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Chapter7
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Applisce，medidoperationsresearch
Wemayassessoriprobabilitiesiwaysagtothetext.But，asDavidHandwroteinhisStatistics:AVeryShortIntrodu，‘...thecalculusisthesame’，i.e.howprobabilitiesaremanipulateddoesnotge.
Keepiralideasofthesubject:theAdditionandMultipliLaws;iheLawseNumberslinkioobjectiveprobabilities;Gaussiandistributionswhensummingrandomquaherfrequentlyarisingdistributions;meansandvariancesasusefulsummaries.
&expeowledgeoftherelevantprobabilitiestohavethepreavailablefortheexamplesinthepreviouschapter，butanapproximateaherightquestionbeareliableguidetogooddes.AsstatistiGeeBoxsaid:‘Allm，butsomeareuseful.’
&tersillustrateapplis，looselygroupeduertitles.
Brownianmotion，andrandom>
In1827，thebotaBrowpollenparticlessuspendedinliquidmovearourandhtyyearslater，AlbertEinsteingaveaioiclesweretlybeihemoletheliquid.Thismovementis，ofthreedimensions，buttobuildasatisfaodel，wefirstovementjustalhtline.
Supposethateachstepisajumpofsomefixeddistaimesleftaimesright，ilyeachtime.Thisnotionistermeda randomositionaftermanyjumpsdependsonlyonthedifferehenumbersofjumpsineachdireeanandvariaahestartpoiionaltothenumberofjumpsmade.
Makeadeliputation:overafixedtimeperiod， ihefrequenps，ahedistahecorrethesetwofactors，thelimitbeesotion，therandomdistancemovedhaviralLimitTheaussiandistributionwhosemeanandvariahproportiohofthetimeperiod.Ifmovemehtareequallylikely，themeanwillbezero.
&eiionforBrowionsisthatparticlesmoveinthreedimensioineasionfollowingaGaussianlawfiveionsabouthowatomsandmoleculesbehave，provokihatremovedanylingeringdoubtsabouttheirexistence.
&erm‘Brownianmotion’oughttobereservedfortheaentofpartialiquid，butitisalsooftehismathematicalmodelofthatmovement.
Randomnumbers
Thephrase‘randomoowoideas.First，asinidealgameswithdiceorroulettewheels，onenumberfromafiis，allofthembeingequallylikely.Sed，asiionofsnappingastickatarandompoiinauousintervalisopartofthatintervalbeingfavouredoverahefacilitytogsequencesofsubers，eachvaluebeiofalltheothers，hasmanyapplis，astheionwillillustrate.
In1955，asplendidbookOneMilliitsublished.Itfollowsitstitleexactly:pageafterpageofthedigitszerotonine，groupedinblocksforease，butsuccessivedigitsareentirelyuable–whatevertheretseques，youhaveoeheoday，modernputershavebuilt-ioachievethesameends.Aninitialvalue（theseed）isfedin，ahematiulaproduextvalue，whichaewseed，andsoohingrandomaboutthisprocessatall，andifthesameinitialseedisused，thesamesequeed.But，withagathematiula，thesequeedpassesabatteryofstatisticaltestsandlooks，toallisahoughitwereraermpseudo-randomsequenceisused.
Nomatterhowmuchcareistakeninthisprocess，therewillalwaysbesfearthathiddenflawsihodusedwillmatteriowhiumbersareput.Withthatdrelyingontheexperienceenumberofrespetists，IampreparedtoayputerproducesacceptablesequennumbersoheobviousdangerofinsiderfraudmeansthatthesemethodshavenoplabersinLotteries，orinUKPremiumBonds.）
Monteethods
HowmanumbersearoivespinsofastandardEuropeaewheel?Icouldbeaweenoneand37，butthoseextremeswouldoccurveryrarely;whatisthemostlikelynumberofdifferentnumbers?
&hisproblemuttome，Ididelyseeaosolveit.Thereare3737（ah59decimaldigits）possibleoutesofspinniimes，arytowritedownallthewaysinwhich，say，28differentnumberscouldarise，youquicklyloseenthusiasAmoreappealingapproaaso-teulation.
&heputer’sstreamofrandomnumberswasusedtosimulatetheoutesof37spinsofawheel，afterwhiputeranumbershadarisen.Thisprocesseatedonemilliontimes，leadingto24differentnumberson203，739os，while23arosejust199，262times.
&rivals，22or25numbers，eaedfewerthan160，000times.TheLawehatthefrequehedifferentouteswillsettledowntotheirrespectiveprobabilities，andthesefiguresessehematter:themostlikelyresultisthat24differentnumberswillarise，andtheceofthisisjustover20%.
Dayslater，Ikickedmyselffornotspottingastandardwaytosolvetheproblem!Icouldcalculatetheexactprobabilityofgettinumbersin37spins，foranyvalueofX，ingthedescribedabove.Butthisdoesnotiheuseofsimulationtoattackthissortofproblem–quiddirtyanswersbeuseful.Ihatthesimulationgaveanswerstwiththeexactcalboostedmygehthattheputer’sraorwasbehavingasintended.
AmoreserioususeofMonteethodsopolymerchemistry.Amolesistseoms，egarandomlytwistingsolyatplaeveid，crusthesameplace.Howfarisitlikelytobefromohemoleculetotheother?
&hiomsasbeingattheplacesvisitedbysomedrunkard，staggeringaroundatrandomonathree-dimensionallatticeforawhile，butsomehthesameplacetwice.Withouttherequirementthatnoplaceberevisited，mathematicalexpertsmakegress，butthatrestristoplicatetheproblembeyoack.
However，eveentputerprogrammerwriteaseionofthisplex，twisting，d，bymakingonemillion，tenmillion，evenabillioions，obtainanansreciseasisrequired.（RecalldeMoivre’srelyasthesquarerootofthesizeofthesimulation.）
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