141
3.9Practice1:ContingencyTables
9
3.9.1StudentLearningOutcomes
 Thestudentwillconstructandinterpretcontingencytables.
3.9.2Given
AnarticleintheNewEnglandJournalofMedicine ,reportedaboutastudyofsmokersinCaliforniaand
Hawaii.Inonepartofthereport,theself-reportedethnicityandsmokinglevelsperdayweregiven.Ofthe
peoplesmokingatmost10cigarettesperday,therewere9886AfricanAmericans,2745NativeHawaiians,
12,831Latinos,8378JapaneseAmericans,and7650Whites. Ofthepeoplesmoking11-20cigarettesper
day,therewere6514AfricanAmericans,3062NativeHawaiians,4932Latinos,10,680JapaneseAmericans,
and9877Whites.Ofthepeoplesmoking21-30cigarettesperday,therewere1671AfricanAmericans,1419
NativeHawaiians,1406Latinos,4715JapaneseAmericans,and6062Whites.Ofthepeoplesmokingatleast
31cigarettesperday,therewere759AfricanAmericans,788NativeHawaiians,800Latinos,2305Japanese
Americans,and3970Whites.((Source:http://www.nejm.org/doi/full/10.1056/NEJMoa033250))
3.9.3CompletetheTable
Completethetablebelowusingthedataprovided.
SmokingLevelsbyEthnicity
Smoking
Level
African
American
Native
Hawaiian
Latino
Japanese
Americans
White
TOTALS
1-10
11-20
21-30
31+
TOTALS
Table3.4
3.9.4AnalyzetheData
Supposethatonepersonfromthestudyisrandomlyselected.
Exercise3.9.1
(Solutiononp.161.)
Findtheprobabilitythatpersonsmoked11-20cigarettesperday.
Exercise3.9.2
(Solutiononp.161.)
FindtheprobabilitythatpersonwasLatino.
9
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142
CHAPTER3. PROBABILITYTOPICS
3.9.5DiscussionQuestions
Exercise3.9.3
(Solutiononp.161.)
Inwords,explainwhatitmeanstopickonepersonfromthestudyandthatpersonis“Japanese
AmericanANDsmokes21-30cigarettesperday.”Also,findtheprobability.
Exercise3.9.4
(Solutiononp.161.)
Inwords,explainwhatitmeanstopickonepersonfromthestudyandthatpersonis“Japanese
AmericanORsmokes21-30cigarettesperday.”Also,findtheprobability.
Exercise3.9.5
(Solutiononp.161.)
Inwords,explainwhatitmeanstopickonepersonfromthestudyandthatpersonis“Japanese
AmericanGIVENthatpersonsmokes21-30cigarettesperday.”Also,findtheprobability.
Exercise3.9.6
Provethatsmokinglevel/dayandethnicityaredependentevents.
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143
3.10Practice2:CalculatingProbabilities
10
3.10.1StudentLearningOutcomes
 Studentswilldefinebasicprobabilityterms.
 Studentswillcalculateprobabilities.
 Studentswilldeterminewhethertwoeventsaremutuallyexclusiveorwhethertwoeventsareinde-
pendent.
NOTE
:Useprobabilityrulestosolvetheproblemsbelow.Showyourwork.
3.10.2Given
48%ofallCaliforniansregisteredvoterspreferlife inprisonwithout parole overthe deathpenaltyfor
apersonconvicted offirst degree murder. . AmongLatino o Californiaregisteredvoters, 55% preferlife
inprisonwithoutparoleoverthedeathpenaltyforapersonconvictedoffirstdegree murder. . (Source:
http://field.com/fieldpollonline/subscribers/Rls2393.pdf).
37.6%ofallCaliforniansareLatino(Source:U.S.CensusBureau).
Inthisproblem,let:
 Californians (registeredvoterspreferringlifeinprisonwithoutparoleoverthedeathpenalty forapersonconvictedof f firstdegreemurder
 LatinoCalifornians
SupposethatoneCalifornianisrandomlyselected.
3.10.3AnalyzetheData
Exercise3.10.1
(Solutiononp.161.)
P(C)=
Exercise3.10.2
(Solutiononp.161.)
P(L)=
Exercise3.10.3
(Solutiononp.161.)
P(CjL)=
Exercise3.10.4
Inwords,whatis"CjL"?
Exercise3.10.5
(Solutiononp.161.)
P(LANDC)=
Exercise3.10.6
Inwords,whatis“LandC”?
Exercise3.10.7
(Solutiononp.162.)
AreLandCindependentevents?Showwhyorwhynot.
Exercise3.10.8
(Solutiononp.162.)
P(LORC)=
Exercise3.10.9
Inwords,whatis“LorC”?
Exercise3.10.10
(Solutiononp.162.)
AreLandCmutuallyexclusiveevents?Showwhyorwhynot.
10
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144
CHAPTER3. PROBABILITYTOPICS
3.11Homework
11
Exercise3.11.1
(Solutiononp.162.)
