Ch.5 Probability
5.1 ProbabilityRules
1 Applytherulesofprobabilities.
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Provideanappropriateresponse.
1) Identifythesamplespaceoftheprobabilityexperiment:tossingacoin
2) Identifythesamplespaceoftheprobabilityexperiment:answeringatrueorfalsequestion
3) Identifythesamplespaceoftheprobabilityexperiment:tossingfourcoinsandrecordingthenumberofheads
4) Identifythesamplespaceoftheprobabilityexperiment:answeringamultiplechoicequestionwithA,B,C,D
andEasthepossibleanswers
5) Identifythesamplespaceoftheprobabilityexperiment:determiningthepuppieʹsgenderforalitterofthree
puppies(UseMformaleandFforfemale.)
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
6) WhichofthefollowingprobabilitiesforthesamplepointsA,B,andCcouldbetrueifA,B,andCaretheonly
samplepointsinanexperiment?
A) P(A)=0,P(B)=1/4
,
P(C)=3
/
4 B) P(A)=1/10
,
P(B)=1/5
,
P(C)=1/4
C) P(A)=–1/4,P(B)=1/2,P(C)=3/4 D) P(A)=1/8
,
P(B)=1/8
,
P(C)=1/8
7) IfA,B,C,andD,aretheonlypossibleoutcomesofanexperiment,findtheprobabilityofDusingthetable
below.
Outcome ABCD
Probability 1/7 1/7 1/7 .
A) 4
/
7 B) 1/7 C) 1/4 D) 3/7
8) Ina1–pondbagofskittlesthepossiblecolorswerered,green,yellow,orange,andpurple.Theprobabilityof
drawingaparticularcolorfromthatbagisgivenbelow.Isthisaprobabilitymodel?AnswerYesorNo.
Color Probability
Red 0.2299
Green 0.1908
Orange 0.2168
Yellow 0.1889
Purple 0.1816
A) Yes B) No
9) Abagcontains25woodenbeads.Thecolorsofthebeadsarered,blue,white,green,black,brown,andgrey.
Theprobabilityofrandomlyselectingabeadofaparticularcolorfromthebagisgivenbelow.Isthisa
probabilitymodel?AnsweryesorNo.
Color Red Blue White Green Black Brown Grey
Probability 0.28 0.24 0.20 0.16 0.12 0.08 0.03
A) No B) Yes
Page145
10) Whichofthefollowingcannotbetheprobabilityofanevent?
A) –59 B) 0 C) 0.001 D) 7
3
11) TheprobabilitythateventAwilloccurisP(A)=Numberofsuccessfuloutcomes
Numberofunsuccessfuloutcomes
A) False B) True
12) TheprobabilitythateventAwilloccurisP(A)=Numberofsuccessfuloutcomes
Totalnumberofallpossibleoutcomes
A) True B) False
13) Intermsofprobability,a(n)___________________isanyprocesswithuncertainresultsthatcanberepeated.
A) Experiment B) Samplespace C) Event D) Outcome
14) A(n)_______________ofaprobabilityexperimentisthecollectionofalloutcomespossible.
A) Samplespace B) Eventset C) Bernoullispace D) Predictionset
15) TrueorFalse:Anoutcomeisanycollectionofeventsfromaprobabilityexperiment.
A) True B) False
16) Anunusualeventisaneventthathasa
A) Lowprobabilityofoccurrence B) Probabilityof1
C) Probabilitywhichexceeds1D)Anegativeprobability
2 Computeandinterpretprobabilitiesusingtheempiricalmethod.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Thetablebelowrepresentsarandomsampleofthenumberofdeathsper100casesforacertainillnessover
time.Ifapersoninfectedwiththisillnessisrandomlyselectedfromallinfectedpeople,findtheprobability
thatthepersonlives3–4yearsafterdiagnosis.Expressyouranswerasasimplifiedfractionandasadecimal.
YearsafterDiagnosis Numberdeaths
1–215
3–435
5–616
7–89
9–10 6
11–12 4
13–14 2
15+13
A) 35
100 ;0.35 B) 1
35 ;0.029 C) 35
65 ;0.538 D) 7
120 ;0.058
Page146
2) Recently,thestockmarkettookbigswingsupanddown.Asurveyof971 adultinvestorsaskedhowoftenthey
trackedtheirportfolio.Thetableshowstheinvestorresponses.Whatistheprobabilitythatanadultinvestor
trackshisorherportfoliodaily?Expressyouranswerasasimplifiedfractionandasadecimalroundedto
threedecimalplaces.
Howfrequently? Response
Daily 236
Weekly 261
Monthly 273
Coupletimesayear 141
Donʹttrack 60
A) 236
971 ;0.243 B) 261
971 ;0.269 C) 273
971 ;0.281 D) 141
971 ;0.145
Thechartbelowshowsthepercentageofpeopleinaquestionnairewhoboughtorleasedthelistedcarmodelsandwere
verysatisfiedwiththeexperience.
ModelA 81%
ModelB 79%
ModelC 73%
ModelD 61%
ModelE 59%
ModelF 57%
3) Withwhichmodelwasthegreatestpercentagesatisfied?Estimatetheempiricalprobabilitythatapersonwith
thismodelisverysatisfiedwiththeexperience.Expresstheanswerasafractionwithadenominatorof100.
A) ModelA;81
100 B) ModelA:0.81
100 C) ModelF;57
100 D) ModelF;0.57
100
4) Theempiricalprobabilitythatapersonwithamodelshownisverysatisfiedwiththeexperienceis81
100 .What
isthemodel?
A) A B) B C) C D) F
Provideanappropriateresponse.
5) TrueorFalse:TheprobabilityofaneventEinanempiricalexperimentmaychangefromexperimentto
experiment.
A) True B) False
3 Computeandinterpretprobabilitiesusingtheclassicalmethod.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Usethespinnerbelowtoanswerthequestion.Assumethatitisequallyprobablethatthepointerwilllandon
anyoneofthefivenumberedspaces.Ifthepointerlandsonaborderline,spinagain.
Findtheprobabilitythatthearrowwilllandon5or4.
A) 2
5B) 5 C) 4
3D) 3
5
Page147
2) Usethespinnerbelowtoanswerthequestion.Assumethatitisequallyprobablethatthepointerwilllandon
anyoneofthefivenumberedspaces.Ifthepointerlandsonaborderline,spinagain.
Findtheprobabilitythatthearrowwilllandonanoddnumber.
A) 3
5B) 2
5C) 1 D) 0
3) Youaredealtonecardfromastandard52–carddeck.Findtheprobabilityofbeingdealtanaceora7.
A) 2
13 B) 4
13 C) 13
2D) 8
4) Adieisrolled.Thesetofequallylikelyoutcomesis{1,2,3,4,5,6}.Findtheprobabilityofgettinga5.
A) 1
6B) 5
6C) 5 D) 0
5) Adieisrolled.Thesetofequallylikelyoutcomesis{1,2,3,4,5,6}.Findtheprobabilityofgettinga10.
A) 0 B) 1 C) 10 D) 10
6
6) Youaredealtonecardfromastandard52–carddeck.Findtheprobabilityofbeingdealtapicturecard.
A) 3
13 B) 1
13 C) 3
26 D) 3
52
7) Afaircoinistossedtwotimesinsuccession.Thesetofequallylikelyoutcomesis{HH,HT,TH,TT}. Findthe
probabilityofgettingthesameoutcomeoneachtoss.
A) 1
2B) 1
4C) 3
4D) 1
8) Asingledieisrolledtwice.Thesetof36equallylikelyoutcomesis{(1,1),(1,2),(1,3),(1,4),(1,5), (1,6), (2,1),
(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),
(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}.Findtheprobabilityofgettingtwonumbers
whosesumisgreaterthan10.
A) 1
12 B) 5
18 C) 1
18 D) 3
9) Asingledieisrolledtwice.Thesetof36equallylikelyoutcomesis{(1,1),(1,2),(1,3),(1,4),(1,5), (1,6), (2,1),
(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),
(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}.Findtheprobabilityofgettingtwonumbers
whosesumislessthan13.
A) 1 B) 0 C) 1
2D) 1
4
Page148
10) Asingledieisrolledtwice.Thesetof36equallylikelyoutcomesis{(1,1),(1,2),(1,3),(1,4),(1,5), (1,6), (2,1),
(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),
(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}.Findtheprobabilityofgettingtwonumbers
whosesumisgreaterthan9andlessthan13.
A) 1
6B) 0 C) 5
36 D) 7
36
11) Thisproblemdealswitheyecolor,aninheritedtrait.Forpurposesofthisproblem,assumethatonlytwoeye
colorsarepossible,brownandblue.WeusebtorepresentablueeyegeneandBabrowneyegene.IfanyB
genesarepresent,thepersonwillhavebrowneyes.Thetableshowsthefourpossibilitiesforthechildrenof
twoBb(brown–eyed)parents,whereeachparenthasoneofeacheyecolorgene.
SecondParent
FirstParent
Bb
BBB Bb
bBb bb
Findtheprobabilitythattheseparentsgivebirthtoachildwhohasblueeyes.
A) 1
4B) 1
2C) 1 D) 0
12) Threefaircoinsaretossedintheairandlandonatable.Theupsideofeachcoinisnoted.Howmany
elementsarethereinthesamplespace?
A) 8 B) 3 C) 6 D) 4
13) Thesamplespacefortossingthreefaircoinsis{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}.Whatisthe
probabilityofexactlytwoheads?
A) 3
8B) 3 C) 1
2D) 5
8
14) InthegameofrouletteintheUnitedStatesawheelhas38slots:18slotsareblack,18slotsarered,and2slots
aregreen.Wewatchedafriendplayroulettefortwohours.Inthattimewenotedthatthewheelwasspun50
timesandthatoutofthose50spinsblackcameup22times.Basedonthisdata,theP(black)=22
50 =0.44.This
isanexampleofwhattypeofprobability?
A) Empirical B) Classical C) Subjective D) Observational
15) InthegameofrouletteintheUnitedStatesawheelhas38slots:18slotsareblack,18slotsarered,and2slots
aregreen.TheP(Red)=18
38 ≈0.47.Thisisanexampleofwhattypeofprobability?
A) Classical B) Empirical C) Simulated D) Subjective
4 KnowConcepts:ProbabilityRules
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
1) (a)Rollapairofdice40times,recordingthesumeachtime.Useyourresultstoapproximatetheprobabilityof
gettingasumof8.
(b)Rollapairofdice100times,recordingthesumeachtime.Useyourresultstoapproximatetheprobability
ofgettingasumof8.
Comparetheresultsof(a)and(b)totheprobabilitythatwouldbeobtainedusingtheclassicalmethod.
Whichanswerwasclosertotheprobabilitythatwouldbeobtainedusingtheclassicalmethod?Isthiswhatyou
wouldexpect?
Page149
2) (a)Simulatetheexperimentofsampling100four–childfamiliestoestimatetheprobabilitythatafour–child
familyhasthreegirls.Assumethattheoutcomesʺhaveagirlʺandʺhaveaboyʺareequallylikely.
(b)Simulatetheexperimentofsampling1000four–childfamiliestoestimatetheprobabilitythatafour–child
familyhasthreegirls.Assumethattheoutcomesʺhaveagirlʺandʺhaveaboyʺareequallylikely.
Theclassicalprobabilitythatafour–childfamilyhasthreegirlsis1
4.
Comparetheresultsof(a)and(b)totheprobabilitythatwouldbeobtainedusingtheclassicalmethod.
Whichanswerwasclosertotheprobabilitythatwouldbeobtainedusingtheclassicalmethod?Isthiswhatyou
wouldexpect?
3) (a)Useagraphingcalculatororstatisticalsoftwaretosimulatedrawingacardfromastandarddeck100times
(withreplacementofthecardaftereachdraw).Useanintegerdistributionwithnumbers1through4anduse
theresultsofthesimulationtoestimatetheprobabilityofgettingaspadewhenacardisdrawnfroma
standarddeck.
