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the lev tarasov world world world is world built on probability mir publishers moscowthe world is built on probability levtarasov 2023lev tarasov the world is built on probability mir publishers moscowtranslatedfromtherussianbymichaelburov firstpublished1988 revisedfromthe1984russianedition thiscompletelydigitalversiontypesetinusinglatexwithebgaramondfontby [email protected] releasedonthewebbyhttp://mirtitles.orgin2023. accessthelatexprojectfilesgitlab.com/mirtitles/twibop creativecommonsbysa4.0license aboutmirtitlesproject themirtitlesprojectisanattempttoconservetheknowledgeintheformofvarious booksthatcameoutduringthesovieteraforthefuturegenerations.thecollection containsbooksonscience,mathematics,philosophy,popularscience,history.the collectionalsohassoviet,russianandchildren’sliterature.theprojectwouldnot havebeenpossiblewithoutthehelpwereceivedfromfriendsandcontributorsfrom acrosstheworld. foreword iamveryhappytoreleasethiscompletelyelectronicversionofoneofmyfavourite books.inthiselectroniceditionallthefigureshavebeenreworkedinthesvgformat usinginkscapeforaclearerpresentation.inlatex.–damitrcontents preface 7 i tamedchance 1 1 mathematicsofrandomness 17 1.1 probability . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 randomnumbers. . . . . . . . . . . . . . . . . . . . . . 28 1.3 randomevents . . . . . . . . . . . . . . . . . . . . . . . 33 1.4 discreterandomvariables . . . . . . . . . . . . . . . . . 38 1.5 continuousrandomvariables . . . . . . . . . . . . . . . 46 2 decisionmaking 53 2.1 thesedifficultdecisions . . . . . . . . . . . . . . . . . . 53 2.2 randomprocesseswithdiscretestates . . . . . . . . . . . 59 2.3 queueingsystems . . . . . . . . . . . . . . . . . . . . . . 66 2.4 methodofstatisticaltesting . . . . . . . . . . . . . . . . 76 2.5 gamesanddecisionmaking . . . . . . . . . . . . . . . . 84 3 controlandself-control 97 3.1 theproblemofcontrol . . . . . . . . . . . . . . . . . . . 97 3.2 fromthe“blackbox”tocybernetics . . . . . . . . . . . . 101 56 contents 3.3 information . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.4 selectionofinformationfromnoise . . . . . . . . . . . . 120 3.5 onthewaytoastochasticmodelofthebrain . . . . . . . 125 ii fundamentalityoftheprobabilitylaws 133 4 probabilityinclassicalphysics 135 4.1 thermodynamicsanditspuzzles . . . . . . . . . . . . . . 136 4.2 moleculesinagasandprobability . . . . . . . . . . . . . 146 4.3 pressureandtemperatureofanidealgas . . . . . . . . . . 158 4.4 fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.5 entropyandprobability . . . . . . . . . . . . . . . . . . . 169 4.6 entropyandinformation . . . . . . . . . . . . . . . . . . 177 5 probabilityinthemicrocosm 183 5.1 spontaneousmicro-processes . . . . . . . . . . . . . . . . 184 5.2 fromuncertaintyrelationsthewavefunction . . . . . . 195 5.3 interferenceandsummingprobabilityamplitudes . . . . . 200 5.4 probabilityandcausality . . . . . . . . . . . . . . . . . . 208 6 probabilityinbiology 213 6.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . 213 6.2 thepatternsaftertherandomcombinationofgenesin crossbreeding . . . . . . . . . . . . . . . . . . . . . . . . 220 6.3 mutations . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6.4 evolutionthroughtheeyesofgeneticists . . . . . . . . . 235 aconcludingconversation 241 recommendedliterature 249preface …innature,wherechancealsoseemstoreign,wehavelongago demonstratedineachparticularfieldtheinherentnecessityand regularitythatassertsitselfinthischance. f.engels avastconcourseofeventsandphenomenaoccurintheworldaround us.theeventsareinterrelated:someareeffectsoroutcomesofotherswhich are,inturn,thecausesofstillothers.gazingintothisgiganticwhirlpoolof interrelatedphenomena,wecancometotwosignificantconclusions.oneis thattherearebothcompletelydetermined(uniquelydefined)outcomesand ambiguousoutcomes.whiletheformercanbepreciselypredicted,thelatter canonlybetreatedprobabilistically.thesecond,nolessessentialconclusion isthatambiguousoutcomesoccurmuchmorefrequentlythancompletely determinedones. supposeyoupressabuttonandthelamponyourdesk lightsup.thesecondevent(thelamplightsup)isthecompletelydetermined resultofthefirstevent(thebuttonispressed). suchaneventiscalleda completelydeterminedone.takeanotherexample:adieistossed.eachface ofthediehasadifferentnumberofdots.thediefallsandthefacewithfour dotsendsupatthetop.thesecondeventinthiscase(fourdotsface-up)is notthecompletelydeterminedoutcomeofthefirstevent(thedieistossed). thetopfacemayhavecontainedone,two,three,five,orsixdots.theevent 78 preface ofappearanceofthenumberofdotsonthetopfaceafteradieistossedisan exampleofa randomevent.theseexamplesclearlyindicatethedifference betweenrandomandcompletelydeterminedevents. weencounterrandomevents(andrandomnessofvariouskinds)very often,muchmorefrequentlythaniscommonlythought.thechoiceofthe winningnumbersinalotteryisrandom.thefinalscoreofafootballmatch israndom.thenumberofsunnydaysatagivengeographicallocationvaries randomlyfromyeartoyear.asetofrandomfactorsunderliesthecompletion ofanyserviceactivity:deliveryambulancearrival,telephoneconnection,etc. mauriceglaymannandtamasvargahavewrittenaninterestingbook called lesprobabilitésàl’école(probabilityingamesandentertainment),in whichtheymakeaninterestingremark: “whenfacingachancesituation,smallchildrenthinkthatitispossible to predictitsoutcome. whentheyareabitolder,thebelievethat nothingcanbepostulated.littlebylittletheydiscoverthatthereare patternshidingbehindtheseemingchaosoftherandomworld,and thesepatternscanbeusedtogettheirbearingsinreality.” therearethreedistinctstageshere:lackofunderstandingoftherandomat first,thenmereconfusion,andfinallyacorrectviewpoint.letusforgetsmall childrenforatimeandtrytoapplythistoourselves.weshallhavetorecognize thatfrequentlywestopatthefirststageinasimple-mindedbeliefthatany outcomecanbepreciselypredicted.themisconceptionthatrandomnessis simplyequaltochaos,ortheabsenceofcausality,haslastedalongtime.and evennownoteverybodyclearlyappreciatesthattheabundanceofrandom eventsaroundusconcealdefinite(probabilistic)patterns. theseideaspromptedmetowritethisbook.iwanttohelpthereader discoverforhimselftheprobabilisticnatureoftheworldaroundus,toin- troducerandomphenomenaandprocesses,andtoshowthatitispossibleto orientoneselfinthisrandomworldandtooperateeffectivelywithinit.9 preface thisbookbeginswithatalkbetweenmyselfandanimaginaryreader abouttheroleofchance,andendswithanothertalkabouttherelationship betweenrandomnessandsymmetry.thetextisdividedintotwomajorparts. thefirstisontheconceptofprobabilityandconsidersthevariousapplica- tionsofprobabilityinpractice,namely,makingdecisionsincomplicated situations,organizingqueues,participatingingames,optimizingthecontrol ofvariousprocesses,anddoingrandomsearches.thebasicnotionsofcyber- netics,informationtheory,andsuchcomparativelynewfieldsasoperations researchandthetheoryofgamesaregiven.theaimofthefirstpartistocon- vincethereaderthattherandomworldbeginsdirectlyinhisownlivingroom because,infact,allmodernlifeisbasedonprobabilisticmethods.thesecond partshowshowfundamentalchanceisinnatureusingtheprobabilisticlaws ofmodernphysicsandbiologyasexamples.elementsofquantummechanics arealsoinvolved,andthisallowsmetodemonstratehowprobabilisticlaws arebasictomicroscopicphenomena.theideawasthatbypassingfromthe firstpartofthebooktothesecondone,thereaderwouldseethatprobability isnotonlyaroundusbutisatthebasisofeverything. inconclusioniwouldliketoexpressmygratitudetoeveryonewhohelped mewhenwritingthisbook.i.i.gurevich,correspondingmemberofthe ussracademyofsciences,gavemetheideaofwritingthistextandgaveme anumberofotherprovokingideasconcerningthematerialandstructureof thebook.b.v.gnedenko,memberoftheussracademyofsciences,g.ya. myakishev,d.sc.(philosophy),ando.f.kabardin.cand.sc.(physicsand mathematics)readthemanuscriptthoroughlyandmadevaluableremarks. v.a.ezhivanda.n.tarasovarenderedmeconstantadviceandunstinting supportthewholetimeiwaspreparingthetext.part i tamed chanceintroduction andchance,inventorgod… a.s.pushkin a discussion on the role of chance author: “youwrotesomenicewordsaboutchanceinthepreface. inspiteofthem,istillthinkchanceplaysanegativeroleonthe whole.naturally,thereisgoodluck,buteverybodyknowsitis betternottocountonit.chanceinterfereswithourplans,so it’sbetternothangonit,weshouldratherwarditoffasmuch aspossible.” author: “thatisexactlythetraditionalattitudetowardstheran- dom.however,itisanattitudewemustclearlyreview.firstof all,isitreallypossibletogetbywithouttherandom?” reader: “idon’tsaythatit’spossible.isaidweshouldtry.” author: “supposeyouworkatanambulancecentre.obviously, youcannotforeseewhenanambulancewillbeneeded,whereit willbenecessarytosenditto,andhowmuchtimethepatient 34 introduction willrequire. butapracticaldecisiondependsonallofthese points.howmanydoctorsshouldbeondutyatanyonetime? ontheonehand,theyshouldnotbeidlewaitingforcallsfor longperiodsoftime,yetontheotherhand,patientsshould nothavetoremainwithoutaidfortoolong.youcannotavoid chance.whatiamtryingtosayis:wecannot eliminatechance, andsowemust takeit intoaccount.” reader: “true,wehavetomakepeacewithchanceinthisexample. however,itstillisanegativefactor.” author: “thus,weseethatsometimeswehavetotakechanceinto considerationratherthancontrolit. butwecangofurther. wecandiscoversituationsinwhichchancebecomesapositive factorratherthananegativeone,sothatitisdesirabletoraise theleveloftherandomthreshold.” reader: “idon’tunderstandyou.” author: “ofcourse,chanceoccasionsinterferewithourplans.at thesametimebecauseitmakesusnewsolutionsandimprove ourabilitytocreate”. reader: “doyoumeananimprovementisobtainedbyovercoming difficulties?” author: “themainpointisthatrandomnesscancreatenewpos- sibilities.anamericanwriterhaswrittenaninterestingscience fictionstory. agroupofscientistswithvariousdisciplinesis officiallyinformedthatasensationaldiscoveryhasbeenmade, butunfortunatelythediscovererdiedinanexplosionduringa demonstrationofthephenomenonandthusthesecretwaslost. inrealityneithertheinventionnortheinventoreverexisted. thescientistswerepresentedwiththeevidenceofatragedy: indistinctfragmentsofrecords,alibrary,andanequippedlabo-5 introduction ratory.inotherwords,thescientistsweregivenavastquantity ofunconnectedinformationwithchancedatafromvarious fieldsofscienceandtechnology.theevidencecouldbecalled informationalnoise.thescientistswerecertainadiscoveryhad beenmade,andthereforethetargetwasachievable.theyuti- lizedalltheinformationattheirdisposaland‘revealed’thesecret ofthenon-existinginvention.wemightsaythattheysucceeded insiftinginformationfromthenoise.” reader: “butthat’sonlyasciencefictionstory.” author: “true. however,theideabehindthestoryisfarfrom beingfiction. anydiscoveryisrelatedtotheuseofrandom factors.” reader: “idon’tthinkanyonecandiscoveranythingimportant unlessheorshehasaprofessionalgraspofthesubject.” author: “ithinksotoo.moreover,adiscoveryrequiresbothex- pertiseonthepartoftheresearcherandacertainlevelofthe developmentwithinthescienceasawhole.andyet…,random factorsplayafundamentalroleinthat.” reader: “asiunderstand,thewordfundamentalmeanssomething primary,somethingatthebasis.canyouapplythetermfunda- mentaltosomethingrandom? iadmitthatrandomnessmay beuseful. butcanitbefundamental? inthelastanalysis,we dealwithrandomvariableswhenthereissomethingwedonot knowandcannottakeintoaccount.” author: “bybelievingthatrandomnessisrelatedtoinadequate knowledge,youmakeit subjective.itfollowsthatyoubelieve thatrandomnessappears,asitwere,onthesurfaceandthatthere isnothingrandomatthebasisofphenomena.isitcorrect?”6 introduction reader: “precisely. thatiswhywecannotassertrandomnessis fundamentality.assciencedevelops,ourabilitytotakedifferent factorsintoaccountincreases,andtheresultisthatthedomain ofrandomvariableswillgraduallyrecede. thereissensein sayingthatscienceistheenemyofchance.” author: “you’renotquiteright.indeed,theadvanceofscienceen- hancesourabilitytomakescientificpredictions,thatis,science isagainsttherandomfactor.butatthesametime,itturnsout thatwhileourscientificknowledgebecomesdeeper,or,more accurately,whilewelookatthemolecularandatomicaspectsof phenomena,randomnessnotonlydoesnotbecomelessimpor- tant,bitonthecontrary,itreignssupreme.itsexistenceproves tobeindependentofthedegreeofourknowledge.randomness revealsits fundametalityatthelevelofthemicrocosm.” reader: “thisisthefirsttimei’veheardsomeonesaythat.please tellmemore.” author: “letmesayatoncethatthistopichashadalonghistory. itwasfirstformalizedinancientgreecewithtwoapproaches totherandombeingstated.thetwoviewsareassociatedwith thenamesofdemocritusandepicurus. democritusidenti- fiedtherandomwiththe unknown,believingthatnatureis completelydeterministic.hesaid:peoplehavecreatedanidol outoftherandomasacoverfortheirinabilitytothinkthings out.epicurusconsideredthattherandomisinherentinvarious phenomena,andthatitis,therefore, objective. democritus’s pointofviewwaspreferredforalongtime,butinthe20thcen- tury,theprogressofscienceshowedthatepicuruswasright. inhisdoctoralthesis differencebetweenthedemocritianand epicurianphilosophyonnature(1841),karlmarxpositively evaluatedepicurus’sviewoftherandomandpointedoutthe7 introduction deepphilosophicalsignificanceoftheteachingsofepicuruson the spontaneousdisplacementofatoms.ofcourse,weshould notexaggeratethecontributionofepicurustoourunderstand- ingoftherandombecausehecouldonlyguess.” reader: “itturnsoutthatipresenteddemocritus’sviewsonthe randomwithoutknowingit. butiwouldliketohavesome concreteexamplesshowingthefundamentalityoftherandom.” author: “consider,forinstance,anuclear-poweredsubmarine. howistheenginestarted?” reader: “asfarasiunderstandit,specialneutron-absorbingrods aredrawnfromthecoreofthereactor.thenacontrolledchain reactioninvolvingthefissionofuraniumnucleibegins…” author: “(interrupting)letustryandseehoweverythingbegins.” reader: “afterenteringauraniumnucleus,aneutrontriggersits disintegrationintotwofragmentsandanotherneutronisre- leased.theneutronssplittwomoreuraniumnuclei;fourneu- tronsarethensetfree,whichinturnsplitfourmorenuclei.the processdevelopslikeanavalanche.” author: “allright.butwheredoesthefirstneutroncomefrom?” reader: “whoknows?say,theycomefromcosmicrays.” author: “thesubmarineisdeepunderwater.thethicklayerof waterprotectsitfromcosmicrays.” reader: “wellthen,idon’tknow…” author: “thefactisthatauraniumnucleusmayeithersplitbe- causeaneutronentersitoritmaydecay spontaneously. the processofspontaneousnuclearfissionisrandom.”8 introduction reader: “butmaybespontaneousnuclearfissioniscausedbyfac- torswedonotknowaboutyet.” author: “thisissomethingphysicistshavebeentryingtosolve. manyattemptshavebeenmadetofindthehiddenparameters whichgoverntheprocessesinthemicrocosm.ithasbeencon- cludedthattherearenosuchparameters,andthereforeran- domnessinthemicrocosmisfundamental.thiscornerstone problemisthoroughlytreatedin quantummechanics,atheory whichappearedintheearly20thcenturyinconnectionwith researchonatomicprocesses.” reader: “theonlythingiknowaboutquantummechanicsisthat itdescribesthelawsgoverningthebehaviourofelementary particles.” author: “weshalltalkaboutquantummechanicsinmoredetail later.letmeonlynoteherethatitdemonstratesthefundamen- talroleofspontaneousprocessesand,therefore,demonstrates thefundamentalityoftherandom. theoperationofanyra- diationgenerator, fromavacuumtubetoalaser, wouldbe impossiblewithoutspontaneousprocesses. theyarefunda- mentalasthetriggerwithoutwhichtheradiationgeneration wouldnotstart.” reader: “andyet,itisdifficultformetobelievethatrandomness isfundamental.youmentionedanuclear-poweredsubmarine. whenthecaptainordersthattheenginesbeturnedon,hedoes notrelyonaluckychance.anappropriatebuttonispressed, andtheenginesstart(iftheyareingoodcondition).thesame canbesaidwhenavacuumtubeisturnedon. whereisthe