零度小说网

Chapter 5 Making sense of probabilities（第2页）

Askyourself:amIassessietythingshappening，orjustthealceofthetweh，giveeenhavehappened?

（e）Neersareoftenprodudergreattimepressure，soitisnthatsomeartitaiherearethreereportsthatshouldhavebeenspiked。

Whenallsixeggsinaboxboughtiweredouble-yolked，itwasclaimedthatatrulyastohadhappened。Ohousandhasthisproperty，sotheceofgettingaboxfulisthistinyfraultipliedtogethersixtimes。Theresultingnumberissosmallthatifyouopenedoneboxeverysed，youwouldexpecttotakeoverthirtybilliooeacrossaboxgonlydouble-yolkedeggs!

Butthatakessenseonlyifalltheeggsinaboxaredepelyfromavastofeggs，ohousandofwhichisdouble-yolked。Thisdoesn’thappen。Eggsaresortedbysizebefboxed。Someboxesareeveainingonlydouble-yolkedeggs。。。。

&ionsabouttheprivatelifeofanEnglaainsurfaced。Areporteruresforthe‘likelihood’ofeachoffourpossibleasbytheteammanager:

（a）expelhimfromtheplayingsquad–110

（b）retaininthesquad，butin–310

（inthesquad，butremovethecy–610

（d）takenoa–810

Ahesefourestimatedprobabilitiesisplausible。Butsietomutuallyexes，theirsummusty:however，these‘probabilities’addupto1。8。

Thirdly，itortedthat，amongpeoplewhohadwo￡50，000oioheurningupmostoftenincludedJohn，David，Michael，Margaret，Susan，andPatricia。Sood:butitwasabsurdtoclaimthat　thereforeyoushouldtrytoihthosenamesinyourLotterysyndicate!

Desgignorance

Drawingoneawell-shuffledpack，IexpectBladRedtobeequallylikely，andItlyattachthefigureof‘12’totheceofRed。Inaslightlyvolutedway，Icouldsaythatthe　distributionIattachtotheceofgetties100%tothequantity‘12’–myceisshownbythechoiceof100%。IfIamsurethataprobabilityissomedefinitefigure，Iattachprobability100%tothatfigure。

Butfrequently，Iotseleglefigure:mybestestimateofthecemytrainwillmissitsaybe34，but23and45earlyasplausible，andeveotheextremesofzeroandupletelyruledout。Iuseauousdistributiefromzerotounitytodescribemyfeelingsaboutthisunknownprobability。

Totalignoranceofaprobability–veryrareces–wouldbedescribedbyusiinuousuniformdistributionofFigure5。Moreoften，thereissomeiefigurewhichisyourbestsiheprobability，andyourinstinctaboutbothhigherandlowervaluesisthattheircesfallawayture　8showsaofgraphsfromwhatiscalledthe　betafamilyofdistributionsthathavethisproperty。

Figure　8airettyignorantofthevalueoftheprobability:weattachhighestbelieftovaluesnear12，butvaluesassmallas15has45arestillquitepossible;withFigure　8b，wearemuchmoretthatthevalueiscloseto12，whilestillnouttheextremes。With8c，hestbeliefisforvaluescloseto13，whilewith8dthevaluesareverystronglygregatednear23，withverylittleexpethatthevalueisbelow12。

Howfarwillmycartravelorol?，aeadyspeed，Iexpecttogetabout90miles，butifImakeshortjourneysoveracoupleofweeks，itwillbemorelike60miles。Ihedistancewillhavesomeuy，expressedviasomeuousdistributioahakindofdistributioiosmallcupsofsize10cc。Therewillbe1，000suchdthetotaldistancecoveredwillbethesumofthedistancesachievedusingthese1，000pohetralLimitTheorem，whichsaysthatthe　sumenumberofrandomquaendtofollowaGaussiandistribution。

&afamilyofdistributions

8。ued

Forsteadymotorwaytravel，IwouldselectaGaussiandistributioredon90，sharplypeakedtoshowlowvariance。ForsporadictravelouldalsouseaGaussiandistributioredon60andreadtreateruy。

Utility

Yodmothermakesyouaonlyoffer。Eithershewillgiveyou￡1，orshewilltossadifyoucallHeadsorTailscorrectly，shewillgiveyou￡10，otherwiseyougetnothing。Whatake?

&ivesare￡1for，orafifty-fiftygambleayoutof￡5。Nearlyeverybodyprefersthelatter。Butscalethemoneyupbyafaillily，preferenge。Tohaveonemillionpouainisfarmoreattrabeingequallylikelytohavezero，ortenmillionpouof　utilityliesbehindthisdifference。

Forsmallsumsofmoney，havingtwiceasmuchusually　isworthtwiceasmuch，butifonemillionpoundswouldgeainlevelofpleasureforyouandyourfamily，doublethatamountwouldodoublethepleasure。Isyoueasurethe‘worth’ofamoaryamount，theusualshapeoftherelationshipwillfollowthatshownihegraphalwaysrises，initiallylikeastraightlihensteadilymoreslowly。

‘Utility’explainswhyahouseholderandaninsuranpanyagreethatasumof￡250isareasonableaoinsureahouseworth￡250，000againstperilssuchasfire，subsidence，。Thesubstaheinsuraneanthat，onanyindividualhouse，itactasthoughtheutilityisidenticaltotherelativelysmallsumsinvolved。Solohatitwillpayoutinanyyearislessthahousaedvalueforthistransaispositive，itrofit。Oheuninsuredhouseholderwouldfaousyoffinding￡250，000ifshelostherhousetoohoseperils，sovoluntarilysurre￡250toremovethatpossibilityisagooddealtohertoo。

9。Thegeneralshapeofautilitycurve

Takingainstbreakdownoftelevisions，microwaveovens，andsoonisalmostalwaysabadidea。Thesumsaremuchsmaller，utilityandmoiallythesame，andthepanyremiumlargeenoughforthemtoexpectaprofit。Insteadofbuyingthisexpensiveinsurance，builduparepairfundbyplagthehypotheticalpremiumintoyourbankat。Veryfewpeoplewillregretthatmove。

Ifyousuccessfullystructyourownutilityfun，youuseittohelpmakeachoitheavailableadersofuy。Forea，calculatethe　expectedutilityoftheoutes，i。e。weighttheutilitiesbytheirrespeselectthataforwhichthisexpectedutilityisaslargeaspossible。

Thatistheprobabilist’suniversalreakiheavailablechoices，whatevertheces。