriinaLotteryis50%，onthegroundsthattherearejusttwoalterheryouwinoryoudonot!

&alevidence–frequencies

&hatdihouseholdgameslikeMonameslikeCraps，willshoweachoftheirsixfacesequallyoften。Butifadieismadefromnon-uniformmaterial，oritswidth，breadth，adiffer，itisoassumethatalloutesareequallylikely。Overaseriesofthrowsmadeuhesames，thefrequenyfacewillfluctuate，butwilleveledowosomeparticularvalue。Youdonotfindthat20%ofthefirstthousandthrowsareSixes，aheproportionamohousandthroto60%。Iableexperimeaynotbeequallylikely，buteahasapropensitytooecharacteristida　frequehisvalueastheprobabilityofthatoute。

Perhapsweget170Sixesihousandthroerfectdie，thehehousand，andsoon。Weeverdedu　exactvaluefortheprobabilityofaSixfromtheseexperiments，butthedataleadtoestimates，aathatarecollected，thebetterweexpecttheestimatetobe。Thefactthatweorobabilitydoessexistence。IfIdrawoneawell-shuffledpack，thereseemsnoreasoobefavouredoveranyother。Eachsuitwouldhaveobjectiveprobabilityof14。Ahecard，reshuffle，ahistaskoimes，Iexpecteachsuittoariseaboutequallyoften，inthiscaseabouttweimes。Similarly，withordinarydicewhereallsixoutesareiobeequallylikely，theceofaFiveonanythrowisobjectivelytakeh:andoversixhuhrows，weexpectaFiveonaboutonehundredos。

Whehequallylikelyoutesarerepeatedofteive　frequenypartieisexpectedtobeaatchtoitsprobability，ascalculatedobjectively。AfairgivesexactlyfiftyHeadsinohrows，butintuitioellyouhowclosetothatidealyoushouldreaso。Frequencyideasareappliedmoreetitionsofthesameexperimeiditions。Willsomeimmihbemaleorfemale?Withnospeationaboutthefamilyiurntodatagatheredfrommanytriesandculturesperiod。Thereisatpatternthat，forevery49femalebirths，thereare51males。Ohatthereisnothingtopickoutthisbirthfromallothersthataretakingplace，afrequentistwillputtheprobabilityofaboyat51%。

Someexperimentsonaheroicscalehavebeendu1894，thezoologistRaphaelWeldoheresultsofmorethahousandthrowsofasetofadozeawerenottwiththeideathatallsixfaceswereequallylikely，asthenumbersfiveandsixoccurredrathertoooften。Hisdicehadsmallholesdrilledineachfatifyitssdthefacesforfiveaetwoaively。Thetresofgravityofthesedicewillbeclosertothefaceswithsmallnumbers，givingaplausibleexplanationfortheobservedexcess。

Aboutseveer，WillardLongetianwithtimeonhishands，offeredhisservicestotopHarvardstatistiFrederickMosteller。Ueller’sguidangcorcollectedovertwohuhreweatwentythousandtimes，regtheoutplyasevenorodd–overfourmilliondatavalues。Tomakethesasnearaspossibleidentical，heusedacarpeteddesk-top，tobouhediceoff。ForcheapdicelikethoseusedbyWeldon，therewasasmallbutdistinctbiastowardstoomanyevennumbers–again，nottotallyuedbecause。However，withthehighqualityprediceasusedinLasVegasos，wherethepipsareeitherlightlypaintedorareextremelythindisosuchbiaswasfouhthosediceweretwiththeclassicalviewofequallylikelyoutes。

BlackjackexpertPeterGriffihat，forasequenceof1，820handsheplayedihedealer’supcardwaseitheraTen-A770os。Theobjeceohosefavourablecardsis513，soGriffiherornothehadbeeed–randomcewouldgivethedealerthesegoodly700timese。

In20023，6，202underfiveyearsoldwereadmittedwithsuspeeumoniatohospitalsinMalawi，aalityrateof8。4%。Providedthattherewerenospecialgthisperiodatypical，afrequentistwouldcludethattheprobabilityofdeathwhenayoungMalawichildeumoniaisabout8–9%。Fromaiveperspective，makiementsabouttheceofdeathamongyoungMalawiwithpneumoniawouldbespe，albeitbasedoallthatbesaidforisthatifohoseparticular6，202wereselectedatrahatchilddiedwas8。4%。

&ioweeaaiveprobabilitieswillbefurtherexploredlater。

&iveiion

Brui，oihihefield，wrote

PROBABILITYDOES

AsProfessoroftheTheoryofProbability，hewasnotdismissinghissubjectasamirage，ratherherejected　absoluteclaimssuchas‘TheprobabilityofHeadsisoohim，everystatementinvolvingaprobabilityisjustanexpressionofopinion，basedonone’sownexperienosgwheionarrives。

siderthefiveassertions:

TheEnglaainwillwiossinEnglaMatch;

WhoeverwinstheOscarforbestaextyearwillalsowier;

NopersonborninOslohasyetiggoldmedal;

RichardIIIohedeathofthePriheTower;

AlGorewouldhavebeeedUSPresidentin2000ifRalphstoodasadidate。

&hem，wereeofbelief，orpersonalprobability，or　subjectiveprobability。Thiswillbesomeivegreaterthaly，itisapertagebetween0%and100%，inclusive。

Zeroa，respectively，thetwoextremesofimpossible，aain。IamthatthesoccerWorldCupwillbehostedbyanAfriagainduriury。Ithinkitisimpossibleforsomeoyyearsofagetorize。

Assessiiveprobabilities

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