randomnesshere?” author: “nevertheless,whenweconsiderphenomenainthemi-9 introduction crocosm,theprocessesaretriggeredbyrandomfactors.” reader: “however,wegenerallydealwithprocessesoccurringin themacrocosm.” author: “firstly,whilestudyingtheworldaroundusandtrying tocomprehendits causeandeffectrelations,wemustaddress theatomiclevel,i.e.,thelevelofmicrocosmphenomena.sec- ondly,therandomnessinmicrocosmicphenomenaisessentially reflectedintheprocessesobservedatthemacrocosmicscale.” reader: “canyougivemeanexamplewhenthefundamentality ofrandomnessrevealsitselfatthemacrocosmicscale?” author: “evolution,whichisacontinuousprocessinboththe plantandanimalkingdoms,mayserveasanexample.evolution reliesonmutation,i.e.,randomchangesinthestructureofgenes. arandommutationmayberapidlyspreadbythereproduction oftheorganismsthemselves.itisessentialthatselectionoccurs simultaneouslywithmutation.theorganismswhichcontain therandomgenearethenselectedtothatthosebestfittedto theirenvironmentsurvive.inconsequence,evolutionrequires the selectionofrandomgenechanges.” reader: “idon’tquiteunderstandthisbusinessofselection.” author: “here’sanexample.theflowersofacertainorchidlook likeafemalewasp. theyarepollinatedbymalewaspswhich taketheflowerstobefemales.supposeamutationoccurs,and theshapeandcolouroftheflowerarechanged.theflowerwill thenremainunpollinated.theresultisthatthemutationisnot passedontothenewgeneration.itmaybesaidthatselection rejectedthemutationwhichchangedtheoutwardappearance oftheflower.therewasaspeciesoforchidwhichbecameaself- pollinator,theflowersofthisspeciesrapidlyacquireddiverse10 introduction shapeandcolourowingtothemutation.” reader: “asfarasiknow,evolutionprogressesinthedirection ofthedifferentiationofspecies. doesn’tthisshowthatthe mutationsunderlyingevolutionarenot,infact,sorandom?” author: “thatargumentdoesn’tstandtoreason. evolutionse- lectsthefittestorganismsratherthanthemorecomplex.some- timesahigherdegreeoforganizationispreferable,butsome- timesthisisnotthecase.thisiswhyhumanbeings,jelly-fish, andtheinfluenzaviruscancoexistintoday’sworld.itisessen- tialthatevolutionleadstotheappearanceofnewspeciesthat areunpredictableinprinciple.itmaybesaidthat anyspeciesis uniquebecauseitoccurredfundamentallybychance.” reader: “ihavetoadmitthattherandomnessdoeslooktobea fundamentalfactorindeed.” author: “sincewearediscussingthefundamentalityofrandom- nessinthepictureofevolution,letmedrawyourattentionto onemoreimportantpoint.modernsciencedemonstratesthat chanceand selectionarethe‘creator’.” reader: “justaspushkinsaid,‘andchance,inventorgod…”’ author: “precisely.thislineisstrikinglyaccurate.” reader: “itappearsthatwhenspeakingaboutchanceandselec- tion,weshouldimplythe selectionofinformationfromnoise, shouldn’twe?thesameselectionthatwediscussedinconnec- tionwiththescience-fictionstory.” author: “absolutely.” reader: “ihavetoagreethatweshouldconsciouslyrecognizethe existenceofrandomnessratherthantryandcontrolit.”11 introduction author: “wecouldsaymore.naturally,therandomnesswhich isduetotheincompletenessofourknowledgeisundesirable. whilestudyingtheworld,manhasfought,isfighting,andwill continuetofightit. itshouldbenotedatthesametimethat thereisan objectiverandomnessunderlyingeveryphenomena alongwiththe subjectiverandomnesswhichisduetolackof dataonaphenomenon.weshouldalsotakeintoaccountthe positive,creativeroleoftherandom.andinthisconnectionit isreallynecessarytorecognizeandcontrolrandomness.man shouldbeable,whennecessary,tocreatespecialsituations,abun- dantwiththerandom,andutilizethesituationtohisownends.” reader: “butisitreallypossibletotreatrandomnessinsuchaway? isn’titliketrying tocontroltheuncontrollable?” author: “bothscienceanddailylifeindicatethatitispossibleto orientourselvesconsciouslyinveryrandomsituations.special calculationmethodshavebeendevelopedthatdependonran- domness.specialtheorieshavebeenproduced,suchasqueueing theory,the theoryofgames,andthe theoryofrandomsearch,to dealwithit.” reader: “itishardformetoimagineascientifictheorybuilton randomness.” author: “letmeemphasizerightawaythatrandomnessdoesnot precludescientificprediction.thefundamentalityofrandom- nessdoesnotmeanthattheworldaroundusischaoticandde- voidoforder.randomnessdoesnotimplytherearenocausal relations.butweshalldealwithallthatlater.itisinterestingto tryandimagineaworldinwhichrandomnessasanobjective factoris completelyabsent.” reader: “thiswouldbeanideallyorderedworld.”12 introduction author: “insuchaworld,thestateofanyobjectatagiventime wouldbeunambiguouslydeterminedbyitspaststatesand,in itsturn,woulddeterminethefuturestatesjustasdefinitely.the pastwouldbestrictlyconnectedwiththepresent,aswouldthe presentwiththefuture.” reader: “anythingoccurringinsuchaworldwouldbepredeter- mined.” author: “pierrelaplace,agreatfrenchscientistofthe17thcen- tury,suggestedinthisconnectionthatweimagineasuperbeing whoknewthepastandthefutureofsuchaworldineverydetail. laplacewrote:” ‘theintellectwhocouldknow,atagivenmoment,every forcethatanimatesthenatureandtherelativepositions ofitseverycomponent,andwould,inaddition,bevast enoughtoanalysethesedata,woulddescribebyasingle formulathemotionsofthegreatestbodiesintheuniverse andthemotionsofthelightestatoms.therewouldbe nothinguncertainforthisbeing,andthefuture,likethe past,beopentohisgaze.’ reader: “anideallyorderedworldisthereforeunreal.” author: “asyousee,itisn’thardtofeelthattherealworldshould admittheexistenceofobjectiverandomness. nowletusre- turntotheproblemofcausalrelations. theserelationsare probabilisticintherealworld.itisonlyinparticularcases(for example,whensolvingmathsproblemsatschool)thatwedeal withunambiguous,strictlydeterminedrelations.hereweap- proachoneofthemostessentialnotionsofthemodernscience, thenotionof probability.” reader: “i’mfamiliarwithit.ifithrowadie,icanequallyexpect anynumberofdotsfromonetosix. theprobabilityofeach13 introduction numberisthesameandequalto1/6.” author: “supposeyoustandatthesideofaroad,withmotor-cars passingby.whatistheprobabilityofthefirsttwodigitsintheir fourdigitnumberbeingequal?” reader: “theprobabilityequals1/10.” author: “therefore,ifyou’repatientandobserveenoughcars, aboutonetenthofthemwillhavenumber-plateswiththesame firsttwodigits,wouldthey?say,aboutthirtycarsoutof300 willhavesuchplates.maybe,27or32,butnot10or100.” reader: “ithinkso.” author: “butthentherewouldbenoneedtostandattheroadside. theresultcouldbepredicted.thisisanexampleofprobabilistic prediction.lookathowmanyrandomfactorsareinvolvedin thissituation. acarcouldturnofftheroadbeforereaching theobserver,oranothercarcouldstoporeventurnback.and nonetheless,bothtodayandtomorrow,about30carsoutof 300wouldhaveplateswiththesamefirsttwodigits.” reader: “so,inspiteofnumerousrandomfactors,thesituation hasacertainconstancy.” author: “thisconstancyiscommonlycalledstatisticalstability.it isessentialthatstatisticalstabilityisobservedbecauseofrandom factorsratherthandespitethem.” reader: “ihadn’tthoughtthatwedealwithprobabilisticpredic- tionseverywhere.theyinclude,forinstance,sportspredictions andweatherforecasts.” author: “you’reabsolutelyright.animportantpointisthatprob- abilistic(statistical)causalrelationsarecommon,whilethose14 introduction leadingtounambiguouspredictionsarejustaspecialcase.while definite predictions only presuppose the necessity of a phe- nomenon,probabilisticpredictionsarerelatedsimultaneously bothwithnecessityandrandomness.thus,mutationsareran- dom,buttheprocessofselectionisgovernedbylaws,thatis,it isanecessaryprerequisite.” reader: “isee.theindividualactsofthespontaneousfissionof uraniumnucleiarerandom,butthedevelopmentofthechain reactionisunavoidable.” author: “takenseparately,anydiscoveryisrandom. however, asituationwhichisfavourablefortheappearanceofsucha chanceshouldexist.thischanceisdeterminedbytheadvance ofscience,theexpertiseoftheresearchers,andthelevelofmea- surementtechnology.adiscoveryisrandom,butthelogicof theprogressleadingtothediscoveryinthelongrunisregular, unavoidable,andnecessary.” reader: “nowiseewhythefundamentalityofrandomnessdoes notresultinthedisorderofourworld.randomnessandneces- sityarealwayscombined.” author: “correct. friedrichengelswrotein theoriginofthe family, private property, and the state (1884): ‘in nature, wherechancealsoseemstoreign, wehavelongagodemon- stratedineachparticularfieldtheinherentnecessityandregu- laritythatassertsitselfinthischance.’thehungarianmathe- maticiana.rényiwroteaboutthesamethinginaninteresting book lettersonprobability”: ‘icameacross contemplationsbyaureliusandacciden- tallyopenedthepagewherehewroteabouttwopossibili- ties:theworldiseitherinvastchaosor,otherwise,order andregularityreignsupreme.andalthoughihadread15 introduction theselinesmanytimes,itwasthefirsttimethatithought overwhymarcusaureliusbelievedthattheworldshould bedominatedbyeitherchanceororder.whydidhebe- lievethatthesetwopossibilitiesarecontradictory?the worldisdominatedbyrandomness,butorderandregular- ityoperateatthesametime,beingshapedoutofthemass ofrandomeventsaccordingtothelawsoftherandom.’ reader: “asfarasiunderstand,orderandregularityareproduced fromamassofrandomevents,andthisleadstotheconcept probability.” author: “you’reabsolutelyright. individualfactorsvaryfrom casetocase.atthesametime,the pictureasawholeremains stable.thisstabilityisexpressedintermsof probability.this iswhyourworldprovestobeflexible,dynamic,andcapableof advancing.” reader: “itfollowsthattheworldaroundusmayjustlybesaidto beaworldofprobability.” author: “itisbettertospeakofthe worldasbeingbuiltonprob- ability.whenweexaminethisworld,weshallconcentrateon twogroupsofquestions.firstly,ishallshowhowman,owing tohisuseofprobabilityinscienceandtechnology,wasableto tamerandomnessandthusturnitfrombeinghisenemyinto anallyandfriend. secondly,usingtheachievementsofmod- ernphysicsandbiology,ishalldemonstratetheprobabilistic featuresofthelawsofnature.inconsequence,ishallshowthat theworldaroundus(includingboththenaturalandartificial world)isreallybuiltonprobability.”chapter 1 mathematics of randomness thisdoctrine,combiningtheaccuracyofmathematicalproofs andtheuncertaintyofchanceoccasionsandmakingpeace betweentheseseeminglycontradictoryelementshasafullright tocontendforthetitleofthemathematicsoftherandom. blaisepascal probability classicaldefinitionofprobability.whenwetossacoin,wedonotknow whichwilllandfaceup,headsortails.however,thereissomethingwedo know.weknowthatthechancesofbothheadsandtailsareequal.wealso knowthatthechancesofanyofthefacesofadielandingfaceupareequal. thatthechancesareequalinbothexamplesisduetosymmetry.boththecoin 1718 mathematicsofrandomness andthediearesymmetrical.whentwoormoreeventshaveequalchances ofoccurring,wecallthemequallypossibleoutcomes. headsortailsare equallypossibleoutcomes.supposeweareinterestedinacertainresultwhile throwingadie,forinstance,afacewithanumberofdotsexactlydivisibleby three.letuscalloutcomessatisfyingsucharequirement favourable.there aretwofavourableoutcomesinourexample,namely,athreeorasix.now letuscalloutcomes exclusiveiftheappearanceofoneinsingletrialmakesit impossiblefortheotherstoappearatthesametrial.adiecannotlandwith severalfacesup,sotheyareexclusiveoutcomes. wecannowformulatetheclassicaldefinitionofprobability: theprobabilityofaneventistheratioofthenumberoffavourable outcomestothetotalnumberofequallypossibleexclusiveoutcomes. suppose𝑃 istheprobabilityofaneven𝐴,𝑚 isthenumberoffavourable 𝛢 𝛢 outcomes,and𝑛thetotalnumberofequallypossibleandexclusiveoutcomes. accordingtotheclassicaldefinitionofprobability 𝑚 𝑃 = 𝛢. (1.1) 𝛢 𝑛 if𝑚 = 𝑛,then𝑃 = 1andtheevent𝐴isa certainevent(italwaysoccurs 𝛢 𝛢 ineveryoutcome).if𝑚 =0,then𝑃 =0,andtheevent𝐴isan impossible 𝛢 𝛢 event(itneveroccurs).theprobabilityofa randomeventliesbetween0and 1. letanevent𝐴bethrowingadieandgettinganumberexactlydivisible bythree. here𝑚 = 2andsotheprobabilityoftheeventis1/3,because 𝛢 𝑛 = 6. consideronemoreexample. wehaveabagwith15identicalbut differentlycolouredballs(sevenwhite,twogreen,andsixred).youdrawa ballatrandom.whatistheprobabilityofdrawingawhite(redorgreen)ball? drawingawhiteballcanberegardedasanevent𝐴,drawingaredballisan event𝐵,anddrawingagreenballisanevent𝐶.thenumberoffavourable19 probability outcomesofdrawingaballofacertaincolourequalsthenumberofballsof thiscolour,i.e.,𝑚 =7,𝑚 =6,and𝑚 =2.using(1.1)andgiven𝑛=15, 𝛢 𝛣 𝐶 wecanfindtheprobabilities: 𝑚 7 𝑚 2 𝑚 2 𝑃 = 𝛢 = , 𝑃 = 𝛣 = , 𝑃 = 𝐶 = . 𝛢 𝑛 15 𝛣 𝑛 15 𝐶 𝑛 15 additionandmultiplicationofprobabilities.whatistheprobability thatarandomlydrawnballwillbeeitherredorgreen? thenumberof favourableoutcomesis𝑚 +𝑚 =6+2=8,andthereforetheprobability 𝛣 𝐶 willbe 𝑚 +𝑚 8 𝑃 = 𝛣 𝐶 = . 𝛣+𝐶 𝑛 15 weseethat𝑃 =𝑃 +𝑃 .theprobabilityofdrawingeitheraredoragreen 𝛣+𝐶 𝛣 𝐶 ballisthesumoftwoprobabilities:theprobabilityofdrawingaredballand thatofdrawingagreenball.theprobabilityofdrawingaballwhichiseither redorgreenorwhiteisthesumofthreeprobabilities,𝑃 +𝑃 +𝑃 .itisequal 𝛢 𝛣 𝐶 tounity(7/15+2/5+2/15=1).thisstandstoreasonbecausetheevent inquestionwillalwaysoccur. theruleforaddingprobabilitiescanbeformulatedasfollows: theprobabilityofoneeventofseveralexclusiveeventsoccurringisthe (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) sumoftheprobabilitiesofeachseparateevent. (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) supposethattwodicearethrown. whatistheprobabilityofgetting (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) twofoursatthesametime?thetotalnumberofequallypossibleexclusive (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) outcomesis𝑛 = 6×6 = 36. eachoneislistedinfigure1.1,wheretheleft figureintheparenthesesisthenumberononedie,andtherightfigureis figure1.1:possibleoutcomesofrollingadie. thenumberontheother. thereisonlyonefavourableoutcome,anditis indicatedinfigure1.1as(4,4).hence,theprobabilityoftheeventis1/36. thisprobabilityistheproductoftwoprobabilities:theprobabilityofafour20 mathematicsofrandomness appearingononedieandthatofafourontheotheri.e. 1 1 1 𝑃 =𝑃 ×𝑃 = × = . 44 4 4 6 6 36 theruleformultiplicationofprobabilitiescanbeformulatedasfollows: theprobabilityofseveraleventsoccurringsimultaneouslyequalsthe productoftheprobabilitiesofeachseparateevent. bythewas,itisnotnecessaryfortheeventstobesimultaneous.instead ofthrowingtwodiceatthesametime,wecouldthrowasingledietwice.the probabilityofgettingtwofoursatthesametimewhentwodicearethrown isthesameastheprobabilityofgettingtwofourswhenonedieisthrown twice. inmanycasesbothrules(additionandmultiplicationofprobabilities)are usedjointlytocalculatetheprobabilityofanevent.supposeweareinterested intheprobability𝑃ofthe samenumbercomingupontwodice.sinceitis onlyessentialthatthenumbersbeequal,wecanapplytheruleforadding probabilities, 𝑃=𝑃 +𝑃 +𝑃 +𝑃 +𝑃 +𝑃 . 11 22 33 44 55 66 eachoftheprobabilities𝑃 is,inturn,aproduct𝑃×𝑃. hence 𝑖𝑖 𝑖 𝑖 1 1 1 𝑃=(𝑃 ×𝑃)+(𝑃 ×𝑃)+…+(𝑃 ×𝑃)=6( × )= . 