Section 5.1 Notes - Completed 1 Section 5.1 Randomness, Probability, and Simulation What does it mean to be random? ~Individual outcomes are unknown ~In the long run some underlying set of outcomes will be equally likely (or at least the likelihood of outcomes will be consistent with actual outcomes in the long run) If someone handed you a coin and told you that it’s biased toward usually landing heads, you’d be justly skeptical. What would you do?
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Section 5.1 Notes - Completed
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Section5.1Randomness,Probability,andSimulation
Whatdoesitmeantoberandom?
~Individualoutcomesareunknown
~Inthelongrunsomeunderlyingsetofoutcomeswillbeequallylikely(oratleastthelikelihoodofoutcomeswillbeconsistentwithactualoutcomesinthelongrun)
Ifsomeonehandedyouacoinandtoldyouthatit’sbiasedtowardusuallylandingheads,you’dbejustlyskeptical.Whatwouldyoudo?
Section 5.1 Notes - Completed
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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
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