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Chapter 1 Fundamentals（第1页）

Chapter1Fuals

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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