1 1 2 2 6 6 6 6 6 thisresultcanbeobtainedrightawayfromfigure1.1,wherethefavourable outcomesareshowninthegray,(1,1),(2,2),(3,3),(4,4),(5,5),and(6,6).the totalnumberofsuchoutcomesissix.consequently,𝑃=6/36=1/6. frequency and probability. theclassicaldefinitionofprobability andtherulesforadditionandmultiplicationofprobabilitiescanbeusedto21 probability mk/100 0.24 00..2222 0.20 0.18 a a 0.16 0.14 0.12 1/6 = 0.167 0.10 k 1 5 10 15 20 25 figure1.2: outcomesofrollingadicemany times. calculatetheprobabilityofarandomevent.however,whatisthepractical valueofsuchcalculations?forinstance,whatdoesitmeaninpracticethat theprobabilityofgettingafourwhenadieisthrownequals1/6?naturally, theassertiondoesnotimplythatafourwillappearonceandonlyoncein anysixtrials.itispossiblethatitwillappearonce,butitisalsopossiblethat itwillappeartwo(ormore)times,orthatitwillnotappearatall.inorder todiscovertheprobabilityofaneventinpracticeweshouldperformalarge numberoftrialsandcalculatehowfrequentlyafourappears. letusperformseveralsetsoftrials,forinstance,throwingthedie100 timesineachset. letusdesignate𝑀 tobethenumberoftimesafour 1 appearsinthefirstset,𝑀 tobethenumberoffoursinthesecondset,etc. 2 theratios𝑀/100,𝑀/100,𝑀/100,… arethefrequencieswithwhicha 1 2 322 mathematicsofrandomness 0.22 et0.21 h s c a e n 0.20 e i m o c ut0.19 o d e r desi0.18 of y c n e0.17 u q a a e r f 0.16 1/6 = 0.167 n 100 500 1000 1500 2000 2500 number of trials figure1.3:frequenciesofoutcomesoftrialsofa dieasafunctionofnumberoftrials.notehow thedeviationofthefrequencyoftheoccurrence fourappearedineachset. havingperformedseveralsetsoftrials,wecan ofaneventfromitsprobabilitydecreasesasthe numberoftrialsincreases. seethatthefrequencyoftheappearanceofafour variesfromsettosetina randomfashioninthevicinityoftheprobabilityofthetrials,wecanseethat thefrequencyoftheappearanceofafour variesfromsettoset inarandom fashioninthevicinityoftheprobabilityofthegivenevent,i.e.inthevicinity of1/6.thisisclearfromfigure1.2,wherethenumber𝑘ofsetsoftrialsis plottedalongtheabscissaaxisandthefrequencieswithwhichafourappears alongtheaxisofordinates. naturally,ifweperformtheexperimentagain,wewillgetothervaluesof23 probability thefrequencies𝑀/100.however,thepatternofoscillationsofthefrequen- 𝑘 ciesoftheeventunderconsiderationwillbestable:thedeviationsupwards anddownwardsfromthestraightline𝐴𝐴,whichisassociatedwiththeproba- bilityoftheevent,willbalance.eventhoughtheamplitudesofthedeviations willvaryfromsettoset,theywillnottendtogrowordecrease. thisisa consequenceofthe equivalenceofeachsetoftrials.thenumberoftrialsin eachsetisthesame,andtheresultsobtainedinagivensetdonotdependon theresultsinanyotherset. letusmakeanimportantchangeinthatwegraduallyincreasethenumber oftrialsineachset.usingtheresultsofourpreviousexperiment,aspresented infigure1.2,letusobtainanewresultby addingthevalueofasetoftrials totheresultoftheprecedingsets.inotherwords,wecalculatethenumberof foursinthefirst100trials(inourcase,𝑀 =22),thenthenumberoffours 1 inthefirst200trials(𝑀 +𝑀 = 22+16 = 38),theinthefirst300trials 1 2 (𝑀 +𝑀 +𝑀 =22+16+18=56)etc.wethenfindthefrequenciesof 1 2 3 gettingafourintheeachnewset:𝑀/100=0.22,(𝑀 +𝑀)/200=0.19, 1 1 2 (𝑀+𝑀+𝑀)/300=0.187,etc.thesefrequenciesareplottedinfigure1.3 1 2 3 againstthenumberoftrialsineachset(100,200,…,2500). thefigure demonstratesacrucialfact:thedeviationofthefrequencyoftheoccurrence ofaneventfromitsprobabilitydecreasesasthenumberoftrialsincreases.in otherwords, frequencyoftheoccurrenceofarandomeventtendstoitsprobability withincreasingnumberoftrials. isitpossibletogiveadefinitionofprobabilitybasedonfrequency? sincethefrequencyoftheoccurrenceofarandomeventtendstoitsprob- abilityasthenumberoftrialsincreases,wemightwellaskwhetherwecan definetheprobabilityofaneventasthelimitoftheratioofthenumberofits occurrencetothenumberoftrialsasthenumberoftrialslendstoinfinity. suppose𝑁isthenumberoftrialsand𝑀 (𝑁)isthenumberofoccurrence 𝛢24 mathematicsofrandomness ofaneventa.wewanttoknowwhetherwecandefinetheprobability𝑃 of 𝛢 theevent𝐴as 𝑀 (𝑁) 𝑃 = lim [ 𝛢 ]. (1.2) 𝛢 𝛮→∞ 𝑁 richardvonmises(1883-1953),agermanmathematicianoftheearly 20thcentury,believedthatequation(1.2)couldbeconsideredadefinitionof theprobabilityofarandomevent,andhecalleditthe frequencydefinitionof probability.vonmisespointedoutthattheclassicaldefinitionofprobability (1.1)only“works”whenthereisafinitenumberofequallypossibleoutcomes. forinstance,situationsinvolvingthethrowingofcoinsordice. however, weoftenencountersituationswithoutthe symmetrythat determineswhethertheoutcomesareequallypossible.thesearethecases whenwecannotapplytheclassicaldefinitionofprobability. vonmises assumedthatthenthefrequencydefinitioncanbeusedbecauseitdoesnot requireafinitenumberofequallypossibleoutcomesand,moreover,doesnot requireanycalculationofprobabilityatall.aprobabilityusingthefrequency approachisdeterminedbyexperimentratherthanbeingcalculated. however,isitpossibletodeterminetheprobabilityofarandomevent inpracticeusing(1.2)?therelationshippresupposesan infinitenumberof identicaltrials.inpractice,wemuststopata finitenumberoftrials,andit isdebatablewhatnumbertostopat.shouldwestopafterahundredtrials, orisitnecessaryfortheretobeathousand,amillion,orahundredmillion? andwhatistheaccuracyoftheprobabilitydeterminedinsuchaway?there arenoanswerstothesequestions. bcsides,itisnotpracticabletoprovide thesameconditionswhileperformingaverylargenumberoftrials,tosay nothingofthefactthatthetrialsmaybeimpossibletorepeat. consequently,relationship(1.2)ispracticallyuseless,moreoveritispos- sibletoprove(thoughishallnotdoso)thatthelimitin(1.2) doesnotstrictly25 probability speaking exist.thismeansthatthevonmises’serrorwasthathemadean unwarrantedgeneralizationfromacorrectproposition:heconcludedthat theprobabilityofarandomevenisthelimitofthefrequencyofitsoccurrence whenthenumberoftrialstendstoinfinityfromthecorrectobservationthat thefrequencyoftheoccurrenceofarandomevenapproachesitsprobability asthenumberoftrialsincreases. geometricaldefinitionofprobability.supposethattwopeoplehave agreedtomeetatacertainplacebetweennineandteno’clock. theyalso agreedthateachwouldwaitforaquarterofanhourand,iftheotherdidn’t arrive,wouldleave.whatistheprobabilitythattheymeet?suppose𝑥isthe momentonepersonarrivesattheappointedplace,and𝑦isthemomentthe otherarrives.letusconsiderapointwithcoordinates(𝑥,𝑦)onaplaneasan outcomeoftherendezvous.everypossibleoutcomeiswithintheareaofa squareeachsideofwhichcorrespondstoonehour(figure1.4).theoutcome isfavourable(thetwomeet)forallpoints(𝑥,𝑦)suchthat|𝑥−𝑦| ≤ 1/4. thesepointsarewithinthebluepartofthesquareinthefigure1.4. alltheoutcomesareexclusiveandequallypossible,andthereforethe probabilityoftherendezvousequalstheratiooftheblueareatothearea ofthesquare. thisisreminiscentoftheratiooffavourableoutcomesto thetotalnumberofequallypossibleoutcomesintheclassicaldefinitionof probability.itshouldbeborneinmindthatthisisacasewherethenumber ofoutcomes(bothfavourableandunfavourable)isinfinite. therefore,in- steadofcalculatingtheratioofthenumberoffavourableoutcomestothe totalnumberofoutcomes,itisbettertoconsiderheretheratioofthearea containingfavourableoutcomestothetotalareaoftherandomevents. itisnotdifficulttousefigure1.4andfindthefavourablearea;itisthe differencebetweentheareaofthewholesquareandtheunhatchedarea,i.e. 1−(3/4)2 =7/16ℎ2.dividing7/16ℎ2by1ℎ2,wefindtheprobabilityof therendezvoustobe7/16. thisexampleillustratesthegeometricaldefinitionofprobability:26 mathematicsofrandomness y, h 10 r u o h 1 1 /4 9 1 /4 1 hour x, h 9 10 figure1.4: findingtheprobabilityusingthe geometricalmethod. theprobabilityofarandomeventistheratiooftheareafavourable foraneventtothetotalareaofevents. thegeometricaldefinitionofprobabilityisageneralizationoftheclassical definitionforthecasewhenthenumberofequallypossibleoutcomesis infinite. thedevelopmentoftheconceptofprobability.althoughprobabilis- ticnotionswereusedbyancientgreekphilosophers(suchasdemocritus,27 probability epicurus,andcaruslucretius),thetheoryofprobabilityasasciencebe- ganinthemid-17thcentury,withtheworkofthefrenchscientistsblaise pascalandpierrefermatandthedutchscientistchristianhuygens.the classicaldefinitionfortheprobabilityofarandomeventwasformulatedby theswissmathematicianjacobbernoulliin arsconjectandi (theartof conjectures).thedefinitionwasgivenitsfinalshapelaterbypierrelaplace. thegeometricaldefinitionofprobabilitywasfirstappliedinthe18thcentury. importantcontributionstoprobabilitytheoryweremadebytherussian mathematicalschoolinthe19thcentury(p.l.chebyshev,a.a.markov,and a.m.lyapunov). theextensiveemploymentofprobabilisticconceptsinphysicsandtech- nologydemonstrated,bytheearly20thcentury,thattherewasaneedfora morerefineddefinitionofprobability.itwasnecessary,inparticular,inorder toeliminatetherelianceofprobabilityon“commonsense”.anunsuccessful attempttogiveageneraldefinitionfortheprobabilityofarandomevent onthebasisofthelimitofitsfrequencyofoccurrencewasmade,aswehave seen,byrichardvonmises.however,an axiomaticapproachratherthana frequencyoneresultedinmorerefineddefinitionofprobability.thenew approachwasbasedonasetofcertainassumptions(axioms)fromwhichall theotherpropositionsarededucedusingclearlyformulatedrules. the axiomatic definitionofprobabilitynowgenerallyacceptedwas elaboratedbythesovietmathematiciana.n.kolmogorov,memberofthe ussracademyofsciences,in thebasicnotionsoftheprobabilitytheory (1936,inrussian).ishallnotdiscusstheaxiomaticdefinitionofprobability becauseitwouldrequiresettheory.letmeonlyremarkthatkolmogorov’s axiomsgaveastrictmathematicalsubstantiationtotheconceptofprobability andmadeprobabilitytheoryafullyfledgedmathematicaldiscipline. theexistenceofseveraldefinitionsforthesamenotion(probability) shouldnotworrythereader.28 mathematicsofrandomness asl.e.maistrovputitin thedevelopmentofthenotionofprobability (nauka,moscow,1980): “therearemanydefinitionsofnotions,andthisisanessentialfeature ofmodernscience.hencethenotionofprobabilityisnoexception. moderndefinitionsinsciencerepresentdiverseviewpoints,ofwhich theremaybeverymanyforafundamentalnotion,andeachviewre- flectsapropertyofthedefinednotion. thisincludesthenotionof probability.” letmeaddthatnewdefinitionsforanotionappearasourunderstandingof itbecomesdeeperanditspropertiesaremadeclearer. random numbers randomnumbergenerators. letusputtenidenticalballsnumbered from0to9intoabox. wetakeoutaballatrandomandwritedownits number.supposeitisfive.thenweputtheballbackintothebox,stirthe ballswell,andtakeoutaballatrandom.supposethistimewegetaone.we writeitdown,puttheballbackintothebox,stirtheballs,andtakeoutaball atrandomagain. thistimewegetatwo. repeatingthisproceduremany times,weobtainadisorderedsetofnumbers,forinstance:5,1,2,7,2,3,0, 2,1,3,9,2,4,4,1,3,…thissequenceisdisorderedbecauseeachnumber appeared atrandom,sinceeachtimeaballwastakenoutatrandomfroma well-stirredsetofidenticalballs. havingobtainedasetofrandomdigits,wecancompileasetofrandom numbers.letusconsider,forinstance,four-digitnumbers.weneedonly separateourseriesofrandomnumbersintogroupsoffourdigitsandconsider eachgrouptobearandomnumber:5127,2302,1392,4413,… anydevicethatyieldsrandomnumbersiscalleda randomnumber generator. therearethreetypesofgenerators: urns, dice,and roulettes29 randomnumbers (figure1.5).ourboxwithballsisanurn. dicearethesimplestrandomnumbergenerators.anexampleofsucha generatorisacubeeachofwhosefacesismarkedwithadifferentnumber. anotherexampleisacoin(oratoken).supposefiveofthefacesofacubeare markedwiththenumbers0,1,2,3,4,whilethesixthfaceisunmarked.now supposewehaveatokenonesideofwhichislabelledwith0andtheother with5.letusthrowthecubeandtokensimultaneouslyandaddtogetherthe numbersthatappearfaceup,thetrialbeingdiscountedwhentheunmarked facelandsfaceup. thisgeneratorallowsustoobtainadisorderedsetof numbersfrom0to9,whichcanthenbeeasilyusedtoproducesetsofrandom numbers. arouletteisacirclemarkedinsectors,eachofwhichismarked withadifferentnumber. aroulettehasarotatingarroworrollingball. a trialinvolvesspinningthearrowandrecordingthenumber arouletteisacirclemarkedinsectors,eachofwhichismarkedwith adifferentnumber. aroulettehasarotatingarroworrollingball. atrial involvesspinningthearrowandrecordingthenumbercorrespondingtothe sectoroftheroulettecirclewithinwhichthearrowstops. notethataroulettemayhaveanynumberofsectors.forinstance.we coulddivideacircleintotensectorsandlabelthemfrom0to9.asarandom numbergenerator,ourrouletteinthiscaseisequivalenttothetwogenerators discussedabove: (1) anurnwithtenballsand (2) adieandatokenthrownatthesametime. adiagramoftheseequivalentrandomnumbergeneratorsisshowninfig- ure1.5. tablesofrandomnumbers.anexampleofarandomnumbertableis showninfigure1.6.thetableconsistsofthreehundredfour-digitnumbers. eachdigitinthetablewaschosenrandomly,asaresultofatrial,e.g.throwing adieandatoken.therefore,itisunderstandablethatthereisnoorderinthe30 mathematicsofrandomness 1 0 2 9 3 2 8 4 0 9 3 8 7 5 1 4 6 5 7 6 anurnwithballs aroulette 4 0 3 0 2 1 5 6 a die a token 0 1 2 3 4 00 55 00 55 00 55 00 55 00 55 total 0 5 1 6 2 7 3 8 4 9 figure1.5:threetypesofrandomnumbergen- erators:urns,diceandroulettes.31 randomnumbers numbers,andthereisnowayofpredictingwhichdigitwillfollowagiven one. youcouldcompilemanytablesaftermanytrials. nevertheless,there willnotbeeventheshadowoforderinthesequenceofdigits. 0655 8453 4467 3234 5320 0709 2523 9224 6271 2607 5255 5161 4889 7429 4647 4331 0010 8144 8638 0307 6314 8951 2335 0174 6993 6157 0063 6006 1736 3775 3157 9764 4862 5848 6919 3135 2837 9910 7791 8941 9052 9565 4635 0653 2254 5704 8865 2627 7959 3682 4105 4105 3187 4312 1596 9403 6859 7802 3180 4499 1437 2851 6727 5580 0368 4746 0604 7956 2304 8417 4064 4171 7013 4631 8288 4785 6560 8851 9928 2439 1037 5765 1562 9869 0756 5761 6346 5392 2986 2018 5718 8791 0754 2222 2013 0830 0927 0466 7526 6610 5127 2302 1392 4413 9651 8922 1023 6265 7877 4733 9401 2423 6301 2611 0650 0400 5998 1863 9182 9032 4064 5228 4153 2544 4125 9654 6380 6650 8567 5045 5458 1402 9849 9886 5579 4171 9844 0159 2260 1314 2461 3497 9785 5678 4471 2873 3724 8900 7852 5843 4320 4553 2545 4436 9265 6675 7989 5592 3759 3431 3466 8269 9926 7429 7516 1126 6345 4576 5059 7746 figure1.6:atableofrandomnumbers. 9313 7489 2464 2575 9284 1787 2391 4245 5618 0146 5179 8081 3361 0109 7730 6256 1303 6503 4081 4754 3010 5081 3300 9979 1970 6279 6307 7935 4977 0501 9599 9828 8740 6666 6692 5590 2455 3963 6463 1609 4242 3961 6247 4911 7264 0247 0583 7679 7942 2482 3585 9123 5014 6328 9659 1863 0532 6313 3199 7619 5950 3384 0276 4503 3333 8967 3382 3016 0639 2007 8462 3145 6582 8605 7300 6298 6673 6406 5951 7427 0456 0944 3058 2545 3756 2436 2408 4477 5707 5441 0672 1281 8897 5409 0653 5519 9720 0111 4745 7979 5163 9690 0413 3043 1014 0226 5460 2835 3294 3674 4995 9115 5273 1293 7894 9050 1378 2220 3756 9795 6751 6447 4991 6458 9307 3371 3243 2958 4738 3996 thisisnotamazing. achanceisachance. butachancehasareverse aspect. forinstance,tryandcounthowmanytimeseachdigitoccursin figure1.6.youwillfindthatdigit0occurs118times(thefrequencyitappears is118/1200 = 0.099),digit1occurs110times(thefrequencyitappearsis 0.090),digit2occurs114times(0.095),digit3occurs125times(0.104),digit32 mathematicsofrandomness 4occurs135times(0.113),digit5occurs135times(0.113),digit6occurs 132times(0.110),digit7occurs116times(0097),digit8occurs93times (0.078),anddigit9occurs122times(0.102).wecanseethattheappearance frequencyforeachdigitisaboutthesame,i.e.closeto0.1.naturally,the readerhascometoaconclusionthat0.1isthe probabilitythatadigitappears. thereadermaysaythattheappearancefrequencyofadigitisclosetothe probabilityofitsappearanceoveralongseriesoftrials(thereare1200trials here). althoughthisisnatural,weshouldwonderonce(againhowanunordered setof randomdigitscanhavean inherentstability.thisisademonstration ofthereverseaspectofchanceandillustratesthedeterminismof probability. iadvisethereaderto“work”alittlewitharandomnumbertable(see figure1.6).forinstance,32numbersoutofthethreehundredonesinthe tablebeginwithzero,20beginwith1,33beginwith2,33beginwith3,38 beginwith4,34beginwith5,34beginwith6,24beginwith7,20begin with8,and32beginwith9. theprobabilitythatanumberbeginswitha certaindigitequals0.1. itiseasytoseethattheresultsofourcountarein arathergoodkeepingwiththisprobability(onetenthofthreehundredis thirty). however,thedeviationsaremorenoticeablethanintheexample consideredearlier.butthisisnaturalbecausethenumberoftrialsabovewas 1200whilehereitismuchless,only300. itisalsointerestingtocounthowmanytimesadigitoccursinthesecond place(thenumberofhundreds),inthethirdplace(tens),andthefourthplace (units).itiseasytoseethatineverycasethefrequencywithwhichagiven digitappearsisclosetotheprobability,i.e.closeto0.1.thus,zerooccursin thesecondplace25times,inthethirdplace33times,andinthefourthplace 28times. anexamplewiththenumber-platesofmotor-carsrandomlypassingthe observerwascitedintheintroduction.itwasnotedthattheprobabilitythat