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2/3/2017 Sections 4-1 and 4-2 Review and Preview and Basic Concepts of Probability RARE EVENT RULE FOR INFERENTIAL STATISTICS If,underagivenassumption(suchasalottery beingfair),theprobabilityofaparticular observedevent(suchasfiveconsecutivelottery wins)isextremelysmall,weconcludethatthe assumptionisprobablynotcorrect. Statisticiansusetherareeventrulefor inferentialstatistics. PROBABILITY Probability isthemeasureofthelikelihood thatagiveneventwilloccur. 1 2/3/2017 EVENTS • Anevent isanycollectionofresultsor outcomesofaprocedure. • Asimpleevent isanoutcomeoreventthat cannotbefurtherbrokendownintosimpler components. • Thesamplespace foraprocedureconsists ofallpossiblesimple events.Thatis,the samplespaceconsistsofalloutcomesthat cannotbebrokendownanyfurther. PROBABILITY Probability isameasureofthelikelihoodthat agiveneventwilloccur. NOTATION: • P denotesaprobability. • A,B,andC denotespecificevents. • denotestheprobabilityofeventA occurring. RULE 1: RELATIVE FREQUENCY APPROXIMATION OF PROBABILITY Conduct(orobserve)aprocedurealargenumber oftimes,andcountthenumberoftimesthat eventA actuallyoccurs.Basedontheseactual isapproximated asfollows: results, numberoftimes occurred numberoftimesprocedurewasrepeated ThisruleusestheLawofLargeNumbers. 2 2/3/2017 THE LAW OF LARGE NUMBERS Asaprocedureisrepeatedagainandagain,the relativefrequencyprobability(fromRule1)ofan eventtendstoapproachtheactualprobability. CAUTION:Thelawoflargenumbersappliesto behavioroveralargenumberoftrials,anditdoes notapplytooneoutcome.Don’tmakethefoolish mistakeoflosingalargesumofmoneyby incorrectlythinkingthatastringoflosses increasesthechancesofawinonthenextbet. EXAMPLE Afairdiewastossed563times.Thenumber “4”occurred96times.Ifyoutossafairdie, whatdoyouestimatetheprobabilityisfor tossinga“4”? RULE 2: CLASSICAL APPROACH TO PROBABILITY Assumethatagivenprocedurehas different simpleeventsandthateachofthosesimple eventshasanequalchanceofoccurring.If eventA canoccurin ofthose ways,then numberofways canoccur numberofdifferentsimpleevents CAUTION:Whenusingtheclassicalapproach, alwaysverifythattheoutcomesareequally likely. 3 2/3/2017 EXAMPLE Findtheprobabilityofgettinga“7”whena pairofdiceisrolled. RULE 3: SUBJECTIVE PROBABILITIES ,theprobabilityofeventA,isestimated by usingknowledgeoftherelevantcircumstances. PROBABILITY LIMITS — Likely • Theprobabilityofan impossibleeventis0. • Theprobabilityofaneventhat iscertaintooccuris1. •0 1 foranyeventA — Unlikely 4 2/3/2017 COMPLEMENTARY EVENTS ThecomplementofeventA,denotedby ̅, consistsofalloutcomesinwhicheventA does not occur. EXAMPLE Whatistheprobabilityofnot rollinga“7” whenapairofdiceisrolled? ROUNDING OFF PROBABILITIES Whenexpressingthevalueofaprobability, eithergivetheexact fractionordecimalor roundofffinaldecimalresultstothree significantdigits. Suggestion: Whentheprobabilityisnota simplefractionsuchas2/3or5/9,expressit asadecimalsothatthenumbercanbebetter understood. 5 2/3/2017 UNLIKELY AND UNUSUAL EVENTS Aneventisunlikely ifitsprobabilityisvery small,suchas0.05orless.Aneventhasan unusuallylownumber ofoutcomesofa particulartypeoranunusuallyhighnumber ofthoseoutcomesifthatnumberisfarfrom whatwetypicallyexpect. SUMMARY: • Unlikely:smallprobability • Unusual:extremeresult ODDS • Theactualoddsagainst eventA occurringarethe ̅ ⁄ ratio ,usuallyexpressedintheformof a:b (or“a tob”),wherea andb areintegershaving nocommonfactors. • Theactualoddsinfavor ofeventA areratio ̅ ,whichisthereciprocaloftheactual / oddsagainstthatevent.IftheoddsagainstA are a:b,thentheoddsinfavorofA areb:a. • Thepayoffodds againsteventA representtheratio ofthenetprofit(ifyouwin)totheamountbet: payoffoddsagainstA =(netprofit):(amountbet) EXAMPLE TheAmericanStatisticalAssociationdecided toinvestsomeofitsmemberrevenueby buyingaracehorsenamedMean.Meanis enteredinaraceinwhichtheactual probabilityofwinningis3/17. (a) FindtheactualoddsagainstMean winning. (b) Ifthepayoffoddsarelistedas4:1,how muchprofitdoyoumakeifyoubet$5 andMeanwins. 6