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Chapter 7（第2页）

Orsupposeyouwaheareaularlyshapedleaf.Drawareglearouhehepositionsenumberofpoirandomwithiangle.Yourestimateultiplyihewhlebytheproportionofpointsthatfallwithintheleaf’sboundaries.

Asafinalillustrativeappli，supposePaulishopiuparolfillingstatioallsfourpumps，theminimumviableherewillberoomforuptoeightothercarstowaitinaqueue;eachextrapumpremovestaces，soifheihemaximumofeightpumps，therewouldbe.Toworkouthoswillmaximizehisprofits，heulationsofwhatealledfour，five，six，sevepumps.

&heinstallationcosts，therunningdtheprins，hewouldokeofarrivalofpotentialersaributiotakesfrupatapumpuntilthepumpbeesfree.Heshouldalsotakeatofthecethatapotentialerwoulddrivepastifnopumpwerefree，orthequeuetoolong.Allthesefiguresarerelativelyeasytofie，anditisfarputersimulationsthaphysicallywithvariousnumbersofpumpsoverseveralmonths.Sinusethesameiime，heallhissimulationsuhesames，therebyimprovingtheparisohediffereesofprofits.

Why‘MonteCarlo’?Despitetheobviousbetweenrandomnumbersandes，thenamewasinallyjustacodewordtorefertotheuseofthesemethodsinmilitarymatters，ingtheearlydevelopmentofnubs.

Errorsincodes

Morseootransmitmessagesusingonlytwosymbols，0and1say.Butsomesymbolsmaygetcorrupted，sothataninitial0arrivesasa1，andviceversa.Evenwithalowerrorrate，themessagereeahingquitedifferentfromthatseodealwiththis?

Supposeeachsymbolsenthas，ily，somesmallprobabilityofbeingcorrupted.Wecouldrepeatthesymbols，butamoment’sthoughtshowsthatsending00and11ratherthan0aivelyisall:if01or10arrives，itisapureguessastowhether00or11hadbee.We’llguesscorrectlyhalfthetime，butdoublinguponthesymbolsseweshouldexpecttwiyerrors，sothesefactelycelout.Butsidertransmitting000or111insteadof0or1.

Using‘majorityvote’todeessages，allof{000，100，010，001}willbeiheotherfourpossibilitieswillbeiedas1.Ifjust1%ofsymbolsseed，thenwhen000issent，usingthebinomialdistributionshowsthatthereisa99.97%eofthosefoursequencesabovearrives.Thatmeansthattheerrorratedropsfrom1%to0.03%，afactorofoverthirty.Wewoulddoeveerifeachdigitwererepeatedfivetimes，butattheexpehemessagelechoicewilldependoheie，aransmission.

Amesis

Prospectiveparents（andstatistis）JuanjuanFanandRichardLeviherornotFanshoaesis–atesttoseewhetherthefoetusshewasgmighthaveDown’sSyheirexperieasatemplateforothersinasimilarsituation.

KnowledgeofFan’sage，andasimplebloodtest，putthe’sas1in80;anultrasoundimagewaseng，leadingtoauseofBayes’Rulethatreducedthe120.Amesisisa–ahollowneedleisiheabdomeasampleofamnioticfluid;iftheextraosome21thatischaracteristi’sispresent，itwilllybedetected，butthetesthasarisk，estimatedinthiscaseat1in200，ofgamiscarriage.Shouldparents，whowouldabortionifDow，takethistest?

FanandLeviheirdebythelogiaximiziedutility.TheworstpossibleoutiscarriageofafoetuswithoutDown’s，wasassigyzero，thebestoute，birthwithoutDown’s，wasgiveyofunity.TogivebirthtoaDown’schild，haviofamesis，wasallocatedutilityx，whiletotakethetestandfindDown’shasasomewhatgreaterutility，　y.（Thepossibilityofamiscarriageisirrelevantinthislastcase，asthefoetuswouldbeabortedany>

&edutilitieswithandwithoutthetestaremade.Thetestshouldbetakeexceedsthesedwhithiscase，reducestthatyshouldexceed（119200）+x;inroundfigures，　yisbiggerthan0.6+x.