thefirsttwodigitsinthelicencenumberwereidenticalis0.1.theprobability33 randomevents thatthetwolastdigitsofthenumberortwomiddledigitsorthefirstandthe lastdigitareidenticalisthesame. inordertoseethis,weneednotobserveasequenceofcarspassingby. wecansimplyusearandomnumbertable(seefigure1.6). thefour-digit randomnumbersinthetablecanbetakenasthelicensenumbersofcars randomlypassingtheobserver.wecanseethat40ofthe300number’shave thesametwofirstdigits,28numbershavethesametwolastdigits,24numbers havethesametwomiddledigits,and32numbershavethesamefirstandlast digits.inotherwords,thefrequencieswithwhichapairofidenticaldigits appearsactuallyvariesaroundtheprobability,i.e.intheneighbourhoodof 0.1. random events whenwethrowadieortakeaballoutofanurnwedealwitha randomevent. thereareseveralinterestingproblemswheretheprobabilityofarandom eventisrequiredtobefound. aproblemwithcolouredballs. therearethreeblueballsandared ballinabox. youtaketwoballsoutoftheboxatrandom. whichismore probable:thatthetwoballsareblueorthatoneisblueandoneisred? peopleoftenanswerthatitismoreprobablethattwoblueballsaretaken outbecausethenumberofblueballsintheboxisthreetimesgreaterthan thenumberofredones. however,theprobabilityoftakingouttwoblue ballsis equaltotheprobabilityoftakingoutablueandaredball.youcan seethisbyconsideringfigure1.7.clearlytherearethreewaysinwhichtwo blueballsmaybechosenandthreewaysofchoosingablueandaredballat thesametime.therefore,theoutcomesareequallyprobable. figure1.7:differentwaysoftakingouttwoout ofthreeblueandoneredballs. wecanalsocalculatetheprobabilityoftheoutcomes.theprobability34 mathematicsofrandomness oftakingouttwoblueballsequalstheproductoftwoprobabilities. the firstoneistheprobabilityoftakingoutablueballfromasetoffourballs (threeblueonesplusaredone),whichis3/4.thesecondprobabilityisthat oftakingoutablueballfromasetofthreeballs(twoblueonesplusared one)whichis2/3.consequently,theprobabilityoftakingouttwoblueballs simultaneouslyis3/4×2/3=1/2. theprobabilityoftakingoutablueandaredballisthesum𝑃 +𝑃 , br rb where𝑃 ,istheprobabilityoftakingoutablueballfromasetoffourballs br (threeblueonesplusaredone)multipliedbytheprobabilityoftakingouta redballfromasetofthreeballs(twoblueonesplusitredone)and𝑃 ,isthe rb probabilityoftakingoutaredballfromasetoffourballs(thesecondallinthis casemustthenbeablueone).inotherwords,𝑃 istheprobabilityoftaking br outablueballfirstandthenaredballwhile𝑃 istheprobabilityoftaking rb outaredballfirstandthenablueball.inasmuchas𝑃 =3/4×1/3=1/4 br and𝑃 = 1/4,theprobabilityoftakingoutapairofdifferentlycoloured rb ballsequals1/4+1/4=1/2. throwingadie: agame.therearetwoplayersinthisgame,player 𝐴andplayer𝐵.thedieisthrownthreetimesinsuccessionduringeachturn. ifacertainfaceturnsupatleastonceduringaturn(letitbea5),player𝐴 scoresapoint.butifthefivedoesnotturnup,apointisscoredbyplayer𝐵. thegameisplayeduntiloneofthemscores,say,ahundredpoints.whohas thechanceofwinninggreater?player𝐴orplayer𝐵? inordertoanswer,wefirstcalculatetheprobabilityofplayer𝐴scoringa pointinaturn(thedieisthrownthreetimesinsuccession).hereceivesapoint inanyofthefollowingthreecases:iffiveturnsupinthefirsttrial,iffivedoes notturnupinthefirsttrialbutturnsupinthesecondone,andiffivedoesnot turnupinthefirsttwotrialsbutturnsupinthethirdone.letusdesignate theprobabilityofthesethreeeventsas𝑃,𝑃,and𝑃,respectively.thesought 1 2 3 probabilityis𝑃=𝑃 +𝑃 +𝑃.notethattheprobabilityoffiveappearing 1 2 3 whenthedieisthrownis1/6,andtheprobabilitythatfivedoesnotappearis35 randomevents 5/6.itisclearthat𝑃 =1/6.tofind𝑃,weshouldmultiplytheprobability 1 2 oftheabsenceofafiveinthefirsttrialbytheprobabilityofitspresencein thesecondtrial,𝑃 =5/6×1/6=5/36.theprobability𝑃 istheproductof 2 3 theprobabilityoftheabsenceofafiveintwotrials(thefirstandthesecond) andtheprobabilityofafiveinthethirdtrial,𝑃 =(5/6)2×1/6=25/216. 3 consequently,𝑃 = 𝑃 +𝑃 +𝑃 = 1/6+5/6+25/216 = 91/216. since 1 2 3 𝑃 < 1/2,player𝐵hasmorechanceofwinningthisgame. wecouldhave reachedthesameconclusioninasimplerwaybyconsideringtheprobability ofplayer𝐵scoringapointafterthreetrials. thisistheprobabilityofthe absenceoffiveinthreetrials:𝑝=5/6×5/6×5/6=125/216.since𝑝>1/2, player𝐵’schancesarebetter. notethat𝑃+𝑝 = 91/216+125/216 = 1. thisisnaturalbecauseoneoftheplayers,𝐴or𝐵,mustscoreapointineach turn. letuschangetherulesofthegamealittle:thedieisthrownfourtimes ratherthanthreetimesineachturn.theotherconditionsremainthesame. theprobabilityofplayer𝐵scoringapointinaturnis5/6×5/6×5/6×5/6= 625/1296.thisislessthan1/2,andthereforenow𝐴hasabetterchanceof winningagame. theproblemofanastrologer.atyrantgotangrywithanastrologer andorderedhisexecution. however,atthelastmomentthetyrantmade uphismindtogivetheastrologerachancetosavehimself. hetooktwo blackandtwowhiteballsandtoldtheastrologertoputthemintotwourns atrandom.theexecutionerwastochooseanurnandpickaballoutofitat random.iftheballwaswhite,theastrologerwouldbepardoned,andifthe ballwasblack,hewouldbeexecuted.howshouldtheastrologerdistribute theballsbetweenthetwournsinordertogivehimselfthegreatestchanceof beingsaved? supposetheastrologerputsawhiteandablackballintoeachurn(fig- ure1.8(a)).inthiscase,nomatterwhichurntheexecutionerchooses,hewill drawawhiteballoutofitwithaprobabilityof1/2.therefore,theprobability36 mathematicsofrandomness (a) (b) (c) (d) figure1.8: differentwaysofarrangingtwo whiteandtwoblackballsfordifferentproba- theastrologerwouldbesavedis1/2. bilitiesofdrawingoutaballofagivencolour. theprobabilityoftheastrologerbeingsavedwillbethesameifheputs thetwowhiteballsintooneurnandthetwoblackballsintotheother(fig- ure1.8(b)).hisdestinywillbedecidedbytheexecutionerwhenhechooses anurn.theexecutionermaychooseeitherurnwithequalprobability. thebestsolutionfortheastrologeristoputawhiteballintooneum andawhiteballandtwoblackonesintotheotherurn(figure1.8(c)).ifthe executionerchoosesthefirsturn,theastrologerwillcertainlybesaved,but iftheexecutionerpicksthesecondurn,theastrologerwillbesavedwitha probabilityof1/3.sincetheexecutionerchooseseitherurnwithprobability 1/2,theoverallprobabilitythattheastrologerwillbesavedis(1/2×1)+ (1/2×1/3)=2/3. bycontrast,iftheastrologerputsablackballintooneurnandablack ballandtwowhiteballsintotheother(figure1.8(d)),theprobabilityofhim beingsavedwillbesmallest:(1/2×0)+(1/2×2/3)=1/3. thus,inordertohavethegreatestchanceofbeingsaved,theastrologer shoulddistributetheballsbetweentheurnsasshownin(figure1.8(c))this

37 randomevents isthebeststrategy.theworststrategyistodistributetheballsasshownin (figure1.8(d)).ofcourse,theselectionofthebeststrategydoesnotguarantee thedesiredoutcome.althoughtheriskisdecreased,itstillremains. wanderinginalabyrinth.alabyrinthwithtreasurehasadeathtrap, asshowninfigure1.9.unluckytreasure-huntersdieinthetrap.whatisthe 5 probabilitythattheywillavoidthetrapandreachthetreasure? afterwalkingawayfromtheentrance𝐴topoint1(seefigure1.9)a 4 treasure-huntermayeithergostraightahead(inwhichcasehewalksdirectly intothetrap)orturntotheleft(inwhichcasehearrivesatpoint2)weshall supposehepickseitherpathatrandom,withequalprobability,i.e. with probability1/2.alterarrivingatpoint2,thetreasure-huntermayeithergo straightaheadorturnrightorturnleftwithprobability1/3.thefirsttwo pathsleadtothetrap,whilethethirdpathleadstopoint3.theprobability ofsomeonegettingfromtheentrance𝐴topoint3istheproductofthe 3 probabilityofturningleftatpoint1andtheprobabilityofturningleftat point2,i.e.,1/2×1/3.itiseasytoseenowthattheprobabilityofreaching point4from𝐴is1/2×1/3×1/2;theprobabilityofreachingpoint5from𝐴 2 is1/2×1/3×1/2×1/3;andfinally,theprobabilityofreachingthetreasure a from𝐴is𝑃+ =1/2×1/3×1/2×1/3×1/2=1/72.theonlywayofgetting fromtheentranceofthelabyrinthtothetreasureisshowninthefigureby thedashline.theprobabilitythatapersonwillfollowitisthus𝑃+ =1/72, 1 whiletheprobabilityofwalkingintothetrapis𝑃− =71/72. theprobability𝑃− wascalculatedfromthefactthat𝑃+ + 𝑃− = 1. figure1.9:theprobabilityoffindingthetrea- sureoratrapinalabyrinth. however,wecancalculate𝑃−directly.letusexpand𝑃−asthesum𝑃− = 𝑃 +𝑃 +𝑃 +𝑃 +𝑃 wherethe𝑃aretheprobabilitiesofarrivingatpoint 1 2 3 4 5 𝑖 𝑖from𝐴multipliedbytheprobabilityofwalkingintothetrapfrompoint38 mathematicsofrandomness 𝑖 (𝑖=1,2,3,4,5). 𝑃 =1/2, 1 𝑃 =1/2×2/3, 2 𝑃 =1/2×1/3×1/2, 3 𝑃 =1/2×1/3×1/2×2/3, 4 𝑃 =1/2×1/3×1/2×1/3×1/2. 5 youcanthenfindthat𝑃 +𝑃 +𝑃 +𝑃 +𝑃 =71/72. 1 2 3 4 5 discrete random variables randomvariables.supposethereisabatchof100manufacturedarticles and11articlesarerejectedasdefective,9articlesarerejectedinanotherbatchof thesamesize,10articlesarerejectedinthethirdone,12articlesarerejectedin thefourthone,etc.weuse𝑛todenotetheoverallnumberofmanufactured articlesinabatchand𝑚todenotethenumberofrejectedarticles. the number𝑛isconstant(here𝑛=100)whilethevalueof𝑚variesfrombatch tobatchinarandommanner. supposethereisa definiteprobabilitythat therewillbe𝑚rejectedarticlesinarandomlyselectedbatchof𝑛articles. thenumberofrejectedarticles(thevariable𝑚)isanexampleofarandom variable. itvariesrandomlyfromonetrialtoanother,andacertainprobability isassociatedwiththeoccurrenceofeachvalueofthevariable.notethatweare dealingwithadiscreterandomvariablehere,i.e.itmayonlytakeadiscrete setofvalues(theintegersfrom0to100inthiscase). therearealso continuousrandomvariables. forinstance,thelength andweightofnewbornbabiesvaryrandomlyfromchildtochildandmay takeanyvaluewithinaparticularintervaltherearesomespecialfeatures ofcontinuousrandomvariableswhichweshalldiscusslater;weshallfirst39 discreterandomvariables considerdiscretevariables. expectedvaluesandvarianceofadiscreterandomvariable.let𝑥 beadiscreterandomvariablewhichmayassume𝑠values:𝑥 ,𝑥 ,…𝑥 ,…𝑥. 1 2 𝑚 𝑠 thesevaluesareassociatedwiththeprobabilities𝑝 ,𝑝 ,…𝑝 ,…𝑝. forin- 1 2 𝑚 𝑠 stance,𝑝 istheprobabilitythatavariableis𝑥 . thesumofalltheprob- 𝑚 𝑚 abilities(𝑝 + 𝑝 + … + 𝑝)istheprobabilitythatatrialwillgiveoneof 1 2 𝑠 thevalues𝑥 ,𝑥 ,…𝑥,(withoutsayingwhichone).thisprobabilityisunity. 1 2 𝑠 consequently, 𝑠 ∑ 𝑝 =1, (1.3) 𝑚 𝑚=1 thesetofprobabilities𝑝 +𝑝 +…+𝑝 (alsocalledthedistributionofthe 𝑠 1 2 𝑠 thenotation∑meansthatthesummationis probabilities)containsalltheinformationneededabouttherandomvariable. 𝑚=1 performedoverall𝑚from1to𝑠. however,wedonotneedalltheprobabilitiesformanypracticalpurposes.it issufficienttoknowtwomostimportantcharacteristicsofarandomvariable: itsexpectedvalue(itsmathematicalexpectation)anditsvariance. the expectedvalueisanaveragevalueoftherandomvariabletakenover alargenumberoftrials.weshallusetheletter𝐸todenotetheexpectedvalue. theexpectedvalueofarandomvariable𝑥isthesumoftheproductsofeach variableanditsprobability,i.e. 𝐸(𝑥)=𝑝 𝑥 +𝑝 𝑥 +…+𝑝𝑥, 1 1 2 2 𝑠 𝑠 orusingthesummationsign, 𝑠 𝐸(𝑥)=∑ 𝑝 𝑥 . (1.4) 𝑚 𝑚 𝑚=1 wealsoneedtoknowhowavariabledeviatesfromtheexpectedvalue,or, inotherwords,howmuchtherandomvariableis scattered.theexpected valueofthedeviationfromtheexpectedvalue(thatisthedifference𝑥−𝐸(𝑥))40 mathematicsofrandomness cannotbeusedbecauseitisequaltozero.wecanshowthisasfollows: 𝑠 𝐸(𝑥−𝐸(𝑥))=∑ 𝑝 (𝑥 −𝐸(𝑥)), 𝑚 𝑚 𝑚=1 𝑠 𝑠 =∑ 𝑝 𝑥 −𝐸(𝑥) ∑ 𝑝 , 𝑚 𝑚 𝑚 𝑚=1 𝑚=1 =𝐸(𝑥)−𝐸(𝑥), =0. thisiswhytheexpectedvalueofthe squareddeviation(ratherthanthe expectedvalueofthedeviationitself)isused,i.e. 𝑠 var=𝜎2 =𝐸(𝑥—𝐸(𝑥))2 =∑ 𝑝 (𝑥 −𝐸(𝑥))2. (1.5) 𝑚 𝑚 𝑚=1 thisisthevarianceofarandomvariableandweshallusevartodenoteit.the squarerootofthevariable√variscalledthe standard(or root-mean-square) deviation𝜎oftherandomvariable.itiseasytoshowthat var=𝐸(𝑥2)−(𝐸(𝑥))2. (1.6) indeed, 𝑠 𝑠 ∑ 𝑝 (𝑥 −𝐸(𝑥))2 =∑ 𝑝 (𝑥2 −2𝑥 𝐸(𝑥)+𝐸(𝑥))2), 𝑚 𝑚 𝑚 𝑚 𝑚 𝑚=1 𝑚=1 𝑠 𝑠 𝑠 =∑ 𝑝 𝑥2 −2𝐸(𝑥)∑ 𝑝 𝑥 +(𝐸(𝑥))2∑ 𝑝 , 𝑚 𝑚 𝑚 𝑚 𝑚 𝑚=1 𝑚=1 𝑚=1 =𝐸(𝑥2)−2𝐸(𝑥)𝐸(𝑥)+(𝐸(𝑥))2, =𝐸(𝑥2)−(𝐸(𝑥))2. twoprobabilitydistributionsareshowninfigure1.10(a). thetwo randomvariablespossessdifferentexpectedvalueswhilehavingthesame41 discreterandomvariables e(x) < e(x) e(x) = e(x) 1 2 1 2 var x = var x var x < var x 1 2 1 2 e(x1), var x1 e(x2), var x2 e(x1), var x1 e(x2), var x2 x x (a) m (b) m figure1.10:distributionsofrandomvariables withdifferentparameters.(a)showstwodistri- butionswithdifferentexpectedvaluesbutthe variance.lookingatfigure1.10(b),wecanseeadifferentpicture:therandom samevariance,while(b)showstwodistributions variablespossessdifferentvarianceswhilehavingthesameexpectedvalues. withdifferentvariancebutthesameexpectedval- ues. bernoulli’sbinomialdistribution.supposeaseriesof𝑛independent identicaltrialsisperformed. thetrialsareindependentinthesensethat theresultsofanytrialdonotinfluencetheresultsofanyothertrial.some trialsproduceadesiredoutcomewhiletherestdonot.letuscallthedesired outcome,“event𝑈”.thisisarandomevent.supposeevent𝑈occursinin trials.thisisarandomvariable.letusconsidertheprobability𝑃 (𝑚)that 𝑛 event𝑈willoccur𝑚timesinaseriesof𝑛trials. thisisacommonlyoccurringsituation.suppose𝑛manufacturedarticles arechecked.thenevent𝑈isarejection,and𝑃 (𝑚)istheprobabilityofin 𝑛 articlesbeingrejectedoutofasetofinarticlessupposeahospitalregisters 𝑛newbornbabiesandtheevent𝑈isthebirthofagirl.hence𝑃 (𝑚)isthe 𝑛 probabilitythattherewillbe𝑚girlsinasetof𝑛newbornbabies.suppose inalottery,𝑛ticketsarechecked,event𝑈isthediscoveryofaprize-winning ticket,and𝑃 (𝑚)istheprobabilitythat𝑚prize-winningticketswillbefound 𝑛 outofatotalof𝑛tickets. supposeinaphysicsexperiment𝑛neutronsare recorded,theevent𝑈istheoccurrenceofaneutronwithanenergywithin42 mathematicsofrandomness acertainrange,and𝑃 (𝑚)istheprobabilitythat𝑚ofthe𝑛neutronswill 𝑛 possessenergiesintherange.inalltheseexamples,theprobability𝑃 (𝑚)is 𝑛 describedbythesameformulawhichisthe binomialdistribution(sometimes namedaftera17thcenturyswissmathematiciancalledjacobbernoulli). thebinomialdistributionisderivedbyassumingthattheprobability thatevent𝑈willoccurinasingletrialisknownanddoesnotvaryfromtrial totrial. letuscallthisprobability𝑝. theprobabilitythatevent𝑈does notoccurinasingletrialis𝑞 = 1−𝑝. itisimportantthattheprobability thatanarticleisrejecteddoesnotdependinanywayonhowmanyrejected articlesthereareinthegivenbatch.theprobabilitythatagirlisborninany actualdeliverydoesnotdependonwhetheragirloraboywasborninthe previousbirth(noronhowmanygirlshavesofarbeenborn).theprobability ofwinningaprizeneitherincreasesnordecreasesasthelotteryticketsare checked.theprobabilitythataneutronhasanenergyinagivenrangedoes notchangeduringtheexperiment. now, oncetheprobability𝑝thatacertainrandomeventwilloccurina singletrialisknown,wefindtheprobability𝑃(𝑚)of𝑚occurrencesinaseries 𝑛 of𝑛independentidenticaltrials. supposetheevent𝑈occurredinthefirst𝑚trialsbutdidnotoccurin 𝑛−𝑚trials,thentheprobabilityofthesituationwouldbe𝑝𝑚𝑞𝑛−𝑚.naturally, otherordersarepossible.forinstance,event𝑈maynotoccurinthefirst𝑛−𝑚 trialsandoccurintherestintrials.theprobabilityofthissituationisalso 𝑝𝑚𝑞𝑛−𝑚.therearealsootherpossiblesituations.thereareasmanysituations astherearewayschoosing𝑛elementstakeninatatime(thisiswritten(𝑛)). 𝑚 theprobabilityofeachsituationisidenticalandequals𝑝𝑚𝑞𝑛−𝑚.theorder inwhichevent𝑈occursisinessential.itisonlyessentialthatitoccursin𝑚 trialsanddoesnotoccurintheremaining𝑛−𝑚trials.thesoughtprobability 𝑃(𝑚)isthesumoftheprobabilitiesofeach(𝑛)situation,i.e.theproduct 𝑛 𝑚43 discreterandomvariables of𝑝𝑚𝑞𝑛−𝑚and(𝑛): 𝑚 𝑛 𝑃(𝑚)=( )𝑝𝑚𝑞𝑛−𝑚. (1.7) 𝑛 𝑚 thereisaformulaforthenumberofcombinationsof𝑛elementstaken𝑚at atime: 𝑛 𝑛! 𝑛(𝑛−1)(𝑛−2)…(𝑛−𝑚+1) ( )= = . (1.8) 𝑚 𝑚!(𝑛−𝑚)! 𝑚! here𝑛!=1⋅2⋅3⋅…⋅𝑛(read𝑛!as“enfactorial”),byconvention0!=1. substituting(1.8)into(1.7),wecanfind 𝑛! 𝑃(𝑚)= 𝑝𝑚𝑞𝑛−𝑚. (1.9) 𝑛 𝑚!(𝑛−𝑚)! thisisthe binomialdistribution,orthedistributionofabinomialrandom variable.ishallexplainthistermbelow,andweshallseethat 𝑛 ∑𝑃 (𝑚)=1. (1.10) 𝑛 𝑚=0 p (m) 20 8 1 0. m 0 4 6 8 10 12 14 16 20 bywayofexample,letuscalculatetheprobabilitythat𝑚girlsarebornin figure1.11:thebinomialdistribution. agroupof20babies.assumethattheprobabilityofdeliveringagirlis1/2,44 mathematicsofrandomness weset𝑝=1/2and𝑛=20inexpression(1.9)andconsidertheintegervalues ofvariable𝑚withintherangefrom0to20.theresultcanbeconveniently presentedasadiagram(figure1.11). weseethatthebirthof10girlsisthe mostprobable;theprobabilityofdelivering,forinstance,6or14girlsissix timessmaller. ifarandomvariablehasabinomialdistribution,thenitsexpectedvalue is 𝑛 𝐸(𝑚)=∑𝑚𝑃(𝑚). 