(b)Simulatedrawingacardfromastandarddeck400times(withreplacementofthecardaftereachdraw).
Estimatetheprobabilityofgettingaspadewhenacardisdrawnfromastandarddeck.
Comparetheresultsof(a)and(b)totheprobabilitythatwouldbeobtainedusingtheclassicalmethod.
Whichsimulationresultedintheclosestestimatetotheprobabilitythatwouldbeobtainedusingtheclassical
method?Isthiswhatyouwouldexpect?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
4) Classifythestatementasanexampleofclassicalprobability,empiricalprobability,orsubjectiveprobability.
Theprobabilitythatitwillsnowtomorrowis53%.
A) subjectiveprobability B) classicalprobability C) empiricalprobability
5) Classifythestatementasanexampleofclassicalprobability,empiricalprobability,orsubjectiveprobability.It
isknownthattheprobabilityofhittingapotholewhiledrivingonacertainroadis1%.
A) empiricalprobability B) classicalprobability C) subjectiveprobability
6) Classifythestatementasanexampleofclassicalprobability,empiricalprobability,orsubjectiveprobability.
Theprobabilitythatcabfareswillriseduringthewinteris0.05.
A) subjectiveprobability B) classicalprobability C) empiricalprobability
7) Classifythestatementasanexampleofclassicalprobability,empiricalprobability,orsubjectiveprobability.
Inonestatelottery,apersonselectsa4–digitnumber.Theprobabilityofwinningthisstateʹslotteryis 1
10,000 .
A) classicalprobability B) empiricalprobability C) subjectiveprobability
8) Classifythestatementasanexampleofclassicalprobability,empiricalprobability,orsubjectiveprobability.
Theprobabilitythatanewbornkittenisamaleis1
2.
A) classicalprobability B) empiricalprobability C) subjectiveprobability
9) The______________probabilityofanoutcomeisaprobabilitybasedonpersonaljudgment.
A) Subjective B) Classical C) Empirical D) Conditional
10) The______________probabilityofanoutcomeisobtainedbydividingthefrequencyofoccurrenceofanevent
bythenumberoftrialsoftheexperiment.
A) Empirical B) Subjective C) Classical D) Conditional
Page150
11) The______________probabilityofanoutcomeisobtainedbydividingthenumberofwaysaneventcanoccur
bythenumberofpossibleoutcomes.
A) Classical B) Subjective C) Empirical D) Conditional
5.2 TheAdditionRuleandComplements
1 UsetheAdditionRuleforDisjointEvents.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Aprobabilityexperimentisconductedinwhichthesamplespaceoftheexperimentis
S={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. LeteventA={2,3,4,5}andeventB={12,13,14}.Assumethat
eachoutcomeisequallylikely.ListtheoutcomesinAandB.AreAandBmutuallyexclusive?
A) {};yes B) {};no
C) {2
,
3
,
4
,
5
,
12
,
13
,
14};no D) {2
,
3
,
4
,
5
,
12
,
13
,
14};yes
2) TheeventsAandBaremutuallyexclusive.IfP(A)=0.2 andP(B)=0.6
,
whatisP(AorB)?
A) 0.8 B) 0 C) 0.12 D) 0.4
3) Thetableliststhedrinkinghabitsofagroupofcollegestudents.Ifastudentischosenatrandom,findthe
probabilityofgettingsomeonewhoisaregularorheavydrinker.Roundyouranswertothreedecimalplaces.
Sex Non–drinker RegularDrinker HeavyDrinker Total
Man 135 38 5 178
Woman 187 21 6 214
Total 322 59 11 392
A) 0.179 B) 0.629 C) 0.201 D) 0.112
4) Thetableliststhedrinkinghabitsofagroupofcollegestudents.Ifastudentischosenatrandom,findthe
probabilityofgettingsomeonewhoisamanorawoman.Roundyouranswertothreedecimalplaces.
Sex Non–drinker RegularDrinker HeavyDrinker Total
Man 135 58 5 198
Woman 187 21 8 216
Total 322 79 13 414
A) 1 B) 0.930 C) 0.778 D) 0.222
5) Thetableliststhedrinkinghabitsofagroupofcollegestudents.Ifastudentischosenatrandom,findthe
probabilityofgettingsomeonewhoisanon–drinker.Roundyouranswertothreedecimalplaces.
Sex Non–drinker RegularDrinker HeavyDrinker Total
Man 135 66 5 206
Woman 187 21 7 215
Total 322 87 12 421
A) 0.765 B) 0.933 C) 1 D) 0.235
6) ThedistributionofBachelorʹsdegreesconferredbyauniversityislistedinthetable.Assumethatastudent
majorsinonlyonesubject.WhatistheprobabilitythatarandomlyselectedstudentwithaBachelorʹsdegree
majoredinPhysicsorPhilosophy?Roundyouranswertothreedecimalplaces.
Major Frequency
Physics 227
Philosophy 205
Engineering 86
Business 176
Chemistry 222
A) 0.472 B) 0.528 C) 0.248 D) 0.224
Page151
7) ThedistributionofBachelorʹsdegreesconferredbyauniversityislistedinthetable.Assumethatastudent
majorsinonlyonesubject.WhatistheprobabilitythatarandomlyselectedstudentwithaBachelorʹsdegree
majoredinBusiness,ChemistryorEngineering?Roundyouranswertothreedecimalplaces.
Major Frequency
Physics 216
Philosophy 207
Engineering 86
Business 174
Chemistry 223
A) 0.533 B) 0.467 C) 0.287 D) 0.341
8) Acardisdrawnfromastandarddeckof52playingcards.Findtheprobabilitythatthecardisapicturecard.
A) 3
13 B) 1
13 C) 4
13 D) 8
13
9) Iftwoeventshavenooutcomesincommontheyaresaidtobe
A) Disjoint B) Independent C) Conditional D) Atodds
10) TrueorFalse:Mutuallyexclusiveeventsarenotdisjointevents.
A) False B) True
11) Thetablebelowshowstheprobabilitiesgeneratedbyrollingonedie50timesandrecordingthenumberrolled.
AretheeventsA={rollanoddnumber}andB={rollanumberlessthanorequaltotwo}disjoint?
Roll 123456
Probability 0.22 0.10 0.18 0.12 0.18 0.20
A) No B) Yes
12) Inthegameofcraps,twodicearetossedandtheupfacesaretotaled.Istheeventgettingatotalof9andone
ofthediceshowinga6mutuallyexclusive?AnswerYesorNo.
A) No B) Yes
13) Usingastandarddeckof52playingcardsaretheeventsofgettinganaceandgettingajackonthecarddrawn
mutuallyexclusive?AnswerYesorNo.
A) Yes B) No
14) Thebelowtableshowstheprobabilitiesgeneratedbyrollingonedie50timesandnotingtheupface.Whatis
theprobabilityofgettinganoddupface?
Roll 123456
Probability 0.22 0.10 0.18 0.12 0.18 0.20
A) 0.58 B) 0.42 C) 0.50 D) 0.55
15) Inthegameofcrapstwodicearerolledandtheupfacesaretotaled.Ifthepersonrollingthediceonthefirst
rollrollsa7oran11totaltheywin.Iftheyrolla2,3,or12onthefirstrolltheylose.Iftheyrollanyothertotal
thenonsubsequentrollstheymustrollthattotalbeforerollinga7towin.Whatistheprobabilityofwinning
onthefirstroll?
A) 0.22 B) 0.17 C) 0.06 D) 0.50
Page152
2 UsetheGeneralAdditionRule.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Aprobabilityexperimentisconductedinwhichthesamplespaceoftheexperimentis
S={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. LeteventA={7,8,9,10}andeventB={9,10,11,12,13}.
Assumethateachoutcomeisequallylikely.ListtheoutcomesinAandB.AreAandBmutuallyexclusive?
A) {9
,
10};no B) {9
,
10};yes
C) {7
,
8
,
9
,
10
,
11
,
12
,
13};no D) {7
,
8
,
9
,
10
,
11
,
12
,
13};yes
2) Aprobabilityexperimentisconductedinwhichthesamplespaceoftheexperimentis
S={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. LeteventA={7,8,9,10}andeventB={9,10,11,12,13}.
Assumethateachoutcomeisequallylikely.ListtheoutcomesinAorB.FindP(AorB).
A) {7,8,9,10,11,12,13};7
15 B) {9,10};2
15
C) {7,8,9,10,11,11,12,13};3
5D) {7,8,9,10,12,13};2
5
3) TheeventsAandBaremutuallyexclusive.IfP(A)=0.1 andP(B)=0.5
,
whatisP(AandB)?
A) 0 B) 0.05 C) 0.5 D) 0.6
4) GiventhatP(AorB)=1
6,P(A)=1
8,andP(AandB)=1
9,findP(B).Expresstheprobabilityasasimplified
fraction.
A) 11
72 B) 23
432 C) 29
72 D) 13
72
5) Thetableliststhedrinkinghabitsofagroupofcollegestudents.Ifastudentischosenatrandom,findthe
probabilityofgettingsomeonewhoisamanoranon–drinker.Roundyouranswertothreedecimalplaces.
Sex Non–drinker RegularDrinker HeavyDrinker Total
Man 135 47 5 187
Woman 187 21 10 218
Total 322 68 15 405
A) 0.923 B) 0.947 C) 0.941 D) 0.832
6) Thetableliststhedrinkinghabitsofagroupofcollegestudents.Ifastudentischosenatrandom,findthe
probabilityofgettingsomeonewhoisawomanoraheavydrinker.Roundyouranswertothreedecimal
places.
Sex Non–drinker RegularDrinker HeavyDrinker Total
Man 135 61 5 201
Woman 187 21 12 220
Total 322 82 17 421
A) 0.534 B) 0.922 C) 0.805 D) 0.173
7) Acardisdrawnfromastandarddeckof52playingcards.Findtheprobabilitythatthecardisaqueenora
club.Expresstheprobabilityasasimplifiedfraction.
A) 4
13 B) 7
52 C) 2
13 D) 3
13
Page153
8) Onehundredpeoplewereasked,ʺDoyoufavorstrongerlawsonguncontrol?ʺOfthe33thatansweredʺyesʺ
tothequestion,14weremale.Ofthe67thatansweredʺnoʺtothequestion,sixweremale.Ifonepersonis
selectedatrandom,whatistheprobabilitythatthispersonansweredʺyesʺorwasamale?Roundthethe
nearesthundredth.
A) 0.39 B) 0.53 C) 0.67 D) 0.13
9) Thebelowtableshowstheprobabilitiesgeneratedbyrollingonedie50timesandnotingtheupface.Whatis
theprobabilityofgettinganoddupfaceandatwoorless?Roundthethenearesthundredth.
Roll 123456
Probability 0.22 0.10 0.18 0.12 0.18 0.20
A) 0.68 B) 0.90 C) 0.66 D) 0.32
10) Yourolltwodiceandtotaltheupfaces.Whatistheprobabilityofgettingatotalof8ortwoupfacesthatare
thesame?Roundthethenearesthundredth.
A) 0.28 B) 0.31 C) 0.33 D) 0.50
11) Considerthedatainthetableshownwhichrepresentsthemaritalstatusofmalesandfemales18yearsorolder
intheUnitedStatesin2003.DeterminetheprobabilitythatarandomlyselectedU.S.resident18yearsorolder
isdivorcedoramale?Roundtothenearesthundredth.
Males
(inmillions)
Females
(inmillions)
Total
(inmillions)
Nevermarried 28.6 23.3 51.9
Married 62.1 62.8 124.9
Widowed 2.7 11.3 14.0
Divorced 9.0 12.7 21.7
Total(inmillions) 102.4 110.1 212.5
Source:U.S.CensusBureau,CurrentPopulationreports
A) 0.54 B) 0.58 C) 0.50 D) 0.04
12) Ifonecardisdrawnfromastandard52cardplayingdeck,determinetheprobabilityofgettingaten,akingor
adiamond.Roundtothenearesthundredth.
