Section 5-3 5% GUIDELINE
5/5/2015 Section 5-3 Binomial Probability Distribution BINOMIAL PROBABILITY DISTRTIBUTION Abinomialprobabilitydistribution resultsfroma procedurethatmeetsallthefollowingrequirements: 1. Theprocedurehasafixednumberoftrials. 2. Thetrialsmustbeindependent.(Theoutcome ofanyindividualtrialdoesn’taffectthe probabilitiesintheothertrials.) 3. Eachtrialmusthavealloutcomesclassifiedinto twocategories.Thesecategoriesareusually “success”or“failure”. 4. Theprobabilityofasuccessremainsthesamein alltrials. 5% GUIDELINE TreatingDependentEventsasIndependent: 5%GuidelineforCumbersomeCalculations Whencalculationswithsamplingarevery cumbersomeandthesamplesizeisnotmore than5%ofthesizeofthepopulation,treatthe selectionsasbeingindependent (evenifthey areactuallydependent). 1 5/5/2015 NOTATION FOR BINOMIAL PROBABILITY DISTRIBUTIONS “S”and“F”(successandfailure)denotethe twopossiblecategoriesofalloutcomes;p will denotetheprobabilityof“S”andq willdenote theprobabilityof“F”.Thatis, 1 ( probabilityofsuccess) ( probabilityoffailure) NOTATION (CONCLUDED) denotesthefixednumberoftrials denotesthenumberofsuccessesin trials, so canbeanynumberbetween0and , inclusive. denotestheprobabilityofsuccess inone of the trials. denotestheprobabilityoffailure inone of the trials. denotestheprobabilityofgettingexactly successesamongthe trials. EXAMPLES Identifysuccess,failure,n,p,andq forthequestions below.Wewillanswerthequestionslater. 1. Iftheprobabilityis0.70thatastudentwith veryhighgradeswillgetintolawschool,what istheprobabilitythatthreeoffivestudents withveryhighgradeswillgetintolawschool? 2. Theprobabilityis0.60thatapersonshopping inacertainmarketwillspend$25ormore. Findtheprobabilitythatamongeightpersons shoppingatthismarket,atleast6willspend $25ormore. 2 5/5/2015 IMPORTANT CAUTIONS • Besurethat and bothrefertothesame categorybeingcalledasuccess. • Whensamplingwithoutreplacement,the eventscanbetreatedasiftheywere independentifthesamplesizeisnomore than5%ofthepopulationsize.Thatis, 0.05 .(Thisisthe5%Guideline.) THREE METHODS FOR FINDING BINOMIAL PROBABILITIES 1. UsingtheBinomialProbabilityFormula. 2. UsingTableA‐1locatedinAppendixAon pages583. 3. UsingtheTI‐83/84calculator. METHOD 1: THE BINOMIAL PROBABILITY FORMULA ! ! ! where · · for 0, 1, 2, . . . , numberoftrials numberofsuccessesin trials probabilityofsuccessinanyonetrial probabilityoffailureinanyonetrial ( 1 ) Thisformulacanalsobewrittenas · · 3 5/5/2015 EXAMPLE Iftheprobabilityis0.70thatastudentwith veryhighgradeswillgetintolawschool,what istheprobabilitythatthreeoffivestudents withveryhighgradeswillgetintolawschool? UsetheBinomialProbabilityFormula. METHOD 2: USING TABLE A-1 IN APPENDIX A PartofTableA‐1isshownbelow.Withn =4andp = 0.2inthebinomialdistribution,theprobabilitiesof0, 1,2,3,and4successesare0.410,0.410,0.154,0.026, and0.002respectively. EXAMPLE Theprobabilityis0.60thataperson shoppinginacertainmarketwillspend$25 ormore.Findtheprobabilitythatamong eightpersonsshoppingatthismarket,at least6willspend$25ormore.UseTableA‐ 1. 4 5/5/2015 METHOD 3: USING THE TI-83/84 CALCULATOR 1. Press2nd VARS (togetDISTR) 2. Selectoption0:binompdf( ontheTI‐83.Use A:binompdf( ontheTI‐84. 3. Completetheentryofbinompdf(n,p,x) with specificvaluesforn,p,and . 4. PressENTER, andtheresultwillbethe probabilityofgetting successesin trials;that is, . NOTE:Formoreinformationon“binompdf(”seethe UsingTechnologyboxonpage217. TI-84 WITH NEW OPERATION SYSTEM IfusingtheTI‐84withthenewestoperating system,completethescreenasfollows Entervaluefor here. Entervaluefor here. Entervaluefor here. HighlightPASTEandpressENTERtocopythe commandtothehomescreen.PressENTERagainto executethecommand. EXAMPLE Useyourcalculatortocomputetheprobability of3successesin10trialsiftheprobabilityof successis0.4. 5 5/5/2015 MATHEMATICAL TRANSLATIONS OF ENGLISH PHRASES Phrase MathSymbol “atleast”or“nolessthan” ≥ “morethan”or“greaterthan” > “fewerthan”or“lessthan” < “nomorethan”or“atmost” ≤ “exactly” = CUMULATIVE BINOMIAL PROBABILITIES USING THE TI-83/84 CALCULATOR Thecumulativebinomialprobability isthesumofthe individualprobabilitieslessthanorequalto .Thatis, 0 1 2 ⋯ Tofindthecumulativebinomialprobabilityonthe calculator: 1. Press2nd VARS (togetDIST) 2. SelectoptionA:binomcdf( ontheTI‐83.Use B:binomcdf( ontheTI‐84. 3. Completetheentryofbinomcdf(n,p,x) withspecific valuesfor , ,and . 4. PressENTER,andtheresultwillbetheprobabilityof gettingatmost successesin trials,thatis, . CUMULATIVE BINOMIAL PROBABILITIES USING THE TI-84 WITH NEW OS Press2nd VARS (togetDIST)andselectB:binomcdf(. Entervaluefor here. Entervaluefor here. Entervaluefor here. HighlightPASTEandpressENTERtocopythe commandtothehomescreen.PressENTERagainto executethecommand. 6 5/5/2015 EXAMPLE AccordingtoNielsonMediaResearch,88%of allUnitedStateshouseholdsin2014hadHD television. (a) Inarandomsampleof15households,what istheprobabilitythatexactly10hadHD? (b) Inarandomsampleof15households,what istheprobabilitythatatleast13hadHD? (c) Inarandomsampleof15households,what istheprobabilitythatfewerthan13had HD? 7
© Copyright 2024