𝑛 𝑚=0 ortheproductofthenumberoftrialsandtheprobabilityoftheeventina singletrial, 𝐸(𝑚)=𝑛𝑝. (1.11) thevarianceofsucharandomvariableistheproductofthenumberof trials,theprobabilityoftheoccurrenceoftheeventinasingletrial,andthe probabilityitdoesnotoccur: var=𝐸(𝑚2)−(𝐸(𝑚))2 =𝑛𝑝𝑞. (1.12) thenormal(gaussian)distribution.probabilitycalculationsusing thebinomialdistributionaredifficultforlarge𝑛.forinstance,inorderto findtheprobabilitythat30girlsweredeliveredfrom50births,youhaveto calculate 50! 𝑃 (50)= (0.5)50. 30 30!20! notethateven20!isa19—digitnumber.insuchcasesonecanuseaformula whichisthelimitofthebinomialdistributionatlarge𝑛: 1 (𝑚−𝐸(𝑚))2 𝑃 (𝑚)= exp(− ), (1.13) 𝑛 √2𝜋var 2var where𝐸(𝑚) = 𝑛𝑝andvar = 𝑛𝑝𝑞,andexp = 2.718…isthebaseofnatu- rallogarithms. thedistributiondefinedin(1.13)iscalledthe normalor gaussiandistribution.45 discreterandomvariables thepoissondistribution.iftheprobabilitythataneventwilloccur inasingletrialisverysmall(𝑝 ≪ 1),thebinomialdistributionatlarge𝑛 becomesthe poisson(ratherthanthenormal)distribution,andisdefinedas (𝑛𝑝)𝑚 𝑃 (𝑚)= exp(−𝑛𝑝). (1.14) 𝑛 𝑚! thisdistributionisalsosometimescalledthe lawofrareevents. itisin- terestingtonotethatthevarianceofarandomvariablewiththepoisson distributionequalsitsexpectedvalue. twodistributionsarecomparedinfigure1.12.theparametersofthefirst distributionare𝑛=30and𝑝=0.3,anditisclosetothenormaldistribution withtheexpectedvalue𝐸(𝑚) = 9. theseconddistribution’sparameters are𝑛 = 30and𝑝 = 0.05,anditisclosetothepoissondistributionwith 𝐸(𝑚)=1.5. p(m) p(m) n n 0.3 0.3 n=30 n=30 p=0.3 p=0.05 0.2 0.2 0.1 0.1 m m 0 1 5 10 15 0 1 5 alittleofmathematics.theexpression(𝑞+𝑝)𝑛,where𝑛isapositive figure1.12:thepoisson(right)andgaussian (left)distributions. integer,iscalledabinomial(two-term)expressionofdegree𝑛.youshould knowaboutthebinomialexpansionsofsecondandthirddegrees: (𝑞+𝑝)2 =𝑞2+2𝑞𝑝+𝑝2, (𝑞+𝑝)3 =𝑞3+3𝑞2𝑢+3𝑞𝑝2+𝑝3.46 mathematicsofrandomness ingeneral(forarandominteger𝑛)thebinomialexpansionis 𝑛(𝑛−1)…(𝑛−𝑚+1) (𝑞+𝑝)𝑛 =𝑞𝑛+𝑛𝑞𝑛−1𝑝+…+ 𝑞𝑛−𝑚𝑝𝑚+…+𝑛𝑞𝑝𝑛−1+𝑝𝑛. 𝑚! usingthenotationgivenin(1.8),wecanrewritethisformulaas 𝑛 𝑛 𝑛 𝑛 𝑛 (𝑞+𝑝)𝑛 =( )𝑞𝑛+( )𝑞𝑛−1𝑝+…+( )𝑞𝑛−𝑚𝑝𝑚+…+( )𝑞𝑝𝑛−1+( )𝑝𝑛. 0 1 𝑚 𝑛−1 𝑛 thusfrom(1.9),wecanconcludethat 𝑛 𝑛 𝑛 (𝑞+𝑝)𝑛 =∑( )𝑞𝑛−𝑚𝑝𝑚 =∑𝑃 (𝑚). 𝑚 𝑛 𝑚=0 𝑚=0 consequently,theprobabilities𝑃 (𝑚)coincidewiththecoefficientsofthe 𝑛 binomialexpansion,andthisiswhythe binomialdistributionissocalled. theprobabilities𝑞and𝑝inabinomialdistributionaresuchthat𝑞+𝑝=1. therefore,(𝑞+𝑝)𝑛 =1.ontheotherhand, 𝑛 (𝑞+𝑝)𝑛 =∑𝑃 (𝑚). 𝑛 𝑚=0 hence(1.10). continuous random variables continuousrandomvariablesareveryunlikediscreteones.acontinuous variablecanassumeanyofinfinitesetofvalues,whichcontinuouslyfilla certaininterval. itisimpossibleinprincipletolisteveryvalueorsucha variableattheveryleastbecausetherersnosuchthingastwoneighbouring values(justasitisimpossibletomarktwoneighbouringpointsonthenumber47 continuousrandomvariables axis). besides,theprobabilityofaconcretevalueofacontinuousrandom variableiszero. canprobabilityofapossibleeventequaltozero?youknownow thatanimpossibleeventhasazeroprobability.however,apossibleeventcan alsohaveazeroprobability. supposeathinneedleisthrownmanytimesatrandomontoastripof paperonwhichanumberaxisismarked. wecanregardthe𝑥-coordinate ofthepointwheretheneedlecrossesthenumberaxis(figure1.13(a))tobe acontinuousrandomvariable.thiscoordinatevariesinarandomfashion fromonetrialtoanother. 0 x x (a) (b) figure1.13:theprobabilitythatacontinuous wecouldalsousearouletteinsteadofthrowinganeedle.astripofpaper randomvariablewilltakeacertainvalueiszero. withanumberedlinecouldbe.pastedtothecircumferenceoftheroulette circle,asshowninfigure1.13(b).whereverthefreelyrotatingarrowofthe rouletteispointingwhenitstops,ityieldsanumberthatwillbeacontinuous randomvariable. whatistheprobabilityofthearrowstoppingatacertainpoint𝑥? in otherwords,whatistheprobabilitythataconcretevalue𝑥ofacontinuous randomvariableischosen?supposetheroulettecircle’sradius𝑅isdivided intoafinitenumberofidenticalsectors,e.g. 10sectors(figure1.14). the48 mathematicsofrandomness lengthofthearccorrespondingtothesectorequalsδ𝑥 = 2𝜋𝑅/10. the probabilitythatthearrowwillstopwithinthesectorhatchedinthefigureis δ𝑥/2𝜋𝑅 = 1/10. thus,theprobabilitythattherandomvariablewilltake avaluefrom𝑥to𝑥+δ𝑥isδ𝑥/2𝜋𝑅. letusgraduallynarrowtherangeof numbers,i.e.dividethecircleintolargernumbersofsectors.theprobability 0 δ𝑥/2𝜋𝑅thatanyvalueisintherangefrom𝑥to𝑥+δ𝑥alsowillfall.inorder toobtaintheprobabilitythatthevariablewilltakethevalue𝑥exactly,we mustfindthelimitasδ𝑥→0.inthiscase,theprobabilityδ𝑥/2𝜋𝑅becomes x zero.thuswecanseethattheprobabilitythatacontinuousrandomvariable willtakeacertainvalueisindeedzero. thateventmaybebothpossibleandpossessazeroprobabilitymayseem figure1.14:aroulettetogeneratecontinuous paradoxical,butitisnot.infactthereareparallelsyouaresurelywellaware randomvariables. of. considerabodyofvolume𝑉withamass𝑀. letusselectapoint 𝐴withinthebodyandconsiderasmallervolume𝑉 whichcontainsthe 1 point(figure1.15)andassignamass𝑀 toit. letusgraduallyshrinkthe 1 smallervolumearoundpoint𝐴.weobtainasequenceofvolumescontaining 𝐴,i.e. 𝑉,𝑉,𝑉,𝑉,…,andacorrespondingsequenceofdecreasingmasses: 1 2 3 v 𝑀,𝑀,𝑀,𝑀,…,.thelimitofthemassvanishesasthevolumearound𝐴 1 1 2 3 contractstozero.wecanseethatabodywhichhasafinitemassconsistsof pointswhichhavezeromasses.inotherwords,thenonzeromassofthebody isthe sumofaninfinitenumberofzeromassesofitsseparatepoints.inthe sameway,thenonzeroprobabilitythataroulettearrowstopswithinagiven a rangeδ𝑥isthe sumofaninfinitenumberofzeroprobabilitiesthatthearrow willstopateachindividualvaluewithintheconsideredrange. figure1.15:afinitenon-zeromasscanbegener- atedfromthesumofaninfinitenumberofzero the density of a probability. thisconceptualdifficultycanbe masses. avoidedbyusingtheidea density. althoughthemassofapointwithina bodyiszero,thebody’sdensityatthepointisnon-zero.ifδ𝑀isthemassof avolumeδ𝑉withinwhichthepointinquestionislocated(weshalldescribe thepointintermsofitspositionvectorr,thenthedensity𝜌(r)atthispoint49 continuousrandomvariables isthelimitoftheratioδ𝑀/δ𝑉asδ𝑉convergestothepointatr,i.e., δ𝑀 𝜌(r)= lim . δ𝑉→0 δ𝑉 ifthevolumeδ𝑉issmallenough,wecansaythatδ𝑀≃𝜌(r)δ𝑉.usinga strictapproach,weshouldsubstituteδ𝑉bythedifferentiald𝑉. themass𝑀ofabodyoccupyingvolume𝑉isthenexpressedbythe integral: 𝑀=∫ 𝜌(r)d𝑉, 𝑉 overthevolumeinquestion. probabilitytheoryusesasimilarapproach.whendealingwithcontinuous randomvariables,the probabilitydensityisusedratherthantheprobability itself.let𝑓(𝑥)betheprobabilitydensityofarandomvariable𝑥,andsoby analogywiththemassdensitywehave δ𝑝 𝑓(𝑥)= lim 𝑥. δ𝑥→0 δ𝑥 hereδ𝑝 istheprobabilitythatarandomvariablewilltakeavaluebetween 𝑥 𝑥and𝑥+δ𝑥. theprobability𝑝thatarandomvariablewillhaveavalue between𝑥 and𝑥 is,intermsofprobabilitydensity,asfollows: 1 2 𝑥 2 𝑝=∫𝑓(𝑥)d𝑥. (1.15) 𝑥 1 iftheintegrationisoverthewholerangeofvaluesarandomvariablemaytake, theintegral(1.15)willevaluatetounity(thisistheprobabilityofacertain event).intheexamplewitharoulettementionedabove,thewholeintervalis from𝑥=0to𝑥=2𝜋𝑅.ingeneral,weassumetheintervalisinfinite,when +∞ ∫𝑓(𝑥)𝑑𝑥=1. (1.16) −∞50 mathematicsofrandomness theintegralisverysimpleintherouletteexamplebecausetheprobabilitythe roulettearrowstopswithinanintervalfrom𝑥to𝑥+δ𝑥 doesnotdependon 𝑥.therefore,theprobabilitydensitydoesnotdependon𝑥,andhence, asimilarsituationisencounteredwhenthedensityofabodyisthesame ateverypoint,i.e.whenthebodyis uniform(𝜌=𝑀/𝑉).moregenerally, density𝜌(r)variesfrompointtopoint,andsodoestheprobabilitydensity 𝑓(𝑥). theexpectedvalueandthevarianceofacontinuousrandom variable.the expectedvalueand varianceofadiscreterandomvariableare expressedassumsovertheprobabilitydistribution(seeequations(1.4)to(1.6). whentherandomvariableiscontinuous,integralsareusedinsteadofsums andtheprobabilitydensitydistributionisusedratherthantheprobability distribution: +∞ 𝐸(𝑥)=∫𝑥𝑓(𝑥)d𝑥, (1.17) −∞ +∞ var=∫(𝑥−𝐸(𝑥))2𝑓(𝑥)d𝑥. (1.18) −∞ the normal distribution of probability density. thenormal distributionofprobabilitydensity.whendealingwithcontinuousrandom variables,weoftenencounterthe normaldistributionofprobabilitydensity. thisdistributionisdefinedbythefollowingexpression(compareitwith (1.13)): 1 (𝑥−𝐸(𝑥))2 𝑓(𝑥)= exp(− ). (1.19) 𝜎√2𝜋 2𝜎2 here𝜎isthestandarddeviation(𝜎=√𝑣𝑎𝑟)thefunction(1.19)iscalled the normalor gaussiandistribution. theprobabilitydensityofacontinuousrandomvariableisalwaysnormal51 continuousrandomvariables ifthevarianceofitsvaluesisduetomanydifferentequallystrongfactors.it hasbeenprovedinprobabilitytheorythatthesumofalargeenoughnumber ofindependentrandomvariablesobeyinganydistributionstendstothe normaldistribution,andthelargerthenumberofsumsthemoreaccurately thenormaldistributionis. forinstance,supposewearedealingwiththeproductionofnutsand bolts. thescatteroftheinsidediameterofthenutisduetorandomde- viationsinthepropertiesofthemetal,thetemperature,vibrationofthe machinetool,changesinthevoltage,wearofthecutter,etc. allofthese effectsactindependentlyandapproximatelywiththesamestrength.they aresuperimposed,andtheresultisthattheinsidediameterofthenutsisa continuousrandomvariablewithanormaldistribution.theexpectedvalue ofthisvariableshouldevidentlybethedesiredinsidediameterofthenuts, whilethevariancecharacterizesthescatteroftheobtaineddiametersaround thedesiredvalue. f(x) e(x) x figure1.16:thethree-sigmaruleforagaussian distribution. 2 3 thethree-sigmarule.anormaldistributionisshowninfigure1.16.52 mathematicsofrandomness ithasamaximumattheexpectedvalue𝐸(𝑥).thecurve(the gaussiancurve) isbell-shapedandissymmetricabout𝐸(𝑥).theareaundertheentirecurve, +∞ i. e. fortheinterval(−∞ < 𝑥 < +∞),isgivenbytheintegral∫ 𝑓(𝑥)𝑑𝑥. −∞ substituting(1.19)here,itcanbeshownthattheareaisequaltounity.this agreeswith(1.16),whosemeaningisthattheprobabilityofacertaineventis unity.letusdividetheareaunderthegaussiancurveusingverticallines(see figure1.16).letusfirstconsiderthesectioncorrespondingtotheinterval 𝐸(𝑥)−𝜎≤𝑥≤𝐸(𝑥)+𝜎.itcanbeshown(pleasebelieveme)that 𝛦(𝑥)+𝜎 ∫ 𝑓(𝑥)𝑑𝑥=0.683. 𝛦(𝑥)−𝜎 thismeansthattheprobabilityof𝑥takingavalueintheintervalfrom𝐸(𝑥)−𝜎 to𝐸(𝑥)+𝜎equals0.683.itcanalsobecalculatedthattheprobabilityof𝑥 takingavaluefrom𝐸(𝑥)−2𝜎to𝐸(𝑥)+2𝜎is0.954,andtheprobabilityof𝑥 takingavalueintherangeof𝐸(𝑥)−3𝜎to𝐸(𝑥)+3𝜎is0.997.consequently, acontinuousrandomvariablewithanormaldistributiontakesavalueinthe interval𝐸(𝑥)−3𝜎to𝐸(𝑥)+3𝜎withprobability0.997.thisprobabilityis practicallyequaltounity.therefore,itisnaturaltoassumeforallpractical purposesthatarandomvariablewillalwaystakeavalueintheintervalfrom 3𝜎ontherightto3𝜎ontheleftof𝐸(𝑥).thisiscalledthe three-sigmarule.chapter 2 decision making practicaldemandsbroughtforthspecialscientificmethodsthat canbecollectedundertheheading“operationsresearch”.we shallusethistermtomeantheapplicationofquantitative mathematicalmethodstojustifydecisionsineveryareaof goal-orientedhumanactivity. e.s.wentzel these difficult decisions decisionmakingunderuncertainconditions. weoftenhavetomake decisionswhennotalltheinformationisavailableandthisuncertaintyalways decreasestosomeextentourabilitytodecide. forexample,wheretogo foravacationorholiday? thishasworriedmemanytimes,sincevarious uncertaintiesconcerningtheweather,thehotel,theentertainmentatthe resort,andsoon,mustbeforeseen. wetryanddecideonthebestvariant 5354 decisionmaking fromourexperienceandtheadviceofourfriends,andweoftenact“by inspiration”.this subjectiveapproachtodecisionmakingisjustifiablewhen theconsequencesinvolveourselvesandrelatives.however,therearemany situationswhenadecisioncanaffectalargenumberofpeopleandtherefore requiresa scientificand mathematicallyjustifiableapproachratherthana subjectiveone. forinstance,modernsocietycannotfunctionwithoutelectricity,stores offood,rawmaterials,etc.thestoresarekepteverywhere:atfactories,shops, hospitals,andgarages. buthowlargeshouldthestoresbeinaparticular case?itisclearthattheyshouldnotbetoosmall,otherwisethefunctionof theenterprisewouldbeinterrupted.neithershouldtheyalsobetoolarge becausetheycostmoneytobuildandmaintain:theywouldbedeadstock. store-keepingisaproblemofexceptionalimportance.itissocomplicated becauseadecisionmustalwaysbemadeinconditionsofuncertainty. twokindsofuncertainty.howshouldwemakedecisionsundercondi- tionsofuncertainty?firstofall,weshoulddiscoverwhichfactorsarecausing theuncertaintyandevaluatetheirnature.therearetwokindsofuncertainty. thefirstkindisduetofactorswhichcanbetreatedusingthetheoryofproba- bility.theseareeither randomvariablesor randomfunctions,andtheyhave statisticalproperties(forinstance,theexpectedvalueandvariance),which areeitherknownorcanbeobtainedovertime.uncertaintyofthiskindis called probabilisticor stochastic.thesecondkindofuncertaintyiscaused byunknownfactorswhicharenotrandomvariables(randomfunctions)be- causethesetofrealizationsofthesefactorsdoesnotpossessstatisticalstability andthereforethenotionofprobabilitycannotbeused. weshallcallthis uncertainty“bad”. “so”,thereadermaysay,“itwouldseemthatnoteveryeventthatcannot bepredictedaccuratelyisarandomevent.” “well,yes,inaway.” letmeexplain. intheprecedingchapterwedis- cussedrandomevents,randomvariables,andrandomfunctions.irepeatedly55 thesedifficultdecisions emphasizedthatthereshouldalwaysbe statisticalstability,whichisexpressed intermsofprobability.however,thereareevents,whichoccurfromtime totime,thatdonothaveanystatisticalstability.thenotionofprobability isinapplicabletosuchevents,andtherefore,theterm“random”cannotbe usedheretoo. forinstance,wecannotassignaprobabilitytotheeventof anindividualpupilgettinganunsatisfactorymarkinaconcretesubject.we cannot,evenhypothetically,deviseasetofuniformtrialsthatmightyield theeventasoneoutcome. therewouldbenosenseinconductingsucha trialwithagroupofpupilsbecauseeachpupilhashisorherownindividual abilitiesandlevelofpreparationfortheexam.thetrialscannotberepeated withthesamepupilbecausehewillobviouslygetbetterandbetterinthe subjectfromtrialtotrial.similarlythereisnowaywecandiscusstheproba- bilityoftheoutcomeofagamebetweentwoequallymatchedchessplayers. inallsuchsituations,therecanbenosetofuniformtrials,andsothereis nostabilitywhichcanbeexpressedintermsofaprobability.wehave“bad” uncertaintyinallsuchsituations. iamafraidwedonotconsiderthenotion“statisticalstability”andoften useexpressionssuchas“improbable”,“probable”,“mostprobable”,and“in allprobability”torefertoeventsthatcannotbeassignedbyanyprobability. weareapttoascribeaprobabilitytoeveryeventeventhoughitmightnotbe predictable.thisiswhyitbecamenecessarytorefinethenotionofprobability earlythiscentury.thiswasdonebya.n.kolmogorovwhenhedeveloped anaxiomaticdefinitionofprobability. optionsandthemeasureofeffectiveness.whenwespeakofdecision making,weassumethatdifferentpatternsofbehaviourarepossible.theyare called options.letmeemphasizethatinthemoreimportantproblemsthe numberofoptionsisverygreat.let𝑋bethesetofoptionsinaparticular situation.adecisionismadewhenweselectoneoption𝑥fromthisset.how dowedeterminewhichoptionisthemostpreferableorthemostefficient?a quantitativecriterionisneededtoallowustocomparedifferentoptionsin termsoftheireffectiveness.letuscallthiscriterionthe measureofeffective-56 decisionmaking ness.thismeasureisselectedforeachparticular purpose,e.g.,nottobelate forschool,tosolveaproblemcorrectlyandquickly,ortoreachthecinema. adoctorwantstofindanefficientmethodoftreatinghispatient.afactory managerisresponsibleforthefulfilmentofaproductionplan. themost efficientoptionistheonethatsuitsitspurposebest. supposeweworkinashopandourtargetistomaximizethereceipts.we couldchooseprofitasthemeasureofeffectivenessandstrivetomaximizethis measure.theselectionofthemeasureinthisexampleisevident.however, therearemorecomplicatedsituations,whenseveralgoalsarepursuedsimul- taneously,forexample,wewishtomaximizeprofit,minimizethedurationof