A) 0.37 B) 0.40 C) 0.31 D) 0.29
13) Ifonecardisdrawnfromastandard52cardplayingdeck,determinetheprobabilityofgettingajack,athree,a
cluboradiamond.Roundtothenearesthundredth.
A) 0.58 B) 0.65 C) 0.50 D) 0.15
14) Twodicearerolled.Whatistheprobabilityofhavingbothfacesthesame(doubles)oratotalof4or10?Round
tothenearesthundredth.
A) 0.28 B) 0.33 C) 0.06 D) 0.15
Page154
3 ComputetheprobabilityofaneventusingtheComplementRule.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Aprobabilityexperimentisconductedinwhichthesamplespaceoftheexperimentis
S={2,3,4,5,6,7,8,9,10,11,12}.LeteventA={5,6,7,8,9}.Assumethateachoutcomeisequallylikely.List
theoutcomesinAc.FindP(Ac).
A) {2,3,4,10,11,12};6
11 B) {5,6,7,8,9};5
11
C) {10,11,12};3
11 D) {2,3,4,9,10,11,12};7
11
2) Youaredealtonecardfroma52–carddeck.Findtheprobabilitythatyouarenotdealta4.Expressthe
probabilityasasimplifiedfraction.
A) 12
13 B) 1
13 C) 9
10 D) 1
10
3) Youaredealtonecardfroma52–carddeck.Findtheprobabilitythatyouarenotdealtaspade.Expressthe
probabilityasasimplifiedfraction.
A) 3
4B) 1
4C) 4
13 D) 2
5
4) In5–cardpoker,playedwithastandard52–carddeck,2,598,960differenthandsarepossible.Ifthereare624
differentwaysaʺfour–of–a–kindʺcanbedealt,findtheprobabilityofnotbeingdealtaʺfour–of–a–kindʺ.
Expresstheprobabilityasafraction,butdonotsimplify.
A) 2,598,336
2,598,960 B) 624
2,598,960 C) 625
2,598,960 D) 1248
2,598,960
5) Acertaindiseaseonlyaffectsmen20yearsofageorolder.Thechartshowstheprobabilitythatamanwiththe
diseasefallsinthegivenagegroup.Whatistheprobabilitythatarandomlyselectedmanwiththediseaseis
notbetweentheagesof55and64?
AgeGroup Probability
20–24 0.004
25–34 0.006
35–44 0.14
45–54 0.29
55–64 0.32
65–74 0.17
75+0.07
A) 0.68 B) 0.32 C) 0.29 D) 0.24
Page155
6) Acertaindiseaseonlyaffectsmen20yearsofageorolder.Thechartshowstheprobabilitythatamanwiththe
diseasefallsinthegivenagegroup.Whatistheprobabilitythatarandomlyselectedmanwiththediseaseis
betweentheagesof35and64?
AgeGroup Probability
20–24 0.004
25–34 0.006
35–44 0.14
45–54 0.29
55–64 0.32
65–74 0.17
75+0.07
A) 0.75 B) 0.14 C) 0.32 D) 0.29
7) Theovernightshippingbusinesshasskyrocketedinthelasttenyears.Thesinglegreatestpredictorofa
companyʹssuccesshasbeenproventimeandagaintobecustomerservice.Astudywasconductedtostudythe
customersatisfactionlevelsforoneovernightshippingbusiness.Inadditiontothecustomerʹssatisfactionlevel,
thecustomerswereaskedhowoftentheyusedovernightshipping.Theresultsareshownbelowinthe
followingtable.Whatistheprobabilitythatarespondentdidnothaveahighlevelofsatisfactionwiththe
company?Roundthethenearesthundredth.
FrequencyofUse High
Satisfactionlevel
Medium Low TOTAL
<2permonth 250 140 10 400
2–5permonth 140 55 5 200
>5permonth 70 25 5 100
TOTAL 460 220 20 700
A) 0.34 B) 0.66 C) 0.57 D) 0.43
8) Asampleof220shoppersatalargesuburbanmallwereaskedtwoquestions:(1)Didyouseeatelevisionad
forthesaleatdepartmentstoreXduringthepast2weeks?(2)DidyoushopatdepartmentstoreXduringthe
past2weeks?Theresponsestothequestionsaresummarizedinthetable.Whatistheprobabilitythata
randomlyselectedshopperfromthe220questioneddidnotshopatdepartmentstoreX?Roundthethenearest
thousandth.
ShoppedatXDidNotShopatX
Sawad 100 35
Didnotseead 35 50
A) 0.386 B) 0.159 C) 0.227 D) 0.614
9) Aftercompletinganinventoryofthreewarehouses,agolfclubshaftmanufacturerdescribeditsstockof12,246
shaftswiththepercentagesgiveninthetable.Supposeashaftisselectedatrandomfromthe12,246currently
instock,andthewarehousenumberandtypeofshaftareobserved.Findtheprobabilitythattheshaftwas
producedinawarehouseotherthanwarehouse1.Roundthethenearesthundredth.
TypeofShaft
Regular Stiff ExtraStiff
1 19% 8% 3%
Warehouse2 14% 6% 14%
3 18% 18% 0%
A) 0.70 B) 0.30 C) 0.51 D) 0.83
Page156
10) Thebreakdownofworkersinaparticularstateaccordingtotheirpoliticalaffiliationandtypeofjobheldis
shownhere.Supposeaworkerisselectedatrandomwithinthestateandtheworkerʹspoliticalaffiliationand
typeofjobarenoted.FindtheprobabilitytheworkerisnotanIndependent.Roundthethenearesthundredth.
PoliticalAffiliation
Republican Democrat Independent
Whitecollar 14% 15% 17%
Typeofjob
BlueCollar 18% 16% 20%
A) 0.63 B) 0.37 C) 0.29 D) 0.34
11) Alocalcountryclubhasamembershipof600andoperatesfacilitiesthatincludean18–holechampionshipgolf
courseand12tenniscourts.Beforedecidingwhethertoacceptnewmembers,theclubpresidentwouldliketo
knowhowmanymembersregularlyuseeachfacility.Asurveyofthemembershipindicatesthat61%regularly
usethegolfcourse,45%regularlyusethetenniscourts,and3%useneitherofthesefacilitiesregularly.What
percentageofthe600useatleastoneofthegolfortennisfacilities?
A) 97% B) 3% C) 103% D) 9%
12) Fillintheblank.TheofaneventAistheeventthatAdoesnotoccur.
A) complement B) intersection C) union D) Venndiagram
13) ThefollowingVenndiagramisforthesixsamplepointspossiblewhenrollingafairdie.LetAbetheevent
rollinganevennumberandletBbetheeventrollinganumbergreaterthan1.Whichofthefollowingevents
describestheeventrollinga1?
A) BcB) AcC) B D) A∪B
14) TrueorFalse:P(E)+P(Ec)>1
A) False B) True
15) Thecomplementof4headsinthetossof4coinsis
A) Atleastonetail B) Alltails C) Exactlyonetail D) Threeheads
16) Agamehasthreeoutcomes.Theprobabilityofawinis0.4,theprobabilityoftieis0.5,andtheprobabilityofa
lossis0.1.Whatistheprobabilityofnotwinninginasingleplayofthegame.
A) 0.6 B) 0.5 C) 0.1 D) 0.33
Page157
5.3 IndependenceandtheMultiplicationRule
1 Identifyindependentevents.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Thereare30chocolatesinabox,allidenticallyshaped.Thereare11filledwithnuts,10filledwithcaramel,
and9aresolidchocolate.Yourandomlyselectonepiece,eatit,andthenselectasecondpiece.Isthisan
exampleofindependence?AnswerYesorNo.
A) No B) Yes
2) Numbereddisksareplacedinaboxandonediskisselectedatrandom.Thereare6reddisksnumbered1
through6,and7yellowdisksnumbered7through13.Inanexperimentadiskisselected,thenumberand
colornoted,replaced,andthenaseconddiskisselected.Isthisanexampleofindependence?AnswerYesor
No.
A) Yes B) No
3) Aftercompletinganinventoryofthreewarehouses,agolfclubshaftmanufacturerdescribeditsstockof12,246
shaftswithpercentagesgiveninthetable.Istheeventofselectingashaftindependentofthewarehouse?
AnswerYesorNo.
A) No B) Yes
4) Twoeventsare__________________iftheoccurrenceiftheoccurrenceofeventEinaprobabilityexperiment
doesnotaffecttheprobabilityofeventFinthesameexperiment.
A) independent B) mutuallyexclusive C) dependent D) disjoint
5) Twoeventsare________________iftheoccurrenceofeventEinaprobabilityexperimentchangesthe
probabilityofeventFinthesameexperiment.
A) dependent B) mutuallyexclusive C) independent D) disjoint
6) TrueorFalse:Mutuallyexclusiveeventsarealwaysindependent.
A) False B) True
2 UsetheMultiplicationRuleforindependentevents.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) SupposethateventsEandFareindependent,P(E)=0.8 andP(F)=0.9.WhatistheP(EandF)?
A) 0.72 B) 1.7 C) 0.072 D) 0.98
2) Asingledieisrolledtwice.Findtheprobabilityofgettinga2 thefirsttimeanda2thesecondtime.Expressthe
probabilityasasimplifiedfraction.
A) 1
36 B) 1
6C) 1
12 D) 1
3
3) Youaredealtonecardfroma52carddeck.Thenthecardisreplacedinthedeck,thedeckisshuffled,andyou
drawagain.Findtheprobabilityofgettingapicturecardthefirsttimeandaclubthesecondtime.Expressthe
probabilityasasimplifiedfraction.
A) 3
52 B) 1
13 C) 3
13 D) 1
4
Page158
4) Ifyoutossafaircoin12times,whatistheprobabilityofgettingallheads?Expresstheprobabilityasa
simplifiedfraction.
A) 1
4096 B) 1
2048 C) 1
8192 D) 1
2
5) Ahumangenecarriesacertaindiseasefromthemothertothechildwithaprobabilityrateof39%.Thatis,
thereisa39%chancethatthechildbecomesinfectedwiththedisease.Supposeafemalecarrierofthegenehas
fourchildren.Assumethattheinfectionsofthefourchildrenareindependentofoneanother.Findthe
probabilitythatallfourofthechildrengetthediseasefromtheirmother.Roundtothenearestthousandth.
A) 0.023 B) 0.977 C) 0.138 D) 0.089
6) Amachinehasfourcomponents,A,B,C,andD,setupinsuchamannerthatallfourpartsmustworkforthe
machinetoworkproperly.Assumetheprobabilityofonepartworkingdoesnotdependonthefunctionalityof
anyoftheotherparts.AlsoassumethattheprobabilitiesoftheindividualpartsworkingareP(A)=P(B)=0.99,
P(C)=0.91,andP(D)=0.93.Findtheprobabilitythatthemachineworksproperly.Roundtothenearest
ten–thousandth.
A) 0.8295 B) 0.8378 C) 0.8919 D) 0.1705
7) Supposeabasketballplayerisanexcellentfreethrowshooterandmakes91%ofhisfreethrows(i.e.,hehasa
91%chanceofmakingasinglefreethrow).Assumethatfreethrowshotsareindependentofoneanother.
Supposethisplayergetstoshootthreefreethrows.Findtheprobabilitythathemissesallthreeconsecutivefree
throws.Roundtothenearestten–thousandth.
A) 0.0007 B) 0.2464 C) 0.7536 D) 0.9993
8) Whatistheprobabilitythatinthreeconsecutiverollsoftwofairdice,apersongetsatotalof7,followedbya
totalof11,followedbyatotalof7?Roundtothenearestten –thousandth.
