Chapter5:Probability
InExercises13–18,assumethatafairdieisrolled.Thesamplespaceis{𝟏,𝟐,𝟑,𝟒,𝟓,𝟔},andall
theoutcomesareequallylikely.
5.1:14.FindP(Evennumber).
FindP(5).
5.1:16.FindP(Greaterthan2).
5.1:18.FindP(Lessthan10).
5.1:20.RoulettewheelsinNevadahave38pockets.Theyarenumbered0,00,and1through
36.Oneachspinofthewheel,aballlandsinapocket,andeachpocketisequallylikely.
a.Ifyouspinaroulettewheel38times,isitcertainthateachnumberwillcomeuponce?
b.Ifyouspinaroulettewheel3800times,isitcertainthateachnumberwillcomeup100
times?
c.Asthewheelisspunmoreandmoretimes,willtheproportionoftimesthateachnumber
comesupapproach1∕38?
5.1:22.Isthefollowingaprobabilitymodelforthisexperiment?Whyorwhynot?
5.1:24Isthefollowingaprobabilitymodelforthisexperiment?Whyorwhynot?
5.1:30Acoinflip:Afaircoinistossedthreetimes.Theoutcomesofthethreetossesare
recorded.
a.Listalleightoutcomesinthesamplespace.
b.Assumingtheoutcomestobeequallylikely,findtheprobabilitythatallthreetossesare
‘‘Heads.’’
c.Assumingtheoutcomestobeequallylikely,findtheprobabilitythatthetossesareallthe
same.
d.Assumingtheoutcomestobeequallylikely,findtheprobabilitythatexactlyoneofthethree
tossesis‘‘Heads.’’
5.1:32EmpiricalMethod:Adieisrolled600times.On85ofthoserolls,thediecomesup6.
UsetheEmpiricalMethodtoapproximatetheprobabilitythatthediecomesup6.
5.1:40Howmanykids?TheGeneralSocialSurveyasked46,349womenhowmanychildren
theyhad.Theresultswereasfollows.
a.Whatistheprobabilitythatasampledwomanhastwochildren?
b.Whatistheprobabilitythatasampledwomanhasfewerthanthreechildren?
c.AssumethisisasimplerandomsampleofU.S.women.UsetheEmpiricalMethodtoestimate
theprobabilitythataU.S.womanhasmorethanfivechildren.
d.Usingacutoffof0.05,isitunusualforawomantohavenochildren?
5.2:14.IfP(A)=0.45,P(B)=0.7,andP(AandB)=0.65,findP(AorB).
5.2:16.IfP(A)=0.7,P(B)=0.1,andAandBaremutuallyexclusive,findP(AorB).
5.2:20.IfP(B)=0.6,findP(Bc).
5.2:24.IfP(A)=P(Ac),findP(A).
InExercises25–30,determinewhethereventsAandBaremutuallyexclusive.
5.2:26.A:JaydenhasamathclassonTuesdaysat2:00; B:JaydenhasanEnglishclasson
Tuesdaysat2:00.
5.2:28.Asampleof75booksisselectedfromalibrary.A:Atleast10oftheauthorsarefemale;
B:Atleast10ofthebooksarefiction.
5.2:30.Areddieandabluediearerolled.A:Thereddiecomesup1;
B:Thetotalis9.
5.2:32.Asampleof700cellphonebatterieswasselected.Findthecomplementsoftheevents.
a.Exactly24ofthebatteriesweredefective.
b.Atleast24ofthebatteriesweredefective.
c.Morethan24ofthebatteriesweredefective.
d.Fewerthan24ofthebatteriesweredefective.
5.2:34.Dice:Twofairdicearerolled.Thefirstdieisredandthesecondisblue.Anexperiment
consistsofobservingthenumbersthatcomeuponthedice.
a.Thereare36outcomesinthesamplespace.Theyareorderedpairsoftheform(Reddie,Blue
die).Listthe36outcomes.
b.LetAbetheeventthatthesamenumbercomesuponbothdice.ListtheoutcomesinA.
c.LetBbetheeventthatthereddiecomesup6.ListtheoutcomesinB.
d.LetCbetheeventthatonediecomesup6andtheothercomesup1.ListtheoutcomesinC.
e.AreeventsAandBmutuallyexclusive?Explain.
f.AreeventsAandCmutuallyexclusive?Explain.
5.3:16.LetAandBbeeventswithP(A)=0.6,P(B)=0.4,andP(B|A)=0.4.FindP(AandB).
5.3:18.LetAandBbeeventswithP(A)=0.5andP(B)=0.7.AssumethatAandBare
independent.FindP(AandB).
5.3:20.LetAandBbeeventswithP(A)=0.3,P(B)=0.5,andP(B|A)=0.7.FindP(AandB).
5.3:24.Afaircoinistossedfourtimes.Whatistheprobabilitythatthesequenceoftossesis
HTHT?
5.3:26.Afairdieisrolledthreetimes.Whatistheprobabilitythatallthreerollsare6?
5.3:34.LetAandBbeeventswithP(A)=0.5,P(B)=0.3,andP(AorB)=0.8.
a.ComputeP(AandB).
b.AreAandBmutuallyexclusive?Explain.
c.AreAandBindependent?Explain
5.3:38.Shuffle:Charleshassixsongsonaplaylist.Eachsongisbyadifferentartist.Theartists
areUsher,Ke$ha,LadyGaga,Eminem,theBlackEyedPeas,andLudacris.Heprogramshis
playertoplaythesongsinarandomorder,withoutrepetition.Whatistheprobabilitythatthe
firstsongisbyLadyGagaandthesecondsongisbyEminem?
5.3:40.U.S.senators:Thefollowingtabledisplaysthe100senatorsofthe113thU.S.Congress
onJanuary3,2013,viewedbypoliticalpartyaffiliationandgender.
Asenatorisselectedatrandomfromthisgroup.
a.Whatistheprobabilitythatthesenatorisawoman?
b.WhatistheprobabilitythatthesenatorisaRepublican?
c.WhatistheprobabilitythatthesenatorisaRepublicanandawoman?
d.Giventhatthesenatorisawoman,whatistheprobabilitythatsheisaRepublican?
e.GiventhatthesenatorisaRepublican,whatistheprobabilitythatthesenatorisawoman?
InExercises11–16,evaluatethefactorial.
5.4:12.5! 5.4:13.0! 5.4:14.12! 5.4:15.1! InExercises17–22,evaluatethepermutation.
5.4:18.8P1
5.4:20.5P4
5.4:22.45P5
InExercises23–28,evaluatethecombination.
5.4:24.7C1
5.4:26.10C9
5.4:28.50C50
5.4:30.Books:Josephinehassixchemistrybooks,threehistorybooks,andeightstatistics
books.Shewantstochooseonebookofeachtypetostudy.Inhowmanywayscanshechoose
thethreebooks?
5.434.Committee:TheStudentCouncilatacertainschoolhastenmembers.Fourmembers
willformanexecutivecommitteeconsistingofapresident,avicepresident,asecretary,anda
treasurer.
a.Inhowmanywayscanthesefourpositionsbefilled?
b.Inhowmanywayscanfourpeoplebechosenfortheexecutivecommitteeifitdoesnot
matterwhogetswhichposition?
c.FourofthepeopleonStudentCouncilareZachary,Yolanda,Xavier,andWalter.Whatisthe
probabilitythatZacharyispresident,Yolandaisvicepresident,Xavierissecretary,andWalteris
treasurer?
d.WhatistheprobabilitythatZachary,Yolanda,Xavier,andWalterarethefourcommittee
members?