thesales,anddistributethegoodstothegreatestnumberofcustomers.in suchcaseswehavetohaveseveralmeasuresofeffectiveness;theseproblems arecalledmulti-criterial. let𝑊beasinglemeasureofeffectiveness.itwouldseemthatourtaskis nowtofindanoption𝑥atwhich𝑊isatamaximum(or,theotherwayround, ataminimum).however,weshouldrememberthatdecisionmakingoccurs underconditionsofuncertainty.thereareunknown(random)factors(let ususe𝜉todenotethem),whichinfluencetheendresultandthereforeaffect themeasureofeffectiveness𝑊.thereisalsoalwaysasetoffactorsknown beforehand(letusdesignatethem𝛼).thereforethemeasureofeffectiveness isdependentonthreegroupsoffactors:knownfactors𝛼,unknown(random) factors𝜉,andtheselectedoption𝑥: 𝑊=𝑊(𝛼,𝜉,𝑥). inthesalesexample,the𝛼setisgoodsonsale,theavailablepremises,the season,etc.the𝜉factorsincludethenumberofcustomersperday(itvaries randomlyfromdaytoday),thetimecustomersarrive(randomcrowdingis possible,whichleadstolongqueues),thegoodschosenbythecustomers (thedemandforagivencommodityvariesrandomlyintime),etc. sincethe𝜉,factorsarerandom,themeasureofeffectiveness𝑊isaran-57 thesedifficultdecisions domvariable. now,howisitpossibletomaximize(minimize)arandom variable?theanswerquiteclearlyisthatitisnaturallyimpossible.whichever option𝑥ischosen,𝑊remainsrandom,anditcannotbemaximizedormini- mized.thisanswershouldnotdiscouragethereader.itistruethatunder conditionsofuncertaintywecannotmaximize(minimize)themeasureof effectivenesswithahundredpercentprobability.however,anadequateselec- tionofanoptionispossiblewithareasonablylargeprobability.thisiswhere weshouldtacklethetechniquesusedindecisionmakingunderconditionsof stochasticuncertainty. substitutionofrandomfactorsbymeans.theeasiesttechniqueis merelytosubstitutetherandomfactors𝜉bytheirmeans.theresultisthat theproblembecomescompletelydeterminedandthemeasureofeffectiveness 𝑊canbecalculatedprecisely.itcan,inparticular,beeithermaximizedor minimized.thistechniquehasbeenwidelyusedtosolveproblemsinphysics andtechnology. almosteveryparameterencounteredinthesefields(e.g., temperature,potentialdifference,illuminance,pressure)is,strictlyspeak- ing,arandomvariable.asarule,weneglecttherandomnatureofphysical parametersandusetheirmeanvaluestosolvetheproblems. thetechniqueisjustifiedifthedeviationofaparameterfromitsmean valueisinsignificant.however,itisnotvalidiftherandomfactorsignificantly affectstheoutcome.forinstance,whenorganizingthejobsinamotor-car repairshop,wemaynotneglecttherandomnessinthewaycarsfail,orthe randomnatureofthefailuresthemselves,ortherandomtimeneededto completeeachrepairoperation.ifwearedealingwiththenoisearisinginan electronicdevice,wecannotneglecttherandombehaviourofelectronflows. intheseexamples,the𝜉factorsmustindeedbeconsideredasrandomfactors, weshallsaytheyareessentiallyrandom. meanvalueoptimization.ifthe𝜉factorsareessentiallyrandom,we canuseatechniquecalled mean-valueoptimization.whatwedoistousethe expectedvalue𝐸(𝑊)asthemeasureofeffectiveness,ratherthantherandom58 decisionmaking variable𝑊andtheexpectedvalueismaximizedorminimized. naturally,thisapproachdoesnotresolvetheuncertainty.theeffective- nessofanoption𝑥forconcretevaluesofrandomparameters𝜉maybevery differentfromtheexpectedone.however,usingmean-valueoptimization meansthatwecanbesurethataftermanyrepeatedoperationsweshallgain overall.itshouldbeborneinmindthatmean-valueoptimizationisonlyad- missiblewhenthegainsofrepeatedoperationsare totalled,sothat“minuses” insomeoperationsarecompensatedbythe“pluses”inothers.mean-value optimizationwouldbejustifiedshouldwebetryingtoincreasetheprofit obtained,forinstance,inasalesdepartment. theprofitondifferentdays wouldbetotalled,sothatrandom“unlucky”dayswouldbecompensatedby the“lucky”days, buthereisanotherexample. supposeweconsidertheeffectivenessof theambulanceserviceinalargecity.letusselecttheelapsedtimebetween summoninghelpandtheambulancearrivingasthemeasureofeffectiveness. itisdesirablethatthisparameterbeminimized.wecannotapplymean-value optimizationbecauseifonepatientwaitstoolongforadoctor,heorsheis notcompensatedbythefactthatanotherpatientreceivedfasterattention. stochasticconstraints.letusputforwardanadditionaldemand.sup- posewedesirethattheelapsedtime𝑊tillthearrivalofhelpafteracallfor anambulancebelessthansomevalue𝑊. since𝑊isarandomvariable, 0 wecannotdemandthattheinequality𝑊<𝑊 bealwaystrue,wecanonly 0 demandthatitbetrueforsomelargeprobability,forinstance,nolessthan 0.99.inordertotakethisintoaccountwedeletefromthe𝑋setthoseoptions 𝑥,forwhichtherequirementisnotsatisfied. these constraintsarecalled stochastic.naturally,theuseofstochasticconstraintsnoticeablycomplicates decisionmaking.59 randomprocesseswithdiscretestates random processes with discrete states a randomprocesscanbethoughtofasthetransitionofasystemfromone statetoanotheroccurringinarandomfashion.weshallconsiderrandom processeswith discretestatesinthischapterandsooursystemwillbesup- posedtohaveasetofdiscretestates,eitherfiniteorinfinite. therandom transitionsofthesystemfromonestatetoanotherareassumedtotakeplace instantaneously. stategraphs.randomprocesseswithdiscretestatescanbeconveniently consideredusingadiagramcalleda stategraph. thediagramshowsthe possiblestatesasystemmaybeinandindicatesthepossibletransitionsusing arrows. s letustakeanexample.supposeasystemconsistsoftwomachinetools, 1 eachofwhichproducesidenticalproducts.ifatoolfailsitsrepairisstarted immediately.thus,oursystemhasfourstates:𝑆 bothtoolsareoperating; 1 𝑆2thefirsttoolisunderrepairafterafailurewhilethesecondisoperating;𝑆3, s s thesecondtoolisunderrepairwhilethefirstisoperating;𝑆,bothtoolsare 2 3 4 beingrepaired. thestategraphisgiveninfigure2.1. thetransitions𝑆 → 𝑆, 𝑆 → 1 2 1 s 𝑆, 𝑆 → 𝑆 and𝑆 → 𝑆 occurasaresultoffailuresinthesystem. the 4 3 2 4 3 4 reversetransitionstakeplaceuponterminationoftherepairs.failuresoccur atunpredictablemomentsandthemomentswhentherepairsareterminated figure2.1:astategraphforsystemwithfour arealsorandom. therefore,thesystem’stransitionfromstatetostateis states. random. notethatthefiguredoesnotshowtransitions𝑆 →𝑆 and𝑆 →𝑆.the 1 4 4 1 formercorrespondstothesimultaneousfailureofbothtoolsandthelatter tothesimultaneousterminationofrepairofbothtools.weshallassumethat theprobabilitiesoftheseeventsarezero. eventarrival. supposethatwehaveasituationinwhicha streamof60 decisionmaking uniformeventsfolloweachotheratrandommoments. theymaybetele- phonedordersfortaxi,domesticappliancesbeingswitchedon,thefailures intheoperationofadevice,etc. t (a) t (b) t (c) t (d) t t+𝛥t figure2.2:arecordoftaxiordersatataxidepot. supposethedispatcheratataxidepotrecordsthetimeeachtaxiorder ismadeoveranintervaloftime,forinstance,from12a.m. to2p.m. we canshowthesemomentsaspointsonthetimeaxis,andsothedispatcher mightgetthepatternillustratedinfigure2.2(a).thisistherealizationof61 randomprocesseswithdiscretestates thetaxi-callarrivalsduringthatintervaloftime.threemoresuchrealizations areshowninfigure2.2(b),(c),and(d),andtheyarepatternsrecordedon differentdays.themomentswheneachtaxiorderismadeineachrealization arerandom.atthesametime,thetaxi-orderarrivalspossessstatisticalstability, thatis,thetotalnumberofeventsineachintervaloftimevariesonlyslightly fromexperimenttoexperiment(fromonearrivalrealizationtoanother).we canseethatthenumberofeventsinthearrivalrealisationspresentedare19, 20,21,and18. intheprecedingchapter,arandomeventinanexperimentwasanout- comewhichhasadefiniteprobability.whenweareconsideringarrivalsof events,wemusthaveanothermeaningfortheterm“event”.thereisnouse speakingabouttheprobabilityofanoutcome(event)becauseeacheventis uniform,i.e.indistinguishablefromtheothers.forinstance,onetaxi-order isasingleeventinastreamandisindistinguishablefromanotherevent.now letusconsiderotherprobabilities,forinstance,theprobabilitiesthatanevent willoccurduringagivenintervaloftime(suppose,from𝑡to𝑡+δ𝑡,asshown inthefigure)exactlyonce,twice,thrice,etc. thenotionof“eventarrival”isappliedtorandomprocessesinsystems withdiscretestates.itisassumedthatthetransitionsofasystemfromone statetoanotheroccurasaresultoftheeffectofeventarrivals. oncean eventarrives,thesysteminstantaneouslychangesstate.forthestategraph infigure2.1transitions𝑆 → 𝑆 and𝑆 → 𝑆 occurduetothearrivalof 1 2 3 4 eventscorrespondingtofailuresinthefirsttool,whiletransitions𝑆 →𝑆 1 3 and𝑆 →𝑆 occurduetofailuresofthesecondtool.thereversetransitions 2 4 arecausedbythearrivalofeventscorrespondingtothe“terminations”of repair:transitions𝑆 →𝑆 and𝑆 →𝑆,arecausedbythearrivalsofrepair 2 1 4 3 terminationsofthefirsttool,andtransitions𝑆 → 𝑆 and𝑆 → 𝑆 tothe 3 1 4 2 arrivalsofrepairterminationsofthesecondtool. thesystemtransfersfromstate𝑆 tostate𝑆 everytimethenextevent 𝑖 𝑗 relatedtothetransitionarrives.thenaturalconclusionisthattheprobability62 decisionmaking oftransition𝑆 →𝑆 atadefinitemomentintime𝑡shouldequaltheproba- 𝑖 𝑗 bilityofaneventarrivalatthismoment.thereisnosenseinspeakingofthe probabilityofatransitionataconcretemoment𝑡.liketheprobabilityof anyconcretevalueofacontinuousrandomvariable,thisprobabilityiszero, andthisresultfollowsfromthecontinuityoftime. itisthereforenatural todiscusstheprobabilityofatransition(theprobabilityofaneventarrival) occurringduringtheintervaloftimefrom𝑡to𝑡+δ𝑡,ratherthanitsoccur- renceattime𝑡. letusdesignatethisprobability𝑃 (𝑡,δ𝑡). asδ𝑡tendsto 𝑖𝑗 zero,wearriveatthenotionofa transitionprobabilitydensityattime𝑡,i.e. 𝑃 (𝑡,δ𝑡) 𝑖𝑗 𝜆 (𝑡)= lim 𝑥= . (2.1) 𝑗 δ𝑡→0 δ𝑡 thisisalsocalledthe arrivalrateofeventscausingthetransitioninquestion. inthegeneralcase,thearrivalratedependsontime.however,itshould berememberedthatthedependenceofthearrivalrateontimeisnotrelated tothelocationof“dense”or“rare”arrivalrealisations.forsimplicity’ssake, weshallassumethatthetransitionprobabilitydensityandthereforetheevent arrivalratedoesnotdependontime. i.e. weshallconsider steady-state 𝜇 s 𝜆 1 1 2 arrivals. 𝜆1 𝜇2 thechapman-kolmogorovequationsforsteadystate.letususe𝑝𝑖 todenotetheprobabilitythatasystemisinstate𝑆 (sinceourdiscussionis 𝑖 s s onlyfor steady-statearrivals,theprobabilities𝑝,areindependentoftime), 2 3 𝑖 letusconsiderthesystemwhosestategraphisgiveninfigure2.1.suppose 𝜆 𝜇 2 1 𝜆 isthearrivalrateforfailuresofthefirsttooland𝜆 thatforthesecondtool; 1 2 let𝜇 bethearrivalrateforrepairterminationsofthefirsttooland𝜇 thatfor 1 2 𝜇2 s4 𝜆1 thesecondtool.wehavelabelledthestategraphwiththeappropriatearrival rates,seefigure2.3. figure2.3:astategraphforsystemwithfour supposethereare𝑁identicalsystemsdescribedbythestategraphin stateswitharrivalrates. figure2.3. let𝑁 ≫ 1. thenumberofsystemswithstate𝑆,is𝑁𝑝 (this 𝑖 𝑖 statementbecomesmoreaccuratethelarger𝑁is).letusconsideraconcrete63 randomprocesseswithdiscretestates state,say,𝑆.transitionsarepossiblefromthisstatetostates𝑆 and𝑆 with 1 2 3 probability𝜆 +𝜆 ,perunittime.(understeadystate,theprobabilitydensity 1 2 istheprobabilityforthefinitetimeintervalδ𝑡dividedbyδ𝑡.) therefore, thenumberofdeparture:fromstate𝑆,perunittimeintheconsideredset 1 ofsystemsis𝑁𝑝 (𝜆 +𝜆 ),wecandiscernageneralrulehere:thenumber 1 1 2 oftransitions𝑆 →𝑆 perunittimeistheproductofthenumberofsystems 𝑖 𝑗 withstate𝑆 (theinitialstate)bytheprobabilityofthetransitionperunittime, 𝑖 wehaveconsidereddeparturesfromstate𝑆.thesystemarrivesatthisstate 1 from𝑆 and𝑆,thenumberofarrivalsat𝑆 perunittimeis𝑁𝑝 𝜇 +𝑁𝑝 𝜇 . 2 3 1 2 1 3 1 sincewearedealingwithsteadystates,thenumberofdeparturesandarrivals foreachparticularstateshouldbebalanced.therefore. 𝑁𝑝 (𝜆 +𝜆 )=𝑁𝑝 𝜇 +𝑁𝑝 𝜇 . 1 1 2 2 1 3 2 bysettingupsimilarbalancesofarrivalsanddeparturesforeachofthefour statesandeliminatingthecommonfactor𝑁intheequations,weobtainthe followingequationsforprobabilities𝑝 , 𝑝 , 𝑝 and𝑝: 1 2 3 4 forstate 𝑆 ∶(𝜆 +𝜆 )𝑝 =𝜇 𝑝 +𝜇 𝑝 , 1 1 2 1 1 2 2 3 forstate 𝑆 ∶(𝜆 +𝜇 )𝑝 =𝜆 𝑝 +𝜇 𝑝, 2 2 1 2 1 1 2 4 forstate 𝑆 ∶(𝜆 +𝜇 )𝑝 =𝜆 𝑝 +𝜇 𝑝, 3 1 2 3 1 1 1 4 forstate 𝑆 ∶(𝜇 +𝜇 )𝑝 =𝜆 𝑝 +𝜆 𝑝 . 4 1 2 4 2 2 1 3 itiseasytoseethatthefourthequationcanbeobtainedbysummingthefirst three.insteadofthisequation,letususetheequation 𝑝 +𝑝 +𝑝 +𝑝 =1, 1 2 3 4 whichmeansthatthesystemmustbeinoneofthefourstates.therefore,we64 decisionmaking havethefollowingsystemofequations: 𝑆 ∶(𝜆 +𝜆 )𝑝 =𝜇 𝑝 +𝜇 𝑝 , 1 1 2 1 1 2 2 3 𝑆 ∶(𝜆 +𝜇 )𝑝 =𝜆 𝑝 +𝜇 𝑝, 2 2 1 2 1 1 2 4 } (2.2) 𝑆 ∶(𝜆 +𝜇 )𝑝 =𝜆 𝑝 +𝜇 𝑝, 3 1 2 3 2 1 1 4 𝑝 +𝑝 +𝑝 +𝑝 =1. 1 2 3 4 thesearethe chapman-kolmogomvequationsforthesystemwhosestate graphisshowninfigure2.3. whichinnovationshouldbechosen?letusanalyzeaconcretesitua- tionusingequations(2.2).thestategraph(seefigure2.3)corresponding totheseequationsdescribesasystemwhich,weassumed,consistsoftwo machinetoolseachproducingidenticalgoodssupposethesecondtoolis moremodernanitsoutputrateistwicethatothefirsttool. thefirsttool generates(perunittime)anincomeoffiveconventionalunits,whilethe secondonegeneratesoneoftenunits.regretfully,thesecondtoolfails,on theaverage,twiceasfrequentlyasdoesthefirsttool:hence𝜆 =1and𝜆 =2. 1 2 thearrivalratesforrepairterminationareassumedtobe𝑢 =2and𝑢 =3. 1 2 usingthesearrivalratesforfailureandrepairtermination.letusrewrite(2.2) thus 3𝑝 =2𝑝 +3𝑝 , 1 2 3 4𝑝 =𝑝 +3𝑝, 2 1 4 } 4𝑝 =2𝑝 +2𝑝, 3 1 4 𝑝 +𝑝 +𝑝 +𝑝 =1. 1 2 3 4 thissystemofequationscanbesolvedtoyield𝑝 = 0.4, 𝑝 = 0.2, 𝑝 = 1 2 3 0.27and𝑝 = 0.13. thismeansthat,ontheaverage,bothtoolsoperate 4 simultaneously(state𝑆 inthefigure)40percentofthetime,thefirsttool 1 operateswhilethesecondoneisbeingrepaired(state𝑆)20percentofthe 2 time,thesecondtooloperateswhilethefirstoneisbeingrepaired(state𝑆) 3 27percentofthetime,andbothtoolsaresimultaneouslybeingrepaired65 randomprocesseswithdiscretestates (state𝑆)13percentofthetime.itiseasytocalculatetheincomethistool 4 systemgeneratesperunittime: (5+10)×0.4+5×0.2+10×0.27 = 9.7 conventionalunits. supposeaninnovationissuggestedwhichwouldreducetherepairtime ofeitherthefirstorsecondtoolbyafactoroftwo.fortechnicalreasons,we canonlyapplytheinnovationtoonetool.whichtoolshouldbechosen,the firstorthesecond?hereisaconcreteexampleofapracticalsituationwhen, usingprobabilitytheory,wemustjustifyourdecisionscientifically supposewechoosethefirsttool. followingtheintroductionofthe innovation,thearrivalrateofitsrepairterminationincreasesbyafactorof two,whence𝑢 =4(theotherratesremainthesame,i.e.𝜆 =1,𝜆 =2and 1 1 2 𝜇 =3).nowequations(2.2)are 2 3𝑝 =4𝑝 +3𝑝 , 1 2 3 6𝑝 =𝑝 +3𝑝, 2 1 4 } 4𝑝 =2𝑝 +4𝑝, 3 1 4 𝑝 +𝑝 +𝑝 +𝑝 =1. 1 2 3 4 aftersolvingthissystem,wefindthat𝑝 = 0.48,𝑝 = 0.12,𝑝 = 0.32,and 1 2 3 𝑝 =0.08.theseprobabilitiescanbeusedtocalculatetheincomeoursystem 4 willnowgenerate:(5+10)×0.48+5×0.12+10×0.32=11conventional units. ifweapplytheinnovationtothesecondtool,therate𝜇 ,willbedoubled. 2 now𝜆 =1, 𝜆 =2, 𝜇 =2and𝜇 =6,andequations(2.2)willbe 1 2 1 2 3𝑝 =2𝑝 +6𝑝 , 1 2 3 4𝑝 =𝑝 +6𝑝, 2 1 4 } 7𝑝 =2𝑝 +2𝑝, 3 1 4 𝑝 +𝑝 +𝑝 +𝑝 =1. 