A) 0.0015 B) 0.1667 C) 0.2876 D) 0.0012
9) Abagcontains10white,12blue,13red,7yellow,and8greenwoodedballs.Aballisselectedfromthebag,its
colornoted,thenreplaced.Youthendrawasecondball,noteitscolorandthenreplacetheball.Whatisthe
probabilityofselecting2redballs?Roundtothenearestten–thousandth.
A) 0.0676 B) 0.5200 C) 0.2600 D) 0.0624
10) Abagcontains10white,12blue,13red,7yellow,and8greenwoodedballs.Aballisselectedfromthebag,its
colornoted,thenreplaced.Youthendrawasecondball,noteitscolorandthenreplacetheball.Whatisthe
probabilityofselectingonewhiteballandoneblueball?Roundtothenearestten–thousandth.
A) 0.0480 B) 0.4400 C) 0.2200 D) 0.0088
3 Computeat–leastprobabilities.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Ahumangenecarriesacertaindiseasefromthemothertothechildwithaprobabilityrateof51%.Thatis,
thereisa51%chancethatthechildbecomesinfectedwiththedisease.Supposeafemalecarrierofthegenehas
fivechildren.Assumethattheinfectionsofthefivechildrenareindependentofoneanother.Findthe
probabilitythatatleastoneofthechildrengetthediseasefromtheirmother.Roundtothenearestthousandth.
A) 0.972 B) 0.029 C) 0.147 D) 0.028
Page159
2) Amachinehasfourcomponents,A,B,C,andD,setupinsuchamannerthatallfourpartsmustworkforthe
machinetoworkproperly.Assumetheprobabilityofonepartworkingdoesnotdependonthefunctionalityof
anyoftheotherparts.AlsoassumethattheprobabilitiesoftheindividualpartsworkingareP(A)=P(B)=0.93,
P(C)=0.98,andP(D)=0.96.Findtheprobabilitythatatleastoneofthefourpartswillwork.Roundtosix
decimalplaces.
A) 0.999996 B) 0.813698 C) 0.000004 D) 0.186302
3) Investingisagameofchance.Supposethereisa36%chancethatariskystockinvestmentwillendupinatotal
lossofyourinvestment.Becausetherewardsaresohigh,youdecidetoinvestinfiveindependentriskystocks.
Findtheprobabilitythatatleastoneofyourfiveinvestmentsbecomesatotalloss.Roundtothenearest
ten–thousandthwhennecessary.
A) 0.8926 B) 0.302 C) 0.0604 D) 0.006
4) Findtheprobabilitythatof25randomlyselectedstudents,atleasttwosharethesamebirthday.Roundtothe
nearestthousandth.
A) 0.569 B) 0.068 C) 0.432 D) 0.995
5) Twocompanies,AandB,packageandmarketachemicalsubstanceandclaim0.15ofthetotalweightofthe
substanceissodium.However,acarefulsurveyof4,000packages(halffromeachcompany)indicatesthatthe
proportionvariesaround0.15,withtheresultsshownbelow.FindthepercentageofallchemicalBpackages
thatcontainasodiumtotalweightproportionabove0.150.
ProportionofSodium
<0.100 0.100–0.149 0.150–0.199 >0.200
A 25% 10% 10% 5%
ChemcalBrand
B 5% 5% 10% 30%
A) 80% B) 40% C) 50% D) 55%
6) Numbereddisksareplacedinaboxandonediskisselectedatrandom.Ifthereare6reddisksnumbered1
through6,and4yellowdisksnumbered7through10,findtheprobabilityofselectingayellowdisk,giventhat
thenumberselectedislessthanorequalto3orgreaterthanorequalto8.
A) 1
2B) 3
4C) 3
5D) 3
10
7) Findtheprobabilitythatof25randomlyselectedstudents,notwosharethesamebirthday.
A) 0.431 B) 0.995 C) 0.569 D) 0.068
8) Theprobabilitythataregionpronetohurricaneswillbehitbyahurricaneinanysingleyearis1
10 .Whatisthe
probabilityofahurricaneatleastonceinthenext5years?
A) 0.40951 B) 1
2C) 0.99999 D) 0.00001
9) Investmentinnewissues(thestockofnewlyformedcompanies)canbebothsuicidalandrewarding.Suppose
thatof500newlyformedcompaniesin2010,only18appearedtohaveoutstandingprospects.Supposethat
youhadselectedtwoofthese500companiesbackin2010.Findtheprobabilitythatatleastoneofyour
companieshadoutstandingprospects.
A) 0.0707735 B) 0.3581242 C) 0.9292265 D) 0.0347735
10) Youtossafaircoin5times.Whatistheprobabilityofatleastonehead?Roundtothenearestten –thousandth.
A) 0.9688 B) 0.7500 C) 0.5000 D) 0.0313
Page160
11) YouareplayingrouletteatacasinointheUnitedStates.Thewheelhas18redslots,18blackslots,andtwo
greenslots.In4spinsofthewheelwhatistheprobabilityofatleastonered?Roundtothenearest
ten–thousandth.
A) 0.9048 B) 0.9375 C) 0.0625 D) 0.0953
5.4 ConditionalProbabilityandtheGeneralMultiplicationRule
1 Computeconditionalprobabilities.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheindicatedprobability.Ifnecessary,roundtothreedecimalplaces.
1) SupposethatEandFaretwoeventsandthatP(EandF)=0.48 andP(E)=0.5.WhatisP(F E)?
A) 0.96 B) 1.042 C) 0.24 D) 0.98
2) SupposethatEandFaretwoeventsandthatN(EandF)=230 andN(E)=740.WhatisP(F E)?
A) 0.311 B) 3.217 C) 0.237 D) 0.031
3) SupposethatEandFaretwoeventsandthatP(E)=0.4 andP(F E)=0.8.WhatisP(EandF)?
A) 0.32 B) 1.2 C) 0.5 D) 0.032
Findtheindicatedprobability.Giveyouranswerasasimplifiedfraction.
4) Theovernightshippingbusinesshasskyrocketedinthelasttenyears.Thesinglegreatestpredictorofa
companyʹssuccesshasbeenproventimeandagaintobecustomerservice.Astudywasconductedtostudythe
customersatisfactionlevelsforoneovernightshippingbusiness.Inadditiontothecustomerʹssatisfactionlevel,
thecustomerswereaskedhowoftentheyusedovernightshipping.Theresultsareshownbelowinthe
followingtable.Acustomerischosenatrandom.Giventhatthecustomerusesthecompanymorethanfive
timespermonth,whatistheprobabilitythattheyexpressedlowsatisfactionwiththecompany?
FrequencyofUse High
Satisfactionlevel
Medium Low TOTAL
<2permonth 250 140 10 400
2–5permonth 140 55 5 200
>5permonth 70 25 5 100
TOTAL 460 220 20 700
A) 1
20 B) 1
4C) 1
140 D) 23
140
5) Themanagersofacorporationweresurveyedtodeterminethebackgroundthatleadstoasuccessfulmanager.
Eachmanagerwasratedasbeingeitheragood,fair,orpoormanagerbyhis/herboss.Themanagerʹs
educationalbackgroundwasalsonoted.Thedataappearbelow.Giventhatamanagerisonlyafairmanager,
whatistheprobabilitythatthismanagerhasnocollegebackground?
EducationalBackground
Manager
Rating H.S.Degree SomeCollege CollegeDegree MasterʹsorPh.D. Totals
Good 8 3 22 6 39
Fair 7 11 44 25 87
Poor 2 4 1 27 34
Totals 17 18 67 58 160
A) 7
87 B) 7
17 C) 7
160 D) 97
160
Page161
6) Themanagersofacorporationweresurveyedtodeterminethebackgroundthatleadstoasuccessfulmanager.
Eachmanagerwasratedasbeingeitheragood,fair,orpoormanagerbyhis/herboss.Themanagerʹs
educationalbackgroundwasalsonoted.Thedataappearbelow.Giventhatamanagerisonlyafairmanager,
whatistheprobabilitythatthismanagerhasacollegedegree?
EducationalBackground
Manager
Rating H.S.Degree SomeCollege CollegeDegree MasterʹsorPh.D. Totals
Good 7 2 22 8 39
Fair 8 19 48 12 87
Poor 3 6 4 21 34
Totals 18 27 74 41 160
A) 16
29 B) 3
10 C) 37
80 D) 74
87
7) Themanagersofacorporationweresurveyedtodeterminethebackgroundthatleadstoasuccessfulmanager.
Eachmanagerwasratedasbeingeitheragood,fair,orpoormanagerbyhis/herboss.Themanagerʹs
educationalbackgroundwasalsonoted.Thedataappearbelow.Giventhatamanagerisagoodmanager,
whatistheprobabilitythatthismanagerhassomecollegebackground?
EducationalBackground
Manager
Rating H.S.Degree SomeCollege CollegeDegree MasterʹsorPh.D. Totals
Good 5 1 28 5 39
Fair 8 16 45 18 87
Poor 9 4 6 15 34
Totals 22 21 79 38 160
A) 1
39 B) 28
39 C) 1
160 D) 1
21
8) Astudywasrecentlydonethatemphasizedtheproblemweallfacewithdrinkinganddriving.Fourhundred
accidentsthatoccurredonaSaturdaynightwereanalyzed.Twoitemsnotedwerethenumberofvehicles
involvedandwhetheralcoholplayedaroleintheaccident.Thenumbersareshownbelow.Giventhatan
accidentinvolvedmultiplevehicles,whatistheprobabilitythatitinvolvedalcohol?
NumberofVehiclesInvolved
DidAlcoholPlayaRole? 123ormore Totals
Yes 55 94 21 170
No 25 170 35 230
Totals 80 264 56 400
A) 23
64 B) 23
80 C) 3
8D) 21
400
9) Aresearcheratalargeuniversitywantedtoinvestigateifastudentʹsseatpreferencewasrelatedinanywayto
thegenderofthestudent.Theresearcherdividedthelectureroomintothreesections(1–front,middleofthe
room,2–front,sidesoftheclassroom,and3–backoftheclassroom,bothmiddleandsides)andnotedwherehis
studentssatonaparticulardayoftheclass.Theresearcherʹssummarytableisprovidedbelow.Supposea
personsittinginthefront,middleportionoftheclassisrandomlyselectedtoansweraquestion.Findthe
probabilitythepersonselectedisafemale.
Area(1) Area(2) Area(3) Total
Males 17 5 11 33
Females 10 14 15 39
Total 27 19 26 72
A) 10
27 B) 10
39 C) 9
13 D) 5
36
Page162
10) Themanagerofausedcarlottookinventoryoftheautomobilesonhislotandconstructedthefollowingtable
basedontheageofhiscaranditsmake(foreignordomestic).Acarwasrandomlyselectedfromthelot.Given
thatthecarselectedwasaforeigncar,whatistheprobabilitythatitwasolderthan2years?AgeofCar(in
years)
Make 0–23–56–10 over10 Total
Foreign 42 22 15 21 100
Domestic 41 20 13 26 100
Total 83 42 28 47 200
A) 29
50 B) 21
50 C) 58
117 D) 14
39
11) Themanagerofausedcarlottookinventoryoftheautomobilesonhislotandconstructedthefollowingtable
basedontheageofhiscaranditsmake(foreignordomestic).Acarwasrandomlyselectedfromthelot.Given
thatthecarselectedwasadomesticcar,whatistheprobabilitythatitwasolderthan2years?
AgeofCar(inyears)
Make 0–23–56–10 over10 Total
Foreign 40 30 10 20 100
Domestic 45 27 11 17 100
Total 85 57 21 37 200
A) 11
20 B) 9
17 C) 9
40 D) 17
40
12) Themanagerofausedcarlottookinventoryoftheautomobilesonhislotandconstructedthefollowingtable
basedontheageofhiscaranditsmake(foreignordomestic).