IfFanathattheutilityofdisgthepresen’s，andthushavinganabortion，wasbelow0.6，therewouldneverbeanypointintakihehighertheutilitytheyattachedtohavingachild，albeitwithDowhatthresholdwouldbee.Ifthatutilitywereabove0.4，theyshould.

gappropriatevaluesforxandyrequiressomethought;andifthebasicparameters–a1in200iscarriagethroughthetest，a1in120’swithoutthetest–weredifferent，thefinalwouldge.（SeetheAppendix.）Plainly，ifthe’swerelessthantheiscarriage，itwouldionaltotakethetest（yes?）.

FanandLevenediscussedtheirdilemma，andtheiragreedchoiceofutilitiesledthemtooptforthetest.Thereyendirae，andne.

Haemophilia

Haemophiliaisthegeoagroupofdisordersthatpreventbloodfromgifacutoccurs.ThegagentisfoundontheXe，andtheceitismissingisunder1in5，000.AsfemaleshavetwoXes，theywouldsufferfromhaemophiliaonlyifitisabsentfrombothcopies，givingader1in25million，butmalespossessonlyoneX（andoneYe），soalmostallihediseaseareinmales.

Ifmalesdohavehaemophilia，thisbeesknowheyhaveacetofather，butafemalemayunknowinglyhaveonenormalXathegagent.Suchfemalesaretermedcarriers，aheypassoedgeoanychildis50%.Adaughterwhoreceivesthisaffeebeesacarrier，asonwillsufferfromhaemophilia.ThatQueenVictoriawasacarrieris，ashersonLeopoldhiliadatleasttwoofherfivedaughterswerecarriers.Shehadthreeothersonswhodiddisease.

SupposeBettyhasabrotherwhosuffersfromthisdisease;Bettyhasseveral，inistheneisacarrier?

ToahisquestioofindtheprobabilitythatBettyisacarrier;Anne’scewillalwaysbehalfthatfigure.WeknowthatBetty’smotherisacarrier，asBetty’sbrotherisahaemophiliathisinformatioheprobabilityBettyisacarrieris50%.Andifahershavethedisease，Bettywouldbetobeacarrier，sowelookattheallAhersarehealthy.

ThissituationistailoredforBayes’Rule.LetGuiltmeayisacarrier:siialprobabilityofGuiltis50%，theprioroddsareunity.IfBettyisIacarrier），theprobabilityoftheEvidehershealthy）isplainly100%.ButifBettyisGuilty，eachbrotherhas，ily，a50%ceofesgthefaultygehybrotherhalvestheLikelihoodRatitheposterioroddsintoprobabilities，theceBettyisacarrierissuccessively13，15，19，117，...，agasshehasohree，four，...brothers，allhealthy.

10.Familyrelationships

Foryourow，supposethatAnnealsohassisters，someofwhomhavesons，ahesenephewsofAnnearesufferers.Howdoesthataffecttheeisacarrier?ChesweragainstthatgivenintheAppendix.

Epidemics

Thephraseherdimmuhefactthatifsuffitlymanypeoplearevaated，thenshouldatrafetertheunity，noepidemicwilloccur.Thuseventhosewhowerenotvaatedareveryuchthedisease.Whyshouldthisbeso，andhowdowediscoverwhat‘suffitlymany’means?

Typically，thoseiransmitthediseasetoothers，butthosewhorecoverhaveacquiredimmunity.SoleasoneofSusceptible，Ied，orRemoved.Thelatterarethoseimmunefromiionbecauseofva，recovery，physicalisolatiomayseemcalloustogivethesefouroutesthesameword，butthebrutaltruthisthat，sofarasthespreadoftheepidemied，theyreallyareequivalent!

Toseehoidemicmightdevelop，letSbetheibleandItheed.Multiplyiwethergivesthetotalnumberofpossibleeweeedaible.Croulationswillbeimoreoftenthanscatteredruralpopulationsofthesamesize，andtheprobabilityoftheSusceptiblebeihroughaerwilldepeioushedisease.Overall，theprobabilitythat，duringatinytimeperiod，someSusceptibleindividualbeesIedwillbeoftheformβ*S*I，whereβdependsoiousnessandhowmuix.