1 2 3 4 thissystemyields:𝑝 =0.5,𝑝 =0.25,𝑝 =0.17,and𝑝 =0.08,whencethe 1 2 3 466 decisionmaking incomeis(5+10)×0.5+5×0.25+10×0.17=10.45conventionalunits. thereforeitisclearlymoreprofitabletoapplytheinnovationtothefirsttoo. queueing systems theproblemofqueueing.modernsocietycannotexistwithoutawhole networkof queueingsystems. theseincludetelephoneexchanges,shops, polyclinics, restaurants, bookingoffices, petrolstations, andhairdressers. despitetheirdiversity,thesesystemshaveseveralthingsincommonand commonproblems. whenweseektheassistanceofadoctororservicefromacafe,restaurant, orbarber,wemustwaitforourturninaqueue,evenifwetelephoneto makeanappointment,thatis,reserveourplaceinaqueuewithoutactually attendingphysically.clearly,wewishtobeservedstraightawayandwaiting canbefrustrating. itisclearthatthesourceoftheproblemisthe randomnatureofthe demandsforattentioninqueueingsystems.thearrivalofcallsatatelephone exchangeisrandomasisthedurationofeachtelephoneconversation.this randomnesscannotbeavoided.however,itcanbetakenintoaccountand,as aconsequence,wecanrationallyorganizeaqueueingsystemforallpractical purposes.theseproblemswerefirstinvestigatedinthefirstquarterofthis century. themathematicalproblemsforsimulatingrandomprocessesin systemswithdiscretestateswereformulatedandconsidered,andanewfield ofinvestigationinprobabilitytheorywasstarted. historically,queueingtheoryoriginatedinresearchontheoverloading oftelephoneexchanges,asevereproblemintheearly20thcentury. the initialperiodinthedevelopmentofthequeueingtheorycanbedatedas correspondingtotheworkofthedanishscientista.erlangin1908-1922. interestintheproblemsofqueueingrapidlyincreased.thedesireformore67 queueingsystems rationalservicingoflargenumbersofpeopleledtoinvestigationsofqueue formation.itsoonbecameevidentthattheproblemsdealtwithinqueueing theorywentwellbeyondthesphereofrenderingserviceandtheresultsare applicabletoawiderrangeofproblems. supposeaworkmanisoperatingseveralmachinetools.failuresrequiring urgentrepairsoccuratrandommoments,andthedurationofeachrepairis arandomvariable.theresultisasituationsimilartoacommonqueueing system.however,thisisaproblemofservicingmanytoolsbyaworkerrather thanservicingmanypeoplebyaqueueingsystem. therangeofpracticalproblemstowhichqueueingtheorycanbeapplied isuncommonlywide.weneedthetheorywhenwewant,say,toorganizethe efficientoperationofamodernseaport,when,forinstance,weanalyzethe servicingrateofalargeberth.weapplytoqueueingtheorywhenwelookat theoperationofageiger-müllercounter.thesedevicesareusedinnuclear physicstodetectandcountionizingparticles.eachparticleenteringatubein thecounterionizesgasinthetube,theionizationbeingroughlyindependent oftheparticle‘snatureandenergy,andsoauniformdischargeacrossthetube isgenerated.butwhenonedischargeisunderway,anewparticlecannotbe registered(“serviced”)bythesamecounter.themomenteachparticleenters thetube5random,asisthedurationofthedischarge(the“servicing”time). thisisasituationtypicalforqueueingsystems. basicnotions.aqueueingsystemissetuptoorganizetheserviceofa streamofrequests.therequestmaybeanewpassengerinabookingoffice, afailureinamachinetool,ashipmooring,oraparticleenteringageiger- müllercounter. thesystemmayhaveeitheroneorseveral servers. when yougotoalargebarbershoporhairdresserandwanttoknowthenumber ofbarbersorhairdressers,youareineffectaskingforthenumberofservers intheestablishment.inothersituations,theserversmaybethenumberof cashiersinabookingoffice,thenumberoftelephonesatapostofficefor makingtrunkcalls,thenumberofberthsinaport,orthenumberofpumps68 decisionmaking atapetrolstation.if,ontheotherhand,wewishtoseeaparticulardoctor, wearedealingwithasingle-serverqueueingsystem. whenweconsidertheoperationofaqueueingsystem,wemustfirsttake intoaccountthenumberofservers,thenumberofrequestsarrivingatthe systemperunittime,andthetimeneededtoservicearequest.thenumberof requestsarrivingatthesystem,themomentstheyarrive,andthetimeneeded toservicearequestare,asarule, randomfactors.therefore,queueingtheory isa theoryofrandomprocesses. randomprocessesofthistype(i.e.with discretestates)werediscussed intheprecedingsection. asystemtransfersfromstatetostatewheneach requestarrivesatthesystemandwhentherequestsareserviced.thelatter isgivenbytherateatwhichrequestscanbeservedbyasingle,continuously occupiedserver. queueingsystems. therearetwosortsofqueueingsystem: systems withlossesand systemswithqueues.ifarequestarrivesatasystemwithlosses whenalltheserversareoccupied,therequestis“refused”andisthenlostto thesystem.forexample,ifwewanttotelephonesomeoneandthenumberis engaged,thenourrequestisrefusedandweputdownthereceiver.whenwe dialthenumberagain,wearesubmittinganewrequest. themorecommontypesofsystemarethosewithqueuesorsystemswith waiting.thisiswhyitiscalledthe theoryofqueueing.insuchasystem,ifa request(orcustomer)arriveswhenalltheserversareoccupied,thecustomer takesaplaceina queueandwaitsforaservertobecomefree. thereare systemswith infinitequeues(aqueueingcustomeriseventuallyservedand thenumberofplacesinthequeueisunlimited)andsystemswith finite queues.therearedifferentsortsofrestriction,i.e.thenumberofcustomers queueingatthesametimemaybelimited(thequeuecannotbelongerthana certainnumberofcustomersandanynewcustomerisrefused);theduration ofacustomer’sstayinthequeuemaybelimited(afteracertainlengthof timequeueing,anunservedcustomerwillleavethequeue);orthetimethe69 queueingsystems systemoperatesformayberestricted(customersmayonlybeservedfora certainintervaloftime). theserviceorderisalsoimportant. customersarecommonlyserved “firstcomefirstserved”. however, priorityservicingisalsopossible,i.e. a newcomertoaqueueisservedfirstirrespectiveofthequeue. acustomer withahighprioritymayarriveatthesystemandinterrupttheservicingofa customerwithalowerpriority,whichmayalreadystart,orthehigherpriority customermayhavetowaituntiltheservicinghasbeencompleted. the priorityis absoluteinthefirstcaseand relativeinthesecond. queueing systemsarealways multi-critical,thatis,theyhavea setofmeasuresbywhich theireffectivenesscanbeestimated. thesemaybetheaveragenumberof customersservedbythesystemperunittime,theaveragenumberofoccupied servers,theaveragenumberofcustomersinthequeue,theaveragetimeof waitingforservicing,theaveragepercentageofrefusedcustomers,andthe probabilityacustomerarrivingatthesystemisimmediatelyserved.thereare othermeasuresofsuchsystems’effectiveness.itisquitenaturalthatwhen organizingtheoperationofaqueueingsystemweshouldstrivetoreduce theaveragenumberofcustomersinthequeue,andtoreducethetimeof waitingforservicing.itisalsodesirabletomaximizetheprobabilitythata customerarrivingatthesstemisservedimmediately,tominimizetheaverage percentageofrefusedcustomers,andsoon. thiseventuallymeansthattheproductivityofthesystemmustbein- creased(i.e. thetimeneededtoserviceeachcustomerbedecreased),the system’soperationberationalized,andthenumberofserversmadeaslargeas possible.however,byraisingthenumberofservers,wecannotavoiddecreas- ingtheaveragenumberofoccupiedservers.thismeansthatthedurationof thetimeforwhichaserverisnotoccupiedwillincrease,i.e.theserverwill beidleforsometime.theresultisthatthesystem’soperationalefficiencyis lowered.thereforewemustinsomeway optimizethesystem’soperation. thenumberofserversshouldnotbetoosmall(toeliminatelongqueuesand tokeepthenumberofrefusalssmall),butitshouldalsonotbetoolarge(so70 decisionmaking thatthenumberanddurationofidleperiodsforeachserverissmall). systemswithlosses.thesimplesttypeofqueueingsystemisa single- serversystemwithlosses. herearesomeexamples: asystemwithonlyone telephonelineoraparticledetectorconsistingofonlyonegeiger-müller counter.thestategraphforsuchasystemisshowninfigure2.4(a).when theserverisunoccupied,thesystemisinstate𝑆,andwhentheserveris 0 occupied,itisinstate𝑆. thecustomer’sarrivalrateis𝜆,andtheservice 1 completionrateisit𝜇.thisstategraphisverysimple.whenthesystemis instate𝑆 acustomerarrivingatthesystemtransfersittostate𝑆,andthe 0 1 servicingstarts.oncetheservicingiscompleted,thesystemreturnstostate 𝑆 andisreadytoserveanewcustomer. 0 (a) (b) figure2.4:stategraphofasystemwithlosses. weshallnotgointodetailonthistypeofsystemandgostraightovertoa regeneralcase,ann-serversystemwithlosses.anexampleisasystemconsisting of𝑛telephonelines.erlang,thefounderofthequeueingtheory,considered preciselythissystem.thecorrespondingstategraphisgiveninfigure2.4(b). thestatesofthesystemaredesignatedasfollows: 𝑆 whenallserversare 0 unoccupied,𝑆 whenoneserverisoccupiedandtheothersareunoccupied, 1 𝑆,whentwoserversareoccupiedwhiletheothersareunoccupied,andso 271 queueingsystems on,and𝑆 isthestatewhenall𝑛serversareoccupied. asinthepreceding 𝑛 example,𝜆isthecustomerarrivalrate,and𝜇istheservice-completionrate. supposethesystemisinstate𝑆.whenacustomerrequestarrives,one 0 oftheserversbecomesoccupied,andthesystemistransferredtostate𝑆. 1 ifthesystemisinstate𝑆,andanewcustomerarrives,twoserversbecome 1 occupied,andthesystemistransferredfrom𝑆 to𝑆.thus,eachcustomer 1 2 (withtherateofarrivals𝜆)transfersthesystemfromonestatetotheadjacent one fromlefttoright(seethestategraphinthefigure).thearrivalofevents leadingtotransitionstoadjacentstates fromrighttoleftissomewhatmore complicated. ifthesystemisinthestate𝑆 (onlyoneserverisoccupied), 1 thenextservice-completioneventwilldisengagetheserverandtransferthe systemtostate𝑆.letmeremindyouthattheservice-completionrateis𝜇. 0 nowsupposethesystemisin𝑆,i.e.twoserversareoccupied.theaverage 1 timeofserviceforeachserveristhesame.eachsewerisdisengagedwiththe rateitwhenservicesarecompleted.astothetransitionofthesystemfrom𝑆 2 to𝑆,itisindifferentastowhichofthetwoserversisunoccupied.therefore. 1 eventswhichtransferthesystemfrom𝑆 to𝑆 arriveattherate2𝜇.astothe 2 1 transitionofthesystemfrom𝑆 to𝑆,itisindifferentastowhichofthethree 3 2 occupiedserversisdisengaged.eventswhichtransferthesystemfrom𝑆 to 3 𝑆 arriveattherate3𝜇,andsoforth. itiseasytoseethattherateofevent 2 arrivalwhichtransfersthesystemfrom𝑆 to𝑆 is𝑘𝜇. 𝑘 𝑘−1 letusassumethatthesystemisinasteadystate. applyingtherule fromtheprecedingsectionandusingthestategraphinfigure2.4(b),we can compile the chapman-kolmogorov equations for the, probabilities 𝑝 , 𝑝 , 𝑝 ,…𝑝 ,(recallthat𝑝 istheprobabilitythatthesystemisinthestate 0 1 2 𝑛 𝑖 𝑆).weobtainthefollowingsystemofequations: 𝑖72 decisionmaking 𝜆𝑝 =𝜇𝑝 , 0 1 (𝜆+𝜇)𝑝 =𝜆𝑝 +2𝜇𝑝 , ⎫ 1 0 2 } (𝜆+2𝜇)𝑝 =𝜆𝑝 +3𝜇𝑝 , } 2 1 3 … … … …, (2.3) (𝜆+𝑘𝜇)𝑝 =𝜆𝑝 +(𝑘+1)𝜇𝑝 , 𝑘 𝑘−1 𝑘+1 ⎬ … … … …, } [𝜆+(𝑛−1)𝜇]𝑝 =𝜆𝑝 +𝑛𝜇𝑝 , } 𝑛−1 𝑛−2 𝑛 𝑝 +𝑝 +𝑝 +… +𝑝 =1. ⎭ 0 1 2 𝑛 thissetofequationscanbesolvedeasily,usingthefirstequation,wecan express𝑝 intermsof𝑝 andsubstituteitintothesecondequation.thenwe 1 0 canexpress𝑝 inthesecondequationintermsof𝑝 andsubstituteitintothe 2 𝑛 thirdone,andsoforth.atthelastbutonestage,weexpress𝑝 intermsof𝑝 . 𝑛 0 andfinally,theresultsobtainedateachstagecanbesubstitutedintothelast equationtofindtheexpressionfor𝑝 .thus 0 𝜆 (𝜆/𝜇)2 (𝜆/𝜇)3 (𝜆/𝜇)𝑛 −1 𝑝 =[1+ + + +…+ 𝑛! ] , 0 𝜇 2! 3! (2.4) (𝜆/𝜇)𝑘 𝑝 = 𝑝 (𝑘=1, 2, 3𝑛). 𝑘 𝑘! 0 acustomer’srequestisrefusedifitarriveswhenall𝑛serversareengaged,i.e. whenthesystemisinstate𝑆.theprobabilitythatthesystemisin𝑆 equals 𝑛 𝑛 𝑝 ,thisistheprobabilitythatacustomerarrivingatthesystemisrefused 𝑛 andtheservice‘isnotrendered.wecanfindtheprobabilitythatacustomer arrivingatthesystemwillheserved, (𝜆/𝜇)𝑛 𝑄=1−𝑝 =1− 𝑝 . (2.5) 𝑛 𝑛! 0 bymultiplying𝑄by𝜆,weobtaintheservice-completionrateofthesystem. eachoccupiedserverserves𝜇customersperunittime,sowecandivide𝑄by73 queueingsystems 𝜇andfindtheaveragenumberofoccupiedserversinthesystem, 𝜆 (𝜆/𝜇)𝑛 𝐸(𝑁)= (1− 𝑝 ). (2.6) 𝜇 𝑛! 0 howmanyserversarerequired?letusconsideraconcreteexample. supposeatelephoneexchangereceives1.5requestsperminuteontheaverage, andtheservicecompletionrateis0.5requestperminute(theaverageservice timeforonecustomeristwominutes). therefore,𝜆/𝜇 = 3. supposethe exchangehasthreeservers(threetelephonelines).usingformulas(2.4)–(2.6) for𝜆/𝜇=3and𝑛=3,wecancalculatethattheprobabilityofservicingthe arrivingcustomersisonly65percent.theaveragenumberofengagedlines is1.96,whichis65percentofthetotalnumberoflines.thus,35percentof thecustomersarerefusedandnotserved.thisistoomuch. wemaydecideonincreasingthenumberofservers.supposeweaddone more,afourthline.nowtheprobabilityofacustomerbeingservedincreases to79percent(theprobabilityofbeingturnedawaydecreasesto21percent). theaveragenumberofengagedlinesbecomes2.38,whichis60percentof thetotalnumberoflines.itwouldappearthatthedecisiontoinstallafourth lineisreasonablebecausearelativelysmallreductioninthepercentageof occupiedservers(from65to60percent)resultsinasignificantriseinthe probabilitytobeserved,from65to79percent.anyfurtherincreaseinthe numberoflinesmaybecomeunprofitablebecausetheeffectivenessofthe systemmayfallduetotheincreasingidlenessofthelines.amoredetailed analysiswouldthenberequiredtoallowforthecostofinstallingeachnew line.letmeremarkthatat𝑛=5weget𝑄=89percentand𝐸(𝑁)/𝑛=53 percent,whilefor𝑛=6, 𝑄=94percentand𝐸(𝑁)/𝑛=47percent. single-serversystemswithfinitequeues. supposethenumberof queueingcustomersisrestricted,andthequeuemayonlyaccommodate𝑚 customers. ifallplacesinthequeueareoccupied,anewcomeristurned away. forexample,apetrolstationwithonlyonepump(onlyoneserver)74 decisionmaking andaparkingareafornomorethan𝑚cars.ifalltheplacesatthestationare occupied,thenextcararrivingatthestationwillnotstopandwillgoontothe next.thestategraphforthissystemisshowninfigure2.5(a).here𝑆 means 0 (a) (b) figure2.5:stategraphforasystem(a)withfi- theserverisunoccupied,𝑆 theserverisoccupied,𝑆 theserverisoccupied nitequeuesand(b)withinfinitequeues. 1 2 andthereisonecustomerinthequeue,𝑆 theserverisoccupiedandthere 3 aretwocustomersinthequeue,…,𝑆 meanstheserverisoccupiedand 𝑚+1 thereare𝑚customersinthequeue.asbefore,𝜆isthecustomerarrivalrate and𝜇istheservicecompletionrate.thechapman-kolmogorovequations forsteadystateare 𝜆𝑝 =𝜇𝑝 , 0 1 ⎫ (𝜆+𝜇)𝑝 =𝜆𝑝 +2𝜇𝑝 , 1 0 2 … … … …, (2.7) (𝜆+𝜇)𝑝 =𝜆𝑝 +𝜇𝑝 ,⎬ 𝑚 𝑚−1 𝑚+1 𝑝 +𝑝 +𝑝 +…+𝑝 +𝑝 =1. ⎭ 0 1 2 𝑚 𝑚+1 bysolvingthissystemandintroducingthedesignation𝜌=𝜆/𝜇weobtain 1 1−𝜌 𝑝 = = , 𝑝 =𝜌𝑘𝑝 . (2.8) 0 1+𝜌+𝜌2+𝜌3+…+𝜌𝑚+1 1−𝜌𝑚+2 𝑘 075 queueingsystems acustomeristurnedawayiftheserverisengagedandthereare𝑚customersin thequeue,i.e.whenthesystemisinthestate𝑆 .therefore,theprobability 𝑚+1 acustomeristurnedawayis𝑝 .theaveragenumberofcustomersinthe 𝑚+1 queueisevidently 𝑚 𝐸(𝑟)=∑𝑘𝑝 𝑘+1 𝑘=1 (𝑝 istheprobabilityof𝑘customersbeinginthequeue). theaverage 𝑘+1 waitingtimeinthequeueistheratio𝐸(𝑟)/𝜆. supposeonecararrivesatthepetrolstationperminute(𝜆=1customer perminute)andacarisfilled,onaverage,withintwominutes(𝜇 = 1/2). therefore,𝑝 = 𝜆/𝜇 = 2. ifthenumberofplacesinthequeue𝑚 = 3,itis easytocalculatethattheprobabilityofacustomerbeingrefusedis51.6per centwhiletheaveragewaitingtimeinthequeueis2.1min.supposethatin ordertodecreasetheprobabilityofacustomerbeingrefusedwedoublethe numberofplacesinthequeue.itturnsoutthatat𝑚=6theprobabilityof refusalis50.2percent,i. e. itis,infact,thesame,butthewaitingtimein thequeuenoticeablyincreasesto5min.itisclearfrom(2.8)thatif𝜌>1, theprobabilityofbeingrefusedstabilizeswithincreasing𝑚andtends,to (𝜌−1)/𝜌.inordertoreducetheprobabilityofbeingrefusedsignificantly,it isnecessary(ifitisnotpossibletodecrease𝜌)tousemulti-serversystems. single-serversystemswithinfinitequeues. thissortofqueueing systemisrathercommon:forexample,adoctorreceivingpatients,asingle publictelephone,oraportwithonlyoneberthatwhichasingleshipcan unload. thestategraphforthesystemisgiveninfigure2.5(b). hereso meansthattheserverisunoccupied,𝑆 theserverisoccupied,𝑆 theserveris 1 2 occupiedandthereisonecustomerinthequeue,𝑆 theserverisoccupied 3 andtherearetwocustomersinthequeue,and𝑆 meansthattheserveris 𝑘 occupiedandthereare𝑘−1customersinthequeue,andsoon. uptillnow,weconsideredgraphswithafinitenumberofstates.however, hereisasystemwithaninfinitenumberofdiscretestates. isitpossibleto76 decisionmaking discussasteadystateforsuchasystem?infactwecan.itisonlynecessarythat theinequality𝜌<1holdstrue.ifso,thenthesum1+𝜌+𝜌2+𝜌3+…+𝜌𝑚+1 in(2.8)canbesubstitutedbythesumofthedecreasinggeometricprogression 1+𝜌+𝜌2+𝜌3+… =1/(1−𝜌).theresultis 𝑝 =1−𝜌 and 𝑝 =𝜌𝑘𝑝 . (2.9) 0 𝑘 0 if𝜌⩾1,thenthesystemdoesnothaveasteadystate,i.e.thequeueincreases infinitelyas𝑡→∞. method of statistical testing a statisticaltestinginvolvesnumerousrepetitionsofuniformtrials. the resultofanyindividualtrialisrandomandisnotofmuchinterest.however, alargenumberofresultsisveryuseful. itshowssomestability(statistical stability)andsothephenomenonbeinginvestigatedinthetrialscanbede- scribedquantitatively. letusconsideraspecialmethodforinvestigating arandomprocessbasedonstatisticaltesting. thetechniqueiscommonly calledthe montecarlomethod. infactneitherthecityofmontecarlo,thecapitaloftheindependent principalityofmonaconoritsinhabitantsnorguestsareinanywayrelated totheconsideredmethod. instead,thecityisknownforitscasinoswhere touristspaygoodmoneyplayingroulette,andaroulettewheelcouldbethe city’semblem.atthesametime,arouletteisageneratorofrandomnumbers andthisiswhatisinvolvedwhenthemontecarlomethodisused. twoexamplesindicatingtheusefulnessofstatisticaltesting. firstexample. lookatfigure2.6. itcontainsasquarewithside𝑟in