AgeofCar(inyears)
Make 0–23–56–10 over10 Total
Foreign 42 24 13 21 100
Domestic 36 23 11 30 100
Total 78 47 24 51 200
Acarwasrandomlyselectedfromthelot.Giventhatthecarselectedisolderthantwoyearsold,findthe
probabilitythatitisnotaforeigncar.
A) 32
61 B) 29
61 C) 16
25 D) 29
50
Findtheindicatedprobability.Giveyouranswerasadecimalroundedtothenearestthousandth.
13) Afast–foodrestaurantchainwith700outletsintheUnitedStatesdescribesthegeographiclocationofits
restaurantswiththeaccompanyingtableofpercentages.Arestaurantistobechosenatrandomfromthe700to
testmarketanewstyleofchicken.GiventhattherestaurantislocatedintheeasternUnitedStates,whatisthe
probabilityitislocatedinacitywithapopulationofatleast10,000?
Region
NE SE SW NW
<10,000 10% 6% 3% 0%
PopulationofCity10,000–100,000 15% 5% 12% 5%
>100,000 20% 4% 9% 11%
A) 0.733 B) 0.543 C) 0.44 D) 0.267
Page163
14) Aftercompletinganinventoryofthreewarehouses,agolfclubshaftmanufacturerdescribeditsstockof12,246
shaftswiththepercentagesgiveninthetable.Supposeashaftisselectedatrandomfromthe12,246currently
instock,andthewarehousenumberandtypeofshaftareobserved.Giventhattheshaftisproducedin
warehouse2,findtheprobabilityithasanextrastiffshaft.
TypeofShaft
Regular Stiff ExtraStiff
1 19% 8% 17%
Warehouse2 14% 8% 14%
3 2% 18% 0%
A) 0.389 B) 0.452 C) 0.264 D) 0.53
15) Thebreakdownofworkersinaparticularstateaccordingtotheirpoliticalaffiliationandtypeofjobheldis
shownhere.Supposeaworkerisselectedatrandomwithinthestateandtheworkerʹspoliticalaffiliationand
typeofjobarenoted.GiventheworkerisaDemocrat,whatistheprobabilitythattheworkerisinawhite
collarjob.
PoliticalAffiliation
Republican Democrat Independent
Whitecollar 18% 8% 15%
Typeofjob
BlueCollar 12% 10% 37%
A) 0.444 B) 0.195 C) 0.157 D) 0.353
16) Alocalcountryclubhasamembershipof600andoperatesfacilitiesthatincludean18–holechampionshipgolf
courseand12tenniscourts.Beforedecidingwhethertoacceptnewmembers,theclubpresidentwouldliketo
knowhowmanymembersregularlyuseeachfacility.Asurveyofthemembershipindicatesthat68%regularly
usethegolfcourse,47%regularlyusethetenniscourts,and6%useneitherofthesefacilitiesregularly.Given
thatarandomlyselectedmemberusesthetenniscourtsregularly,findtheprobabilitythattheyalsousethe
golfcourseregularly.
A) 0.447 B) 0.309 C) 0.183 D) 0.223
Provideanappropriateresponse.
17) TheconditionalprobabilityofeventG,giventheknowledgethateventHhasoccurred,wouldbewrittenas
.
A) P(G|H) B) P(G) C) P(H|G) D) P(H)
18) Computingtheprobabilityoftheeventʺdrawingasecondredballfromabagofcoloredballsafterhavingkept
theredballthatwasdrawnfirstfromthebagʺisanexampleof
A) conditionalprobability. B) independenceofevents.
C) mutualexclusiveness. D) disjointevents.
19) TrueorFalse:Conditionalprobabilitiesleavethesamplespacethesamewhenconsideringsequentialevents.
A) False B) True
2 ComputeprobabilitiesusingtheGeneralMultiplicationRule.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.Expressyouranswerasasimplifiedfractionunlessotherwisenoted.
1) Thereare36chocolatesinabox,allidenticallyshaped.There6arefilledwithnuts,14withcaramel,and16 are
solidchocolate.Yourandomlyselectonepiece,eatit,andthenselectasecondpiece.Findtheprobabilityof
selecting2solidchocolatesinarow.
A) 4
21 B) 16
81 C) 4
315 D) 5
27
Page164
2) Thereare32chocolatesinabox,allidenticallyshaped.There11 arefilledwithnuts,8withcaramel,and13 are
solidchocolate.Yourandomlyselectonepiece,eatit,andthenselectasecondpiece.Findtheprobabilityof
selecting2nutcandies.
A) 55
496 B) 121
1024 C) 13
992 D) 55
512
3) Thereare36chocolatesinabox,allidenticallyshaped.There10 arefilledwithnuts,12withcaramel,and14
aresolidchocolate.Yourandomlyselectonepiece,eatit,andthenselectasecondpiece.Findtheprobability
ofselectingasolidchocolatecandyfollowedbyanutcandy.
A) 1
9B) 7
72 C) 1
90 D) 35
324
4) Considerapoliticaldiscussiongroupconsistingof6 Democrats,3 Republicans,and7Independents.Suppose
thattwogroupmembersarerandomlyselected,insuccession,toattendapoliticalconvention.Findthe
probabilityofselectinganIndependentandthenaDemocrat.
A) 7
40 B) 21
128 C) 1
40 D) 7
240
5) Considerapoliticaldiscussiongroupconsistingof4 Democrats,6 Republicans,and5Independents.Suppose
thattwogroupmembersarerandomlyselected,insuccession,toattendapoliticalconvention.Findthe
probabilityofselectinganIndependentandthenaRepublican.
A) 1
7B) 2
15 C) 2
105 D) 1
42
6) Anicechestcontains4cansofapplejuice,8 cansofgrapejuice,6 cansoforangejuice,and5cansofpineapple
juice.Supposethatyoureachintothecontainerandrandomlyselectthreecansinsuccession.Findthe
probabilityofselectingnograpejuice.
A) 65
253 B) 1125
3542 C) 2730
12167 D) 8
253
7) Numbereddisksareplacedinaboxandonediskisselectedatrandom.Ifthereare7reddisksnumbered1
through7,and6yellowdisksnumbered8through13,findtheprobabilityofselectingadisknumbered3,
giventhatareddiskisselected.
A) 1
7B) 7
13 C) 1
13 D) 6
13
8) Numbereddisksareplacedinaboxandonediskisselectedatrandom.Ifthereare2reddisksnumbered1
through2,and6yellowdisksnumbered3through8,findtheprobabilityofselectingareddisk,giventhatan
odd–numbereddiskisselected.
A) 1
4B) 3
4C) 1
8D) 3
8
9) AgroupofstudentswereaskediftheycarryanATMcardTheresponsesarelistedinthetable.Ifastudentis
selectedatrandom,findtheprobabilitythatheorsheownsanATMcardgiventhatthestudentisafreshman.
Roundyouranswertothreedecimalplaces.Roundyouranswertothenearestthousandth.
Class
ATMCard
Carrier
NotanATMCard
Carrier Total
Freshman 27 33 60
Sophomore 19 21 40
Total 46 54 100
A) 0.450 B) 0.550 C) 0.587 D) 0.270
Page165
10) Fouremployeesdrivetoworkinthesamecar.Theworkersclaimtheywerelatetoworkbecauseofaflattire.
Theirmanagersasktheworkerstoidentifythetirethatwentflat;frontdriverʹsside,frontpassengerʹsside,rear
driverʹsside,orrearpassengerʹsside.Iftheworkersdidnʹtreallyhaveaflattireandeachrandomlyselectsa
tire,whatistheprobabilitythatallfourworkersselectthesametire?
A) 1
64 B) 1
4C) 1
256 D) 1
8
11) Findtheprobabilitythatof25randomlyselectedhousewives,notwosharethesamebirthday.Roundyour
answertothenearestthousandth.
A) 0.431 B) 0.995 C) 0.569 D) 0.068
12) Afast–foodrestaurantchainwith700outletsintheUnitedStatesdescribesthegeographiclocationofits
restaurantswiththeaccompanyingtableofpercentages.Arestaurantistobechosenatrandomfromthe700to
testmarketanewstyleofchicken.GiventhattherestaurantislocatedintheeasternUnitedStates,whatisthe
probabilityitislocatedinacitywithapopulationofatleast10,000?Roundyouranswertothenearest
thousandth.
Region
NE SE SW NW
<10,000 6% 6% 3% 0%
PopulationofCity10,000–100,000 15% 9% 12% 5%
>100,000 20% 4% 1% 19%
A) 0.8 B) 0.565 C) 0.48 D) 0.2
13) Aftercompletinganinventoryofthreewarehouses,agolfclubshaftmanufacturerdescribeditsstockof12,246
shaftswiththepercentagesgiveninthetable.Supposeashaftisselectedatrandomfromthe12,246currently
instock,andthewarehousenumberandtypeofshaftareobserved.Giventhattheshaftisproducedin
warehouse2,findtheprobabilityithasanextrastiffshaft.Roundyouranswertothenearestthousandth.
TypeofShaft
Regular Stiff ExtraStiff
1 19% 8% 3%
Warehouse2 14% 18% 12%
3 8% 18% 0%
A) 0.273 B) 0.8 C) 0.255 D) 0.47
14) Abagcontains10white,12blue,13red,7yellow,and8greenwoodedballs.Aballisselectedfromthebag
andkept.Youthendrawasecondballandkeepitalso.Whatistheprobabilityofselectingonewhiteballand
oneblueball?Roundyouranswertofourdecimalplaces.
A) 0.0490 B) 0.0480 C) 0.0090 D) 0.0088
15) Abagcontains10white,12blue,13red,7yellow,and8greenwoodedballs.Aballisselectedfromthebag
andkept.Youthendrawasecondballandkeepitalso.Whatistheprobabilityofselectingtwoblueballs?
Roundyouranswertofourdecimalplaces.
A) 0.0539 B) 0.0588 C) 0.0528 D) 0.0576
16) Fivecardsarerandomlyselectedwithoutreplacementfromastandarddeckof52playingcards.Whatisthe
probabilityofgetting5hearts?Roundyouranswertofourdecimalplaces.
A) 0.0005 B) 0.0012 C) 0.0010 D) 0.0004
Page166
17) CampusFestisastudentfestivalwherelocalbusinessescomeoncampustoselltheirgoodstostudentsat
vastlyreducedprices.Aspartofagive–awaypromotion,alocalcellularphonecompanygaveaway300
cellularphonestostudentswhosignedupfortheircallingservice.Unbeknownsttothecompanyisthat20of
thesecellularphoneswerefaultyandwillcauseasmallexplosionwhendialedoutsidethelocalcallingarea.
Supposeyouandyourroommateeachreceivedoneofthegiveawayphones.Findtheprobabilitythatbothof
youreceivedfaultyphones.Roundtofivedecimalplaceswhennecessary.
A) 0.00424 B) 0.00444 C) 0.13333 D) 0.06243
18) Acountywelfareagencyemploys34welfareworkerswhointerviewprospectivefoodstamprecipients.
Periodically,thesupervisorselects,atrandom,theformscompletedbytwoworkerstoauditforillegal
deductions.Unknowntothesupervisor,sixoftheworkershaveregularlybeengivingillegaldeductionsto
applicants.Giventhatthefirstworkerchosenhasnotbeengivingillegaldeductions,whatistheprobability
thatthesecondworkerchosenhasbeengivingillegaldeductions?Roundtothenearestthousandth.
A) 0.182 B) 0.152 C) 0.147 D) 0.176
19) Acountywelfareagencyemploys31welfareworkerswhointerviewprospectivefoodstamprecipients.
Periodically,thesupervisorselects,atrandom,theformscompletedbytwoworkerstoauditforillegal
deductions.Unknowntothesupervisor,nineoftheworkershaveregularlybeengivingillegaldeductionsto
applicants.Whatistheprobabilitybothworkerschosenhavebeengivingillegaldeductions?Roundtothe
nearestthousandth.
