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Section5.1Randomness,Probability,andSimulation

Whatdoesitmeantoberandom?

~Individualoutcomesareunknown

~Inthelongrunsomeunderlyingsetofoutcomeswillbeequallylikely(oratleastthelikelihoodofoutcomeswillbeconsistentwithactualoutcomesinthelongrun)

Ifsomeonehandedyouacoinandtoldyouthatit’sbiasedtowardusuallylandingheads,you’dbejustlyskeptical.Whatwouldyoudo?

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You’dprobablytestthecoinbytossingitafewtimes.Howmanytimesareenough?

Would3headsinarowconvinceyouthatthecoinwasbiased?

Would7of10headsbeconvincing?

Imagineweuse100tosses.Would54headsoutof100convinceyouthatthecoinisbiased?

Whatabout60outof100?

Howabout95outof100?

Howabout80outof100?or70?

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Let’sdecideasaclasswhatourrulewillbe.

Let’suseasimulationtoanswerthisquestion.

randInt(0,1)–simulatingasinglecointoss(with0=tails,1=heads)

randInt(0,1,5)–simulatingtossing5coins

sum(randInt(0,1,100))–sinceeachcoinis0or1,thesumofthese100valuesisequivalenttocountingthenumberofheads

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Let’stryrunningacoupleofthese–howmanypeoplegotmorethan________heads?

Keepinmindthatevenifwecomeupwitharule,tossingacoin100timesanddeclaringitokayorbiasedwouldstillnotgiveusthetruthforsure–justasuggestionthatthecoinisbiasedornot.

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Component–thebasicunitofoursimulation,eachrandomlygenerateddigitgivestheoutcomeofONEcomponent Identifythecomponenttoberepeated

Explainhowyouwillmodelthecomponent’soutcome

Trial–acombinationofcomponentoutcomesthatgiveasimulatedanswertoourquestion

ExplainhowyouwillcombinethecomponentstomodelatrialRunseveraltrials–ifbyhand~20isgood

Responsevariable–theoutcomeofthetrial,whatwearemeasuringwitheachtrial

Statistic–whatwillweusetosummarizeourresults

Conclusions-thesimulationSUGGESTS...

Thestepsofasimulation

Supposeabasketballplayerhasan80%freethrowsuccessrate.Howcanweuserandomnumberstosimulatewhetherornotshemakesafoulshot?Howmanyshotsmightshebeabletomakeinarowwithoutmissing?Describethesimulationofhavinghershootfreethrowsuntilshemisses,countingthenumberofsuccesses.

ComponentTrial

Responsevariable

Statistic

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Howwouldthesimulationchangeifhersuccessratewereonly72%?

ComponentTrial

Responsevariable

Statistic

Howwouldatrialandourresponsevariablechangeifwewanttoknowhowmanyshotsshemightmakeoutof5chances?

ComponentTrial

Responsevariable

Statistic

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ComponentTrial

Responsevariable

Statistic

Howwouldatrialandourresponsevariablechangeifwewanttoknowherchancesofhittingbothshotswhenshegoestothelinetoshoottwo?

RandomDigitsTable

Anotherwaytogeneraterandom(orasrandomaswecanget)isusingatableofrandomdigits.

Thereisoneinthebackofyourbook.

Eachdigit0through9isequallylikelytooccuratanytime,thereforeanypatternoftwoormoredigitsisequallylikelytooccuratanypointinthetable.

Inordertousethistabletogeneraterandomdigits-pickarowtostarton.Readthedigitslefttorightbeingsuretoreadingroupsappropriatetothesizeofthenumbersassignedforyoursimulation.Ifyournumbersgofrom1-15youmustreadtwodigitsatatime,etc.

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ComponentTrial

Responsevariable

Statistic

SupposewearetorandomlyselectthreestudentsfromtheclasstospeakatParents’NightabouttakingAPStats.Howlikelyisitwe’llgetthreeboys?

Samisabasketballplayerwhotypicallymakes85%ofherfreethrows.However,whenshootingtwofreethrowsinarow,thelikelihoodofhermakingthesecondshotdependsuponhowshedoesonthe]irstshot.Ifshemissesthe]irstshot,sheonlymakes70%ofhersecondfreethrows.Butifshemakesher]irstshot,shemakes90%ofhersecondfreethrows.DescribeasimulationthatwilldeterminethechanceofSammakingtwofreethrowsinarow.

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Homework:

p.293#s1,3,7,9,11,15,17,19,23,25