whichaquartercircleofradius𝑟isinscribed.theratiooftheyellowareato theareaofthesquareis(𝜋𝑟2)/4𝑟2 =𝜋/4.thisratioand,therefore,thevalue of𝑛canbeobtainedusingthefollowingstatisticaltest.letusplaceasheet77 methodofstatisticaltesting ofpaperwiththefigureonahorizontalsurfaceandletusthrowsmallgrains onthispaper. weshouldnotaimsothatanygraincanfallonanypartof thepaperwithequalprobability.itispossible,forinstance,toblindfoldthe personthrowingthegrains.thegrainswillbedistributedoverthesurfaceof thepaperinarandomfashion(figure2.6(b)).somewilllandoutsidethe square,butweshallnotconsiderthem.wenowcountthenumberofgrains withinthesquare(andcallthisnumber𝑁)andcountthegrainswithinthe 1 yellowarea(callingit𝑁).sinceanygrainmaylandwithequalprobability 2 onanypartofthefigure,theratio𝑁/𝑁 whenthenumberoftrialsislarge, 2 1 willapproximatetheratiooftheyellowareatotheareaofthesquare,i.e.the number𝜋/4.thisapproximationwillbecomemoreaccurateasthenumber oftrialsincreases.thisexampleisinterestingbecauseadefinitenumber(the (a) (b) figure2.6:findingoutvalueof𝜋usingaran- number𝜋)canbefoundfollowingastatisticaltesting. itcanbesaidthat domdistribution. randomnessisusedheretoobtainadeterministicresult,anapproximation oftherealnumber𝜋. secondexample. statisticaltestingisusedmuchmorecommonlyto investigate randomeventsand randomprocesses.supposesomeoneassembles78 decisionmaking adeviceconsistingofthreeparts(𝐴,𝐵,and𝐶).theassemblerhasthreeboxes containingparts𝐴,𝐵,and𝐶,respectively.supposehalfthepartsofeachtype arelargerthanthestandardandtheotherhalfaresmaller.thedevicecannot operatewhenallthreepartsarelargerthanthenorm.theassemblertakes thepartsfromtheboxesatrandom.whatistheprobabilitythatanormally operatingdevicewillbeassembled? naturally,thisexampleisrathersimpleandtheprobabilitycaneasilybe calculated.theprobabilityofassemblingadevicethatdoesnotworkisthe probabilitythatallthreepartswillbelargerthanthenorm,andthisequals 1/2×1/2×1/2=1/8.therefore,theprobabilitythatanormallyoperating devicewillbeassembledis1−1/8=0.875. letusforgetforatimethatwecancalculatetheprobabilityandinstead usestatisticaltesting.weshouldchoosetrialssuchthateachonehasequally probableoutcomes,forinstance,tossingacoin.letustakethreecoins:𝐴,𝐵, and𝐶.eachcoincorrespondstoapartusedtoassemblethedevice.heads willmeanthattherespectivepartislargerthanthenormwhiletailswillmean thatitissmaller.havingagreedonthis,letusstartthestatisticaltesting.each trialinvolvestossingallthreecoins. supposeafter𝑁trials(𝑁 ≫ 1)three headswererecordedin𝑛trials.itiseasytoseethattheratio(𝑁−𝑛)/𝑁is theapproximationoftheprobabilityinquestion. naturally,wecoulduseanyotherrandomnumbergeneratorinsteadof coins. itwouldalsobepossible,forinstance,tothrowthreedice,having agreedtorelatethreefacesofeachdiewithlargerthannormalpartsandthree faceswithsmallerparts. letmeemphasizethattherandomnessintheseexampleswasapositive factorratherthananegativeone,andwasatoolwhichallowedustoobtaina neededquantity.herechanceworksforusratherthanagainstus. randomnumbertablescomeintoplay.nobodyusesstatisticaltesting insimplepracticalsituationsliketheonesdescribedabove.itisusedwhenit79 methodofstatisticaltesting isdifficultorevenimpossibletocalculatetheprobabilityinquestion.nat- urallyyoumightaskwhetherastatisticaltestingwouldbetoocomplicated andcumbersome. wethrewgrainsorthreecoinsintheexamples. what willberequiredincomplicatedsituations?maybe,therewillbepractically unsurmountableobstacles? inreality,itisnotnecessarytostageastatisticalexperimentwithrandom trials. insteadofrealtrials(throwinggrains,dice,etc.),weneedonlyuse randomnumbertables.letmeshowhowthiscanbedoneintheabovetwo examples. y 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x figure2.7:findingoutvalueof𝜋usingaran- domnumbertable. firstexample. letusagaindiscussthepictureinfigure2.6. wenow plottwocoordinateaxesalongthesidesofthesquareandselectthescales80 decisionmaking suchthatthesideofthesquareequalsunity(figure2.7). nowinsteadof throwinggrains,wetaketherandomnumbertableinfigure1.6anddivide eachnumberby10000sothatweobtainasetofrandomnumbersbetween (0.0655,0.5255) 0and1.wetakethenumbersintheoddlinesas𝑥-coordinatesandtheones (0.6314,0.3157) directlybelowasthe𝑦-coordinatesofrandompoints. weplotthepoints (0.9052,0.4105) ontothediagram,systematicallymovingalongtherandomnumbertable(for (0.1437,0.4064) instance,firstdownthefirstcolumnfromtoptobottom,andthendownthe (0.1037,0.5718) (0.5127,0.9401) secondcolumn,andsoon).thefirstfifteenrandompointsareshowninthe (0.4064,0.5458) pictureinred,andtheyhavethecoordinatesasshownintable2.1.thefigure (0.2461,0.4320) contains85randompointsinblack.fromthediagram,itiseasytocalculate (0.3466,0.9313) thatusingthefirstfifteenpoints𝑁/𝑁 = 13/15andtherefore𝜋 = 3.47 (0.5179,0.3010) 2 1 (0.9599,0.4242) whileforahundredpoints𝑁2/𝑁1 =78/100andtherefore𝜋=3.12. (0.3585,0.5950) (0.8462,0.0456) secondexample.insteadoftossingcoins,wecanusethesamerandom (0.0672,0.5163) numbertable(seefigure1.6).eachnumberover5000canbereplacedbya (0.4995,0.6751) “+”signandtherestreplacedbya“−”sign.theresultisatableconsistingofa randomsetofplusesandminuses.wedividethesesignsintotriplesasshown table2.1:coordinatesoffifteenrandomnum- bersshowninredinthefigure2.7. infigure2.8.eachtriplecorrespondstoasetofthreeparts.a“+”signmeans thatapartislargerthanthenormwhilea“−”signmeansitissmaller.the approximationofthesoughtprobabilityistheratio(𝑁−𝑛)/𝑁,where𝑁is thetotalnumberoftriplesand𝑛isthenumberoftripleswiththreepluses (theyareshadedinthefigure).itcanbeseenthat(𝑁−𝑛)/𝑁=0.9inthis case,andthisiscloseenoughtotheaccuratevalue0.875. thus,wehavereducedstatisticaltestingtooperationsonarandomnum- bertableandusedourdeskinsteadofanexperimentalbench.ratherthan performingverymanytrials,wejustlookatarandomnumbertable. computerscomeintoplay.insteadofsweatingoverarandomnumber table,wecouldprogramacomputertodothejob.weplacearandomnum- bertableinthecomputer’smemoryandprogramittosearchtherandom numbersandsortthemasnecessary.inourtwoexamples,wewoulddothe following.81 methodofstatisticaltesting – + – – + – – + + – + + – + – – – + + – + + – – + + – + – – – + – + + – – + + + + – – – – + + – + – – + – – – + + – – – – – + + – – – + – + – – + – + – + + + – – + – + – + + + – – + + – – – – – – + + + – – – + + – + + – + – + – – – + – + + – + – – – + + + + + + – + + + – + – – – – – + + – – – + + + – – – – + + + + – – – + + + + – + – + + + + – – + – – – + – ++– +++ +–– ++– ++– +++ +–– ++– +–– ––– – – + – + – – + + – – + + + + – – + – + + – – – – + – – – – + – + + + + + + + + – – – – – – – – + + – – + + + + + – – + + + – – – – + – – – – + + – + + – – – + + + – + + – – – – – figure2.8: usingarandomnumbertablein- steadofcointossesforstatisticaltesting. firstexample.thecomputerhastocheckthecoordinatesofeachran- dompointtoseewhether𝑥2+𝑦2 < 1.itcountsthenumberofpointsfor whichthisistrue(thenumberis𝑁)andthenumberofpointsforwhichit 2 isfalse(thisnumberofpointswillbethedifference𝑁 −𝑁). 1 2 secondexample.allrandomnumbersinthecomputer’smemorymust bedividedintotriplesandthetriplescheckedtofindonesinwhichallthree numbersareover5000.thenumberofsuchtriplesis𝑛.82 decisionmaking themontecarlomethod. theworldchangedwhenthecomputer cameintoplay.byprocessingarandomnumbertablethecomputersimu- latesthestatisticaltestinganditcandothismanytimesfasterthancouldbe doneeitherexperimentallyorbyworkingmanuallywitharandomnumber table.andnowwecometothemontecarlomethod,averyusefulandeffi- cientmethodofprobabilisticcalculationwhichisappliedtomanyproblems, primarilythosethatcannotbesolvedanalytically. letmeemphasizetwopoints. firstly,themontecarlomethodutilizes randomnessnotchance.wedonottrytoanalyzethecomplicatedrandom processes,norevensimulatethem.instead,weuserandomness,asitwere,to dealwiththecomplicationschancehasengendered. chancecomplicatesourinvestigationandsorandomnessisusedtoin- vestigateit. secondly,thismethodis universalbecauseitisnotrestrictedby anyassumption,simplification,ormodel.therearetwobasicapplications. thefirstistheinvestigationofrandomprocesseswhichcannotbedealtwith analyticallyduetotheircomplexity.thesecondistoverifythecorrectness andaccuracyofananalyticalmodelappliedinconcretesituations. themontecarlomethodwasfirstwidelyusedinoperationsresearch,in lookingforoptimaldecisionsunderconditionsofuncertainty,andintreating complicatedmulti-criterialproblems.themethodisalsosuccessfullyused inmodernphysicstoinvestigatecomplexprocessesinvolvingmanyrandom events. amontecarlosimulationofaphysicalprocess. letusconsider theflowofneutronsthroughthecontainmentshieldofanuclearreactor. uraniumnucleisplitinthecoreofthereactorandthisisaccompaniedby thecreationofhigh-energyneutrons(oftheorderofseveralmillionelectron volts).thereactorissurroundedbyashieldtoprotecttheworkingareas(and therefore,thepersonnel)fromtheradiation.thewallisbombardedbyan intenseflowofneutronsfromthereactorcore.theneutronspenetrateinto thewallandcollidewiththenucleioftheatomsofthewall.theresultisthat83 methodofstatisticaltesting theneutronsmayeitherbeabsorbedorscattered.ifscattered,theygiveup someoftheirenergytothescatteringnuclei. thisisacomplicatedphysicalprocess involvingmanyrandomevents. theenergyandthedirectionofaneutronwhenitleavesthereactorcore andentersthewallarerandom,thelengthoftheneutronpathbeforeit firstcollidesisrandom,thenatureofcollision(absorptionorscattering)is random,theenergyandthedirectionofthescatteredneutronarerandom,etc. letmeshowingeneralhowthemontecarlomethodisappliedtoanalyze theprocess.obviouslythecomputerisfirstprogrammedwithdataonthe elementarycollisionsbetweenneutronsandthewallnuclei(theprobabilities ofabsorptionandscattering)theparametersoftheneutronflowintothe wall,andthepropertiesofthewall.thecomputermodelsimulatesaneutron witharandomlyselectedenergyanddirection(whenitleavesthereactorcore andentersthewall)inlinewithappropriateprobabilities.thenitsimulates (bearinginmindtherelevantprobabilities)theflightoftheneutronuntil itfirstcollides. thenthefirstcollisionissimulated. iftheneutronisnot absorbed,subsequenteventsaresimulated,i.e.theneutron’sflightuntilits secondcollision,thecollisionitself,andsoon.the“history”oftheneutronis determinedfromthemomentitpenetratesthewalluntilitiseitherabsorbed, scatteredbackintothereactorcore,orscatteredintotheworkingarea. thecomputersimulationisrepeatedforverymanyneutronsuntilaset ofpossibletrajectoriesofneutronswithinthewallisobtained(figure2.9). eachtrajectoryistheresultofonestatisticaltrialsimulatingthe“history”of chancecomplicatesourinvestigationandsorandomnessisusedtoinvesti- gateit.secondly,thismethodisuniversalbecauseitisnotrestrictedbyany assumption,simplification,ormodel.therearetwoanindividualneutron. givenanenormoussetoftrialstheneutronflowthroughthecontainment wallasawholecanbeanalyzedandrecommendationsforthethicknessof thewallanditscompositioncanbemadesoastoguaranteethesafetyofthe personnelworkingatthereactor.84 decisionmaking figure2.9:asetofallpossibletrajectoriesfor theneutron. modernphysicsrequiresthemontecarlomethodonmanyoccasions. physicistsuseittoinvestigatecosmic-rayshowersintheearth’satmosphere, thebehaviouroflargeflowsofelectronsinelectrondischargedevices,and theprogressofvariouschainreactions. games and decision making whatisthetheoryofgames? supposewemustmakeadecisionwhen ourobjectivesareopposedbyanotherparty,whenourwillisinconflict withanotherwill.suchsituationsarecommon,andtheyarecalled conflict situations. theyaretypicalformilitaryactions,games,andevery-daylife. theyoftenariseineconomicsandpolitics. ahockeyplayermakesadecisionthattakesintoaccountthecurrent situationandthepossibleactionsoftheotherplayers. everytimeachess playermakesadecision,he(orshe)hastoconsiderthecounteractionofthe opponent.amilitarydecisionshouldallowfortheretaliationoftheenemy. inordertodecideatwhatpricetosellaproduct,asalesmanmustthinkover theresponsesofthebuyer.inanyelectioncampaign,eachpoliticalpartyin acapitalistcountrytriestoforeseetheactionsoftheotherpartiesthatare85 gamesanddecisionmaking competingforpower.ineachcase,thereisacollisionofopposinginterests, andthedecisionmustberelatedwith overcomingaconflict. decisionmakinginaconflictsituationishamperedby uncertaintyabout thebehaviouroftheopponent.weknowthattheopponentwilltrytoactina waythatisleastadvantageousforusinordertoensurethegreatestadvantage forhimself.however,wedonotknowtowhatextentouropponentisable toevaluatethesituationandthepossibleconsequencesand,inparticular, howheevaluatesouroptionsandintentions.wecannotpredicttheactions oftheopponentaccurately,andtheopponentcannotpredictouractions. butnonetheless,webothhavetomakedecisions. becausesomewayofjustifyingan optimaldecisionwasneededinconflict situations,anewmathematicaldisciplinearose,the theoryofgames. the “game”hereisamathematicalmodelofaconflictsituation. unlikeareal conflict,agamehasdefiniteruleswhichclearlyindicatetherightsandduties oftheparticipantsandthepossibleoutcomesofthegame(againorlossfor eachparticipant).longbeforetheemergenceofgametheory,simplemodels ofconflictswereusedwidely.imeangamesintheliteralsenseoftheword: chess,checkersordraughts,dominoes,cardgames,etc.infact,thenameof thetheoryandthevarioustermsusedinitareallderivedfromthesesimple models.forinstance,theconflictingpartiesarecalledplayers,arealizationof agameisamatch,theselectionofanactionbyaplayer(withintherules)isa move. therearetwokindsofmove,personalandchanceones.a personalmove iswhentheplayerconscientiouslyselectsanactionaccordingtotherulesof thegame. a chancemovedoesnotdependontheplayer’swill: itmaybe determinedbytossingacoin,throwingadie,takingacardfromapack,etc. gamesconsistingofonlychancemovesarecalled gamesofchance,or games ofhazard. typicalexamplesarelotteriesandbingo. gameswithpersonal movesarecalled strategic.therearestrategicgamesconsistingexclusivelyof personalmoves,forinstance,chess.therearealsostrategicgamesconsisting86 decisionmaking ofbothpersonalandchancemoves,forinstance,certaincardgames. let meremarkthattheuncertaintyingameswithbothpersonalandchance movesinvolvebothsortsofrandomness: theuncertaintyoftheresultof thechancemovesandtheuncertaintyoftheopponent’sbehaviourinhis personalmoves. gametheoryisnotinterestedingambles. itonlydealswithstrategic games.theaimofthegametheoryistodeterminetheplayer’sstrategysoasto maximizehischancesofwinning.thefollowingbasicassumptionunderlies thesearchforoptimalstrategies.itisassumedthattheopponentisasactive andasreasonableastheplayer,andheorshealsotakesattemptstosucceed. naturally,thisisnotalwaystrue.veryoftenouractionsinrealconflicts arenotasgoodastheycouldbewhenweassumereasonablebehaviourfrom ouradversary;itisoftenbettertoguessatthe“softspots”oftheopponent andutilizethem.ofcourse,wetakeariskwhendoingso.itisriskytorelytoo muchonthesoftspotsoftheopponent,andgametheorydoesnotconsider risk.itonlydetectsthemostcautious,“safe”versionsofbehaviourinagiven situation. itcanbesaidthatgametheorygiveswiseadvice. bytakingthis advicewhenwemakeapracticaldecision,weoftentakeaconscientiousrisk. e.s.wentzelwritesin operationsresearch: “gametheoryisprimarilyvaluableintermsoftheformulationofthe problem,whichteachesusnevertoforgetthattheopponentalsothinks andtotakeintoaccounthispossibletricksandtraps.therecommen- dationsfollowingfromthegameapproacharenotalwaysconcreteor realizable,butitisstilluseful,whiletakingadecision,toutilizeagame modelasoneofseveralpossibleones.buttheconclusionsproceeding fromthismodelshouldnotberegardedasfinalandindisputable.” thepayoffmatrixofagame finitetwo-personzero-sumgamesare thebestinvestigatedtypesingametheory.a two-persongameisagamein whichthereareexactlytwoplayersorconflictinginterests.agameis finiteif bothplayershaveafinitenumberofpossiblestrategies,i.e.afinitenumber ofbehaviours.whenmakingapersonalmove,aplayerfollowsastrategy.a

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