A) 0.077 B) 0.084 C) 0.087 D) 0.075
20) TrueorFalse:IfAandBareindependentevents,thenAandBaremutuallyexclusivealso.
A) False B) True
21) TrueorFalse:Twoevents,AandB,areindependentifP(AandB)=P(A)·P(B).
A) True B) False
22) AssumethatP(A)=0.7andP(B)=0.2.IfAandBareindependent,findP(AandB).
A) 0.14 B) 0.76 C) 0.90 D) 1.00
23) IfP(A)=0.45,P(B)=0.25,andP(B|A)=0.45,areAandBindependent?
A) no B) yes C) cannotdetermine
24) IfP(A)=0.72,P(B)=0.11,andAandBareindependent,findP(A|B).
A) 0.72 B) 0.0792 C) 0.83 D) 0.11
25) AssumethatP(E)=0.15andP(F)=0.48.IfEandFareindependent,findP(EandF).
A) 0.072 B) 0.15 C) 0.558 D) 0.630
26) IftwoeventsAandBare__________,thenP(AandB)=P(A)P(B).
A) independent B) simpleevents C) mutuallyexclusive D) complements
27) TrueorFalse:FortwoeventsAandB,supposeP(A)=0.35,P(B)=0.65,andP(B|A)=0.35.ThenAandBare
independent.
A) False B) True
28) TrueorFalse:FortwoeventsAandB,supposeP(A)=0.1,P(B)=0.8,andP(A|B)=0.1.ThenAandBare
independent.
A) True B) False
29) GiventhateventsAandBaremutuallyexclusiveandP(A)=0.5andP(B)=0.7,areAandBindependent?
A) no B) yes C) cannotbedetermined
Page167
30) GiventhateventsCandDareindependent,P(C)=0.3,andP(D)=0.6,areCandDmutuallyexclusive?
A) no B) yes C) cannotbedetermined
31) GiveneventsAandBwithprobabilitiesP(A)=0.5,P(B)=0.4,andP(AandB)=0.2,areAandBindependent?
A) yes B) no C) cannotbedetermined
32) GiveneventsCandDwithprobabilitiesP(C)=0.3,P(D)=0.2,andP(CandD)=0.1,areCandDindependent?
A) no B) yes C) cannotbedetermined
33) GiveneventsAandBwithprobabilitiesP(A)=0.75andP(B)=0.15,areAandBmutuallyexclusive?
A) cannotbedetermined B) no C) yes
5.5 CountingTechniques
1 SolvecountingproblemsusingtheMultiplicationRule.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Evaluatethefactorialexpression.
1) 4!
2!
A) 12 B) 2! C) 4
2D) 4
2) 500!
499!
A) 500 B) 249,500 C) 1 D) 499
Provideanappropriateresponse.
3) Apersoncanorderanewcarwithachoiceof8 possiblecolors,withorwithoutairconditioning,withor
withoutheatedseats,withorwithoutanti–lockbrakes,withorwithoutpowerwindows,andwithorwithouta
CDplayer.Inhowmanydifferentwayscananewcarbeorderedintermsoftheseoptions?
A) 256 B) 128 C) 512 D) 16
4) Youaretakingamultiple–choicetestthathas9 questions.Eachofthequestionshas5choices,withonecorrect
choiceperquestion.Ifyouselectoneoftheseoptionsperquestionandleavenothingblank,inhowmany
wayscanyouanswerthequestions?
A) 1,953,125 B) 45 C) 59,049 D) 14
5) Licenseplatesinaparticularstatedisplay3 lettersfollowedby4 numbers.Howmanydifferentlicenseplates
canbemanufactured?(Repetitionsareallowed.)
A) 175,760,000 B) 12 C) 36 D) 260
6) Howmanydifferentfour–lettersecretcodescanbeformedifthefirstlettermustbeanSoraT?
A) 35,152 B) 456,976 C) 72 D) 421,824
7) Thereare6performerswhoaretopresenttheiractsatavarietyshow.Howmanydifferentwaysarethereto
scheduletheirappearances?
A) 720 B) 6 C) 36 D) 30
8) Thereare4performerswhoaretopresenttheiractsatavarietyshow.Oneoftheminsistsonbeingthefirstact
oftheevening.Ifthisrequestisgranted,howmanydifferentwaysaretheretoscheduletheappearances?
A) 6 B) 24 C) 16 D) 12
Page168
9) Youwanttoarrange12ofyourfavoriteCDʹsalongashelf.Howmanydifferentwayscanyouarrangethe
CDʹsassumingthattheorderoftheCDʹsmakesadifferencetoyou?
A) 479,001,600 B) 39,916,800 C) 144 D) 132
10) TrueorFalse:0!=1!
A) True B) False
11) Howmanydifferentbreakfastscanyouhaveatthelocaldinerifyoucanselect3differenteggdishes
(scrambled,fried,poached),4choicesofmeat(steak,ham,bacon,sausage),5breads(white,wheat,rye,muffin,
bagel),6juices(tomato,grape,apple,orange,mixedberry,vegetablecocktail),and4beverages(water,milk,
coffee,tea)?
A) 1440 B) 5 C) 22 D) 360
12) Amedicalsalespersonistovisitthevariousmembersofthestaffataclinic.Hemustsee8doctors,6
physiciansassistants,12nurses,3medicaltechnologists,and3receptionists.Howmanydifferentwayscan
thesepeoplebevisitedbythesalespersoniftheorderisnotimportant?
A) 5184 B) 32 C) 25,920 D) 6,220,800
2 Solvecountingproblemsusingpermutations.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthevalueofthepermutation.
1) 7P4
A) 840 B) 210 C) 1260 D) 5040
2) 4P0
A) 1 B) 40 C) 0 D) 24
3) 9P9
A) 362,880 B) 1 C) 181,440 D) 2
Provideanappropriateresponse.
4) Achurchhas8bellsinitsbelltower.Beforeeachchurchservice3 bellsarerunginsequence.Nobellisrung
morethanonce.Howmanysequencesarethere?
A) 336 B) 6720 C) 56 D) 13,440
5) Aclubelectsapresident,vice–president,andsecretary–treasurer.Howmanysetsofofficersarepossibleif
thereare15membersandanymembercanbeelectedtoeachposition?Nopersoncanholdmorethanone
office.
A) 2730 B) 1365 C) 910 D) 32,760
6) Inacontestinwhich7contestantsareentered,inhowmanywayscanthe5 distinctprizesbeawarded?
A) 2520 B) 42 C) 21 D) 84
7) Howmanyarrangementscanbemadeusing2 lettersofthewordHYPERBOLASifnoletteristobeusedmore
thanonce?
A) 90 B) 1,814,400 C) 45 D) 3,628,800
8) TheEnvironmentalProtectionAgencymustinspectninefactoriesforcomplaintsofwaterpollution.Inhow
manydifferentwayscanarepresentativevisitfiveofthesetoinvestigatethisweek?
A) 15,120 B) 362,880 C) 5 D) 45
Page169
9) Howmanywayscanfivepeople,A,B,C,D,andE,sitinarowataconcerthallifAandBmustsittogether?
A) 48 B) 120 C) 12 D) 24
10) Howmanywayscanfivepeople,A,B,C,D,andE,sitinarowataconcerthallifCmustsittotherightofbut
notnecessarilynexttoB?
A) 60 B) 24 C) 20 D) 48
11) Howmanywayscanfivepeople,A,B,C,D,andE,sitinarowataconcerthallifDandEwillnotsitnextto
eachother?
A) 72 B) 24 C) 48 D) 60
12) Inhowmanydifferentwayscanaskiclubconsistingof20peopleselectapersonforitsofficers?Thepositions
availablearepresident,vicepresident,treasurer,andsecretary.Nopersoncanholdmorethanonepositions
andtheeachofficeisfilledinorder.
A) 116,280 B) 4845 C) 1440 D) 74
3 Solvecountingproblemsusingcombinations.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthevalueofthecombination.
1) 9C8
A) 9 B) 8 C) 72 D) 362,880
2) 14C8
A) 3003 B) 2,162,160 C) 60,540,480 D) 1440
3) 6C0
A) 1 B) 6 C) 180 D) 360
4) 10C1
A) 10 B) 3,628,800 C) 5 D) 725,760
5) 4C4
A) 1 B) 24 C) 6 D) 0.5
6) 10C3
6C4
A) 8 B) 2 C) 4 D) 40,320
Provideanappropriateresponse.
7) From9namesonaballot,acommitteeof3 willbeelectedtoattendapoliticalnationalconvention.Howmany
differentcommitteesarepossible?
A) 84 B) 504 C) 60,480 D) 252
8) TowinatLOTTOinacertainstate,onemustcorrectlyselect6numbersfromacollectionof52numbers(one
through52.)Theorderinwhichtheselectionsismadedoesnotmatter.Howmanydifferentselectionsare
possible?
A) 20,358,520 B) 18,009,460 C) 312 D) 720
Page170
9) Inhowmanywayscanacommitteeofthreemenandfourwomenbeformedfromagroupof12 menand12
women?
A) 108,900 B) 15,681,600 C) 6,652,800 D) 165
10) Aphysicsexamconsistsof9multiple–choicequestionsand6open–endedproblemsinwhichallworkmustbe
shown.Ifanexamineemustanswer7ofthemultiple–choicequestionsand4oftheopen–endedproblems,in
howmanywayscanthequestionsandproblemsbechosen?
A) 540 B) 1512 C) 261,273,600 D) 65,318,400
11) Aprofessorwantstoarrangehisbooksonashelf.Hehas30booksandonlyspaceontheshelffor20ofthem.
Howmanydifferent20–bookarrangementscanhemakeusingthe30books?Thisisanexampleofaproblem
thatcanbesolvedusingwhichmethod?
A) Permutations B) Combinations
C) Conditionalprobability D) Randomness
12) ProfessorAlleWhetteachesFrenchandhasaclassof24students.Partofhisgradingsystemincludesan
observationofgroupsof3studentsengagedinaconversationinFrench.Thisisanexampleofaproblemthat
canbesolvedusingwhichmethod?
A) Combinations B) Permutations
C) Conditionalprobability D) Randomness
4 Solvecountingproblemsinvolvingpermutationswithnondistinctitems.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Howmanydistinctarrangementscanbeformedfromallthelettersofʺstudentsʺ?
A) 10,080 B) 1680 C) 720 D) 40,320
2) Howmanydistinctarrangementscanbeformedfromallthelettersofʺstatisticsʺ?
A) 50,400 B) 201,600 C) 259,200 D) 72 E) 3,628,800
3) Giventhesetsofdigits{1,3,4,6,8},howmanydifferentnumbersbetween40,000and80,000canbewritten
usingthesedigitsifrepetitionofdigitsisallowed?
A) 1250 B) 3125 C) 120 D) 1875 E) 22
4) HowmanydistinctarrangementsofthelettersinthewordMississippiarepossible?
A) 34,650 B) 39,916,800 C) 1152 D) 7920
5) Howmanydistinctarrangementsofthelettersinthewordfootballarepossible?
A) 10,080 B) 1680 C) 720 D) 40,320
6) Amanhas12coinsthatconsistof3pennies,4nickels,and5quarters.Howmanydistinctarrangementsofthe
coinscanhemakeifhelaystheminarowoneatatime?
A) 27,720 B) 479,001,600 C) 9240 D) 83,160
Page171
5 Computeprobabilitiesinvolvingpermutationsandcombinations.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Amy,Jean,Keith,Tom,Susan,andDavehaveallbeeninvitedtoabirthdayparty.Theyarriverandomlyand
eachpersonarrivesatadifferenttime.Inhowmanywayscantheyarrive?InhowmanywayscanJeanarrive
firstandKeithlast?FindtheprobabilitythatJeanwillarrivefirstandKeithwillarrivelast.
