Statistical Phys_Ch3_Concepts of Probability

Chapter3ConceptsofProbabilityc2010byHarveyGouldandJanTobochnik5October2010Weintroducethebasicconceptsofprobabilityandapplythemtosimplephysicalsystemsandeverydaylife.WediscusstheuniversalnatureofthecentrallimittheoremandtheGaussiandistributionforthesumofalargenumberofrandomvariables.Becauseoftheimportanceofprobabilityinmanycontexts,ourdiscussiongoesbeyondwhatweneedfortheapplicationsofstatisticalmechanicsthatwediscussinlaterchapters.3.1ProbabilityinEverydayLifeOneofourgoals,whichwewillconsiderinChapter4andsubsequentchapters,istorelatethebehaviorofvariousmacroscopicquantitiestotheunderlyingmicroscopicbehavioroftheindividualatomsorotherconstituents.Todoso,weneedtointroducesomeideasfromprobability.Wealluseideasofprobabilityineverydaylife.Forexample,everymorningmanyofusdecidewhattowearbasedontheprobabilityofrain.Wecrossstreetsknowingthattheprobabilityofbeinghitbyacarissmall.Youcanmakearoughestimateoftheprobabilityofbeinghitbyacar.Itmustbelessthanoneinathousand,becauseyouhavecrossedstreetsthousandsoftimesandhopefullyyouhavenotbeenhit.Youmightbehittomorrow,oryoumighthavebeenhitthefirsttimeyoutriedtocrossastreet.Thesecommentsillustratethatwehavesomeintuitivesenseofprobability,andbecauseitisausefulconceptforsurvival,weknowhowtoestimateit.AsexpressedbyLaplace(1819),“Probabilitytheoryisnothingbutcommonsensereducedtocalculation.”AnotherinterestingthoughtisduetoMaxwell(1850):“Thetruelogicofthisworldisthecalculusofprobabilities.”Thatis,probabilityisanaturallanguagefordescribingmanyrealworldphenomena.However,ourintuitiononlytakesussofar.Considerairplanetravel.Isitsafetofly?Supposethatthereisonechanceinfivemillionofaplanecrashingonagivenflightandthatthereareabout50,000flightsaday.Thenevery100daysorsothereisareasonablelikelihoodofaplanecrashsomewhereintheworld.Thisestimateisinroughaccordwithwhatweknow.Foragivenflight,yourchancesofcrashingareapproximatelyonepartin5×106,andifyouflytentimesayearfor100years,itseemsthatflyingisnottoomuchofarisk.Supposethatinsteadofliving106CHAPTER3.CONCEPTSOFPROBABILITY107100years,youcouldlive50,000years.Inthiscaseyouwouldtake500,000flights,anditwouldbemuchmoreriskytoflyifyouwishedtoliveyourfull50,000years.Althoughthislaststatementseemsreasonable,canyouexplainwhy?Muchofthemotivationforthemathematicalformulationofprobabilityarosefromtheprofi-ciencyofprofessionalgamblersinestimatingbettingoddsandtheirdesiretohavemorequantitativemeasuresofsuccess.Althoughgamesofchancehavebeenplayedsincehistoryhasbeenrecorded,thefirststepstowardamathematicalformulationofgamesofchancebeganinthemiddleoftheseventeenthcentury.Someoftheimportantcontributorsoverthefollowing150yearsincludePas-cal,Fermat,Descartes,Leibnitz,Newton,Bernoulli,andLaplace,namesthatareprobablyfamiliartoyou.Giventhelonghistoryofgamesofchanceandtheinterestinestimatingprobabilityinavarietyofcontexts,itisremarkablethatthetheoryofprobabilitytooksolongtodevelop.Onereasonisthattheideaofprobabilityissubtleandiscapableofmanyinterpretations.Anunderstandingofprobabilityiselusivedueinparttothefactthattheprobablydependsonthestatusoftheinformationthatwehave(afactwellknowntopokerplayers).Althoughtherulesofprobabilityaredefinedbysimplemathematicalrules,anunderstandingofprobabilityisgreatlyaidedbyexperiencewithrealdataandconcreteproblems.Totestyourcurrentunderstandingofprobability,solveProblems3.1–3.6beforereadingtherestofthischapter.TheninProblem3.7formulatethelawsofprobabilitybasedonyoursolutionstotheseproblems.Problem3.1.MarblesinajarAjarcontainstwoorange,fiveblue,threered,andfouryellowmarbles.Amarbleisdrawnatrandomfromthejar.Findtheprobabilitythat(a)themarbleisorange;(b)themarbleisred;(c)themarbleisorangeorblue.Problem3.2.PiggybankApiggybankcontainsonepenny,onenickel,onedime,andonequarter.Itisshakenuntiltwocoinsfalloutatrandom.Whatistheprobabilitythatatleast$0.30fallsout?Problem3.3.TwodiceApersontossesapairofdiceatthesametime.Findtheprobabilitythat(a)bothdiceshowthesamenumber;(b)bothdiceshowanumberlessthan5;(c)bothdiceshowanevennumber;(d)theproductofthenumbersis12.Problem3.4.FreethrowsApersonhits16freethrowsoutof25attempts.Whatistheprobabilitythatthispersonwillmakeafreethrowonthenextattempt?CHAPTER3.CONCEPTSOFPROBABILITY108Problem3.5.TossofadieConsideranexperimentinwhichadieistossed150timesandthenumberoftimeseachfaceisobservediscounted.1ThevalueofA,thenumberofdotsonthefaceofthedieandthenumberoftimesthatitappearedisshowninTable3.1.(a)WhatisthepredictedaveragevalueofAassumingafairdie?(b)WhatistheaveragevalueofAobservedinthisexperiment?valueofAfrequency123228330421523625Table3.1:ThenumberoftimesfaceAappearedin150tosses(seeProblem3.5).Problem3.6.What’sinyourpurse?Acoinistakenatrandomfromapursethatcontainsonepenny,twonickels,fourdimes,andthreequarters.Ifxequalsthevalueofthecoin,findtheaveragevalueofx.Problem3.7.RulesofprobabilityBasedonyoursolutionstoProblems3.1–3.6,statetherulesofprobabilityasyouunderstandthematthistime.Thefollowingproblemsarerelatedtotheuseofprobabilityineverydaylife.Problem3.8.ChoicesSupposethatyouareofferedthefollowingchoice:(a)Aprizeof$50,or(b)youflipa(fair)co