A) 720;24;1
30 B) 720;15;1
48 C) 120;6;1
20 D) 120;10;1
12
2) Sixstudents,A,B,C,D,E,F,aretogivespeechestotheclass.Theorderofspeakingisdeterminedbyrandom
selection.Findtheprobabilitythat(a)Ewillspeakfirst(b)thatCwillspeakfifthandBwillspeaklast(c)that
thestudentswillspeakinthefollowingorder:DECABF(d)thatAorBwillspeakfirst.
A) 1
6;1
24 ;1
720 ;1
3B) 1
6;1
12 ;1
720 ;1
3C) 1
6;1
36 ;1
720 ;1
12 D) 1
6;1
36 ;1
360 ;1
3
3) Agroupconsistsof6menand5women.Four peopleareselectedtoattendaconference.Inhowmanyways
can4peoplebeselectedfromthisgroupof11?Inhowmanywayscan4menbeselectedfromthe6men?Find
theprobabilitythattheselectedgroupwillconsistofallmen.
A) 330;15;1
22 B) 7920;360;1
22 C) 330;15;1
15,840 D) 330;15;1
1,814,400
4) Toplaythelotteryinacertainstate,apersonhastocorrectlyselect5outof45numbers,paying$1foreach
five–numberselection.Ifthefivenumberspickedarethesameastheonesdrawnbythelottery,anenormous
sumofmoneyisbestowed.Whatistheprobabilitythatapersonwithonecombinationoffivenumberswill
win?Whatistheprobabilityofwinningif100differentlotteryticketsarepurchased?
A) 1
1,221,759 ;100
1,221,759 B) 1
146,611,080 ;10
14,661,108
C) 1
8,145,060 ;1
814,506 D) 1
5,864,443,200 ;1
58,644,432
5) Aboxcontains20widgets,4ofwhicharedefective.If4aresoldatrandom,findtheprobabilitythat(a)allare
defective(b)nonearedefective.
A) 1
4845 ;364
969 B) 1
5;4
5C) 1
116,280 ;1
29,070 D) 1
20 ;1
5
6) Acommitteeconsistingof6peopleistobeselectedfromeightparentsandfourteachers.Findtheprobability
ofselectingthreeparentsandthreeteachers.
A) 8
33 B) 2
33 C) 100
231 D) 10
11
7) Ifyouaredealt5cardsfromashuffleddeckof52cards,findtheprobabilitythatall5cardsarepicturecards.
A) 33
108,290 B) 3
13 C) 1
2,598,960 D) 1
216,580
8) Ifyouaredealt5cardsfromashuffleddeckof52cards,findtheprobabilitythatnoneofthe5cardsarepicture
cards.
A) 108,257
108,290 B) 3
13 C) 33
108,290 D) 1
216,580
Page172
9) Ifyouaredealt6cardsfromashuffleddeckof52cards,findtheprobabilityofgetting3jacksand3aces.
A) 2
2,544,815 B) 2
13 C) 1
1,017,926 D) 3
26
10) Aclubelectsapresident,vice–president,andsecretary–treasurer.Howmanysetsofofficersarepossibleif
thereare15membersandanymembercanbeelectedtoeachposition?Nopersoncanholdmorethanone
office.
A) 2730 B) 1365 C) 910 D) 32,760
11) From9namesonaballot,acommitteeof4 willbeelectedtoattendapoliticalnationalconvention.Howmany
differentcommitteesarepossible?
A) 126 B) 3024 C) 15,120 D) 1512
12) TowinatLOTTOinacertainstate,onemustcorrectlyselect6numbersfromacollectionof55numbers(one
through55.)Theorderinwhichtheselectionsismadedoesnotmatter.Howmanydifferentselectionsare
possible?
A) 28,989,675 B) 25,827,165 C) 330 D) 720
13) Inhowmanywayscanacommitteeofthreemenandfourwomenbeformedfromagroupof9 menand9
women?
A) 10584 B) 1,524,096 C) 5040 D) 42
14) Aphysicsexamconsistsof9multiple–choicequestionsand6open–endedproblemsinwhichallworkmustbe
shown.Ifanexamineemustanswer6ofthemultiple–choicequestionsand4oftheopen–endedproblems,in
howmanywayscanthequestionsandproblemsbechosen?
A) 1260 B) 1296 C) 261,273,600 D) 21,772,800
5.6 PuttingItTogether:WhichMethodDoIUse?
1 Determinetheappropriateprobabilityruletouse.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheindicatedprobability.
1) FindP(AorB)giventhatP(A)=0.3
,
P(B)=0.5
,
andAandBaremutuallyexclusive.
A) 0.8 B) 0 C) 0.15 D) 0.2
2) FindP(AandB)giventhatP(A)=0.5
,
P(B)=0.4
,
andAandBareindependent.
A) 0.2 B) 0 C) 0.9 D) 0.1
3) SupposethatthesamplespaceisS={a,b,c,d,e,f,g,h}andthatoutcomesareequallylikely.Findthe
probabilityoftheeventE={b,f,h}.
A) 0.375 B) 0.3 C) 3 D) 0.33
4) SupposethatthesamplespaceisS={a,b,c,d,e,f,g,h,i,j}andthatoutcomesareequallylikely.Findthe
probabilityoftheeventF=ʺavowelʺ.
A) 0.3 B) 0.2 C) 3 D) 0.33
5) IfP(A)=0.3
,
P(B)=0.6
,
andP(AandB)=0.1
,
findP(AorB).
A) 0.8 B) 0.7 C) 1 D) 0.9
6) IfP(A)=0.5
,
P(B)=0.4
,
andP(AorB)=0.7
,
findP(AandB).
A) 0.2 B) 0.1 C) 0.7 D) 0.9
Page173
7) IfP(B)=0.3
,
P(AorB)=0.4
,
andP(AandB)=0.5
,
findP(A).
A) 0.6 B) 0.3 C) 0.5 D) 0.7
8) SupposethatSandTaretwoevents,P(S)=0.59 andP(T S)=0.16.WhatisP(SandT)?
A) 0.0944 B) 0.75 C) 0.6556 D) 0.4956
9) SupposethatMandNaretwoevents,P(M)=0.20
,
P(N)=0.05
,
andP(MandN)=0.02.WhatisP(M N)?
A) 0.400 B) 0.100 C) 0.030 D) 0.230
10) Of1454peoplewhocameintoabloodbanktogiveblood,357 wereineligibletogiveblood.Estimatethe
probabilitythatthenextpersonwhocomesintogivebloodwillbeineligibletogiveblood.
A) 0.246 B) 0.297 C) 0.214 D) 0.165
11) Astudyconductedatacertaincollegeshowsthat54%oftheschoolʹsgraduatesmovetoadifferentstateafter
graduating.Findtheprobabilitythatamong7randomlyselectedgraduates,atleastonemovestoadifferent
stateaftergraduating.
A) 0.996 B) 0.987 C) 0.143 D) 0.540
12) Inonetown,51%ofadultsareemployedinthetourismindustry.Whatistheprobabilitythatfive adults
selectedatrandomfromthetownareallemployedinthetourismindustry?
A) 0.035 B) 0.965 C) 0.028 D) 0.029
13) Theagedistributionofmembersofagymnasticsassociationisshowninthetable.Amemberofthe
associationisselectedatrandom.Findtheprobabilitythatthepersonselectedisbetween26and35inclusive.
Roundyouranswertothreedecimalplaces.
Age(years) Frequency
Under21 402
21–25 417
26–30 201
31–35 50
Over35 24
1094
A) 0.229 B) 0.184 C) 251 D) 0.046
14) Theagedistributionofmembersofagymnasticsassociationisshowninthetable.Amemberofthe
associationisselectedatrandom.Findtheprobabilitythatthepersonselectedisatleast31.Roundyouranswer
tothreedecimalplaces.
Age(years) Frequency
Under21 407
21–25 401
26–30 214
31–35 54
Over35 28
1104
A) 0.074 B) 0.049 C) 82 D) 0.926
15) AtBillʹscommunitycollege,41.7%ofstudentsareCaucasianand2.7%ofstudentsareCaucasianmathmajors.
WhatpercentageofCaucasianstudentsaremathmajors?
A) 6.5% B) 15.4% C) 39.0% D) 44.4%
16) 66%ofstudentsatonecollegedrinkalcoholregularly,and13%ofthosewhodrinkalcoholregularlysuffer
fromdepression.Whatistheprobabilitythatarandomlyselectedstudentdrinksalcoholregularlyandsuffers
fromdepression?
A) 0.0858 B) 0.79 C) 0.7042 D) 0.5742
Page174
17) Atacertaincollege,13%ofstudentsspeakSpanish,7%speakItalian,and3%speakbothlanguages.Astudent
ischosenatrandomfromthecollege.WhatistheprobabilitythatthestudentspeaksSpanishgiventhatheor
shespeaksItalian?
A) 0.429 B) 0.231 C) 0.040 D) 0.170
18) SupposethattheprobabilitythatSuewillpassherstatisticstestis0.62
,
theprobabilitythatshewillpassher
physicstestis0.48,andtheprobabilitythatshewillpassbothtestsis0.290.Whatistheprobabilitythatshe
passesherphysicstestgiventhatshepassedherstatisticstest?
A) 0.468 B) 0.604 C) 0.190 D) 0.810
19) Thetablebelowdescribestheexercisehabitsofagroupofpeoplesufferingfromhighbloodpressure.Ifa
womanisselectedatrandomfromthegroup,findtheprobabilitythatshedoesnotexercise.
No
exercise
Occasional
exercise
Regular
exercise Total
Men 358 66 87 511
Women 352 86 76 514
Total 710 152 163 1025
A) 0.685 B) 0.343 C) 0.496 D) 0.501
20) Thetablebelowdescribestheexercisehabitsofagroupofpeoplesufferingfromhighbloodpressure.Ifoneof
the1046subjectsisrandomlyselected,findtheprobabilitythatthepersonselectedisfemalegiventhatthey
exerciseoccasionally.
No
exercise
Occasional
exercise
Regular
exercise Total
Men 384 85 74 543
Women 345 69 89 503
Total 729 154 163 1046
A) 0.448 B) 0.066 C) 0.137 D) 0.147
21) 390votersareclassifiedbyincomeandpoliticalparty.Theresultsareshowninthetable.Ifapersonis
selectedatrandomfromthesample,findtheprobabilitythatthepersonhashighincome.
Democrat Republican Total
LowIncome 108 69 177
MediumIncome 91 74 165
HighIncome 18 17 35
SuperHighIncome 8513
Total 225 165 390
A) 0.090 B) 0.046 C) 0.080 D) 0.514
22) 390votersareclassifiedbyincomeandpoliticalparty.Theresultsareshowninthetable.Ifapersonis
selectedatrandomfromthesample,findtheprobabilitythatthepersonhasmediumincomeandvotes
Democrat.
Democrat Republican Total
LowIncome 102 77 179
MediumIncome 94 66 160
HighIncome 23 14 37
SuperHighIncome 6814
Total 225 165 390
A) 0.241 B) 0.418 C) 0.588 D) 0.746
Page175
23) 390votersareclassifiedbyincomeandpoliticalparty.Theresultsareshowninthetable.Ifapersonis
selectedatrandomfromthesample,findtheprobabilitythatthepersonhasmediumincomeorvotes
Democrat.
Democrat Republican Total
LowIncome 107 75 182
MediumIncome 92 67 159
HighIncome 19 19 38
SuperHighIncome 7411
Total 225 165 390
A) 0.749 B) 0.409 C) 0.579 D) 0.985
2 Determinetheappropriatecountingtechniquetouse.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Evaluate.
1) 7P3
A) 210 B) 840 C) 1680 D) 5040
2) 8C4
A) 70 B) 1680 C) 840 D) 48
3) 8!
6!
A) 56 B) 2! C) 8
6D) 8
Solvetheproblem.
4) InhowmanywayscanIrischoose4of13 bookstobringonvacation?
A) 715 B) 17,160 C) 24 D) 28,561
5) Inhowmanywayscanaboardofsupervisorschooseapresident,atreasurer,andasecretaryfromits6
members?
A) 120 B) 216 C) 720 D) 20
6) Kareniscompletingafitnesscircuit.Thereare11 fitnessstations.Ateachstationshecanchoosefrom4
differentactivities.Ifshechoosesoneactivityateachfitnessstation,inhowmanywayscanshecompletethe
circuit?
A) 4,194,304 B) 44 C) 14,641 D) 15
7) Acompanymakesskirtsin5differentstyles.Eachstylecomesintwodifferentfabricsand3differentcolors.
Howmanyskirtsareavailablefromthiscompany?
A) 30 B) 15 C) 8 D) 10
8) Licenseplatesaremadeusing2lettersfollowedby2 digits.Howmanyplatescanbemadeifrepetitionof
lettersanddigitsisallowed?
A) 67,600 B) 10,000 C) 456,976 D) 6760
9) Ifacoinistossed9times,howmanyhead–tail sequencesarepossible?
A) 512 B) 18 C) 362,880 D) 81
Page176
10) Apoolofpossiblecandidatesforastudentcouncilconsistsof10 freshmenand8 sophomores.Howmany
differentcouncilsconsistingof5freshmenand7sophomoresarepossible?
A) 2016 B) 18,564 C) 206,841,600 D) 3,628,800
11) HowmanyarrangementarethereofthelettersDISAPPOINT?
A) 907,200 B) 3,628,800 C) 1,814,400 D) 1024
12) Howmanydifferentarrangementsarepossibleusing6 lettersfromthewordPAYMENT?
A) 5040 B) 2520 C) 7 D) 42
13) Thereare12runnersinarace.Inhowmanywayscanthefirst,second,andthirdplacefinishesoccur?(Assume
therearenoties.)
A) 1320 B) 220 C) 1324 D) 218
14) Apoetwillread4ofherpoemsatanawardceremony.Howmanywayscanshechoosethe4poemsfrom9
poemsgiventhatthesequenceisimportant?
A) 3024 B) 15,120 C) 126 D) 30,240
15) Alicenseplateistoconsistof3lettersfollowedby5 digits.Determinethenumberofdifferentlicenseplates
possibleifthefirstlettermustbeanK,L,orMandrepetitionoflettersandnumbersisnotpermitted.
A) 54,432,000 B) 9,072,000 C) 272,160,000 D) 54,522,000
16) Inhowmanywayscan7womenand4 menbeseatedinarowof11 seatsatamovietheaterassumingthatall
thewomenmustsittogetherandallthemenmustsittogether?
A) 241,920 B) 120,960 C) 39,916,800 D) 2048
17) Inhowmanywayscanaclubchooseapresident,atreasurer,asecretary,andthreeothercommitteemembers
(withidenticalduties)fromagroupof16candidates?
A) 960,960 B) 5,765,760 C) 16,777,216 D) 8008
5.7 BayesʹsRule(onCD)
1 Usetheruleoftotalprobability.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) OneoftheconditionsforasamplespaceStobeportionedintonsubsetsisthat
A) Eachofthesubsetsismutuallyexclusive.
B) Eachofthesubsetsisindependentoftheothersubsets.
C) Atleastonesubsetmustbeempty.
D) Eachsubsetisorthogonaltotheothersubsets.
2) OneoftheconditionsforasamplespaceStobeportionedintonsubsetsisthat
A) EachofthesubsetsmustcontainatleastoneelementofS.
B) Thesubsetsareindependentofeachother.
C) Eachsubsetisorthogonaltotheothersubsets.
D) Theelementsinthesubsetsarerandomlyselectedforinclusion.
Page177
Usetheruleoftotalprobabilitytofindtheindicatedprobability.
3) UsethetreediagrambelowtofindP(E).Roundtothenearestthousandthwhennecessary.
P(E|A)=0.3
P(A)=0.25
P(EC|A)=0.7
P(E|B)=0.4
P(B)=0.3
P(EC|B)=0.6
P(E|C)=0.2
P(C)=0.45
P(EC|C)=0.8
A) 0.285 B) 0.075 C) 0.297 D) 0.195
4) SupposethateventsA1andA2formapartitionofthesamplespaceSwithP(A1)=0.65andP(A2)=0.35.IfB
isaneventthatisasubsetofSandP(B A1)=0.11andP(B A2)=0.23,findP(B).Roundtothenearest
ten–thousandthwhennecessary.
A) 0.152 B) 0.0715 C) 0.17 D) 0.0805
5) SupposethateventsA1
,
A2
,
andA3formapartitionofthesamplespaceSwithP(A1)=0.25
,
P(A2)=0.4
,
and
P(A3)=0.35.IfBisaneventthatisasubsetofSandP(B A1)=0.15,P(B A2)=0.15,andP(B A3)=0.21,find
P(B).Roundtothenearestten–thousandthwhennecessary.
A) 0.171 B) 0.0375 C) 0.1683 D) 0.0975
6) Supposethattherearetwobuckets.Bucket1contains3tennisballsand5ping –pongballs.Bucket2contains4
tennisballs,3ping–pongballs,and5baseballs.Anunfaircoinwilldecidefromwhichbucketwedraw.Heads
implieswedrawfromBucket1andtailsimplieswedrawfromBucket2.Theprobabilityofheadsis1
3andthe
probabilityoftailsis2
3.Whatistheprobabilityofdrawingabaseball?Roundtothenearestthousandthwhen
necessary.
A) 0.277 B) 0.723 C) 0.25 D) 0.417
7) Twostoressellacertainproduct.StoreAhas39%ofthesales,5%ofwhichareofdefectiveitems,andstoreB
has61%ofthesales,3%ofwhichareofdefectiveitems.Thedifferenceindefectiveratesisduetodifferent
levelsofpre–salecheckingoftheproduct.Apersonreceivesoneofthisproductasagift.Whatisthe
probabilityitisdefective?Roundtothenearestthousandthwhennecessary.
A) 0.038 B) 0.04 C) 0.019 D) 0.44
8) Ateacherdesignsatestsoastudentwhostudieswillpass90%ofthetime,butastudentwhodoesnotstudy
willpass5%ofthetime.Acertainstudentstudiesfor93%oftheteststaken.Onagiventest,whatisthe
probabilitythatstudentpasses?Roundtothenearestthousandthwhennecessary.
A) 0.841 B) 0.837 C) 0.475 D) 0.035
Page178
9) Inonetown,8%of18–29yearoldsownahouse,asdo27%of30–50yearoldsand50%ofthoseover50.
Accordingtoarecentcensustakeninthetown,28.1%ofadultsinthetownare18–29yearsold,39.0%are
30–50yearsold,and32.9%areover50.Whatistheprobabilitythatarandomlyselectedadultownsahouse?
Roundtothenearestthousandthwhennecessary.
A) 0.292 B) 0.283 C) 0.029 D) 0.187
10) Acompanymanufacturesshoesinthreedifferentfactories.FactoryOmahaProduces25%ofthecompanyʹs
shoes,FactoryChicagoproduces60%,andfactorySeattleproduces15%.Onepercentoftheshoesproducedin
Omahaaremislabeled,0.5%oftheChicagoshoesaremislabeled,and2%oftheSeattleshoesaremislabeled.
Ifyoupurchaseonepairofshoesmanufacturedbythiscompanywhatistheprobabilitythattheshoesare
mislabeled?Roundtothenearestthousandth.
A) 0.009 B) 0.036 C) 0.043 D) 0.020
11) AsurveyconductedinoneU.S.citytogetherwithinformationfromthecensusbureauyieldedthefollowing
table.Thefirsttwocolumnsgiveapercentagedistributionofadultsinthecitybyethnicgroup.Thethird
columngivesthepercentageofpeopleineachethnicgroupwhohavehealthinsurance.Roundtothenearest
thousandth.
EthnicGroup PercentagePercentagewith
ofadults healthinsurance
Caucasian 46.5 73
AfricanAmerican 13.5 49
Hispanic 19.4 56
Asian 13.3 67
Other 7.3 40
Determinetheprobabilitythatarandomlyselectedadulthashealthinsurance.
A) 0.633 B) 0.570 C) 0.063 D) 0.603
2 UseBayesʹsRuletocomputeprobabilities.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Provideanappropriateresponse.
1) Supposethattherearetwobuckets.Bucket1contains3tennisballsand5ping –pongballs.Bucket2contains4
tennisballs,3ping–pongballs,and5baseballs.Anunfaircoinwilldecidefromwhichbucketwedraw.Heads
implieswedrawfromBucket1andtailsimplieswedrawfromBucket2.Theprobabilityofaheadsis1
3and
theprobabilityofatailsis2
3.Giventhataping–pongballwasselectedwhatistheprobabilitythatitcame
fromBucket2?
A) 0.444 B) 0.555 C) 0.625 D) 0.250
2) Acompanymanufacturesshoesinthreedifferentfactories.FactoryOmahaProduces25%ofthecompanyʹs
shoes,FactoryChicagoproduces60%,andfactorySeattleproduces15%.Onepercentoftheshoesproducedin
Omahaaremislabeled,0.5%oftheChicagoshoesaremislabeled,and2%oftheSeattleshoesaremislabeled.
Ifyoupurchaseonepairofshoesmanufacturedbythiscompanyandyoudeterminetheyaremislabeledwhat
istheprobabilitytheyweremadeinOmaha?
A) 0.294 B) 0.353 C) 0.333 D) 0.070
Page179
3) Acompanymanufacturesshoesinthreedifferentfactories.FactoryOmahaProduces25%ofthecompanyʹs
shoes,FactoryChicagoproduces60%,andfactorySeattleproduces15%.Onepercentoftheshoesproducedin
Omahaaremislabeled,0.5%oftheChicagoshoesaremislabeled,and2%oftheSeattleshoesaremislabeled.
Ifyoupurchaseonepairofshoesmanufacturedbythiscompanywhatistheprobabilitythatitwaslabeled
correctly?
A) 0.991 B) 0.035 C) 0.9 D) 0.08
Page180
Ch.5 Probability
AnswerKey
5.1 ProbabilityRules
1 Applytherulesofprobabilities.
2 Computeandinterpretprobabilitiesusingtheempiricalmethod.
3 Computeandinterpretprobabilitiesusingtheclassicalmethod.
4 KnowConcepts:ProbabilityRules
Page181
5.2 TheAdditionRuleandComplements
1 UsetheAdditionRuleforDisjointEvents.
2 UsetheGeneralAdditionRule.
3 ComputetheprobabilityofaneventusingtheComplementRule.
Page182
5.3 IndependenceandtheMultiplicationRule
1 Identifyindependentevents.
2 UsetheMultiplicationRuleforindependentevents.
3 Computeat–leastprobabilities.
5.4 ConditionalProbabilityandtheGeneralMultiplicationRule
1 Computeconditionalprobabilities.
Page183
2 ComputeprobabilitiesusingtheGeneralMultiplicationRule.
5.5 CountingTechniques
1 SolvecountingproblemsusingtheMultiplicationRule.
Page184
2 Solvecountingproblemsusingpermutations.
3 Solvecountingproblemsusingcombinations.
4 Solvecountingproblemsinvolvingpermutationswithnondistinctitems.
5 Computeprobabilitiesinvolvingpermutationsandcombinations.
5.6 PuttingItTogether:WhichMethodDoIUse?
1 Determinetheappropriateprobabilityruletouse.
Page185
2 Determinetheappropriatecountingtechniquetouse.
5.7 BayesʹsRule(onCD)
1 Usetheruleoftotalprobability.
2 UseBayesʹsRuletocomputeprobabilities.
Page186
Page187