Supposethatyouhave8cards. 5aregreenand3areyellow. . The5greencardsarenumbered
1,2,3,4,and5. The3yellowcardsarenumbered1,2,and3. Thecardsarewellshuffled. You
randomlydrawonecard.
 G=carddrawnisgreen
 E=carddrawniseven-numbered
a. Listthesamplespace.
b. P(G) =
c. P(GjE) =
d. P(GANDE) =
e. P(GORE) =
f. AreGandEmutuallyexclusive?Justifyyouranswernumerically.
Exercise3.11.2
Refertothepreviousproblem. Supposethatthistimeyourandomlydrawtwocards,oneata
time,andwithreplacement.
 G
1
=firstcardisgreen
 G
2
=secondcardisgreen
a. Drawatreediagramofthesituation.
b. P(G
1
ANDG
2
)
=
c. P(atleastonegreen)=
d. P(G
2
jG
1
)=
e. AreG
2
andG
1
independentevents?Explainwhyorwhynot.
Exercise3.11.3
(Solutiononp.162.)
Refertothepreviousproblems. Supposethatthistimeyourandomlydrawtwocards,oneata
time,andwithoutreplacement.
 G
1
=firstcardisgreen
 G
2
=secondcardisgreen
a. Drawatreediagramofthesituation.
b. P(G
1
ANDG
2
) =
c. P(atleastonegreen)=
d. P(G
2
jG
1
) =
e. AreG
2
andG
1
independentevents?Explainwhyorwhynot.
Exercise3.11.4
Rolltwofairdice.Eachdiehas6faces.
a. Listthesamplespace.
b. LetAbetheeventthateithera3or4isrolledfirst,followedbyanevennumber.FindP(A).
c. LetBbetheeventthatthesumofthetworollsisatmost7.FindP(B).
11
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145
d. Inwords,explainwhat“P(AjB)”represents.FindP(AjB).
e. Are Aand Bmutuallyexclusiveevents? Explainyour r answer in1-3completesentences,
includingnumericaljustification.
f. AreAandBindependentevents?Explainyouranswerin1-3completesentences,including
numericaljustification.
Exercise3.11.5
(Solutiononp.162.)
Aspecialdeckofcardshas10cards. Fouraregreen,threeareblue,andthreearered. . Whena
cardispicked,thecolorofitisrecorded. Anexperimentconsistsoffirstpickingacardandthen
tossingacoin.
a. Listthesamplespace.
b. LetAbetheeventthatabluecardispickedfirst,followedbylandingaheadonthecointoss.
FindP(A).
c. LetBbetheeventthataredorgreenispicked,followedbylandingaheadonthecointoss.Are
theeventsAandBmutuallyexclusive? Explainyouranswerin1-3completesentences,
includingnumericaljustification.
d. LetCbetheeventthataredorblueispicked,followedbylandingaheadonthecointoss.Are
theeventsAandCmutuallyexclusive? Explainyouranswerin1-3completesentences,
includingnumericaljustification.
Exercise3.11.6
Anexperimentconsistsoffirstrollingadieandthentossingacoin:
a. Listthesamplespace.
b. LetAbetheeventthateithera3or4isrolledfirst,followedbylandingaheadonthecointoss.
FindP(A).
c. LetBbetheeventthatanumberlessthan2isrolled,followedbylandingaheadonthecoin
toss. AretheeventsAandBmutuallyexclusive? Explainyouranswerin1-3complete
sentences,includingnumericaljustification.
Exercise3.11.7
(Solutiononp.162.)
Anexperimentconsistsoftossinganickel,adimeandaquarter. Ofinterestisthesidethecoin
landson.
a. Listthesamplespace.
b. LetAbetheeventthatthereareatleasttwotails.FindP(A).
c. Let Bbe theevent thatthefirst andsecondtosseslandonheads. . Are e theeventsAandB
mutuallyexclusive?Explainyouranswerin1-3completesentences,includingjustification.
Exercise3.11.8
Considerthefollowingscenario:
 LetP(C)=0.4
 LetP(D)=0.5
 LetP(C|D)=0.6
a. FindP(CANDD).
b. AreCandDmutuallyexclusive?Whyorwhynot?
c. AreCandDindependentevents?Whyorwhynot?
d. FindP(CORD).
e. FindP(D|C).
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146
CHAPTER3. PROBABILITYTOPICS
Exercise3.11.9
(Solutiononp.162.)
EandFmutuallyexclusiveevents.P(E)=0.4;P(F)=0.5.FindP(EjF).
Exercise3.11.10
JandKareindependentevents.P(J|K) 0.3.FindP(J).
Exercise3.11.11
(Solutiononp.162.)
UandVaremutuallyexclusiveevents.P(U)=0.26;P(V)=0.37.Find:
a. P(UANDV)=
b. P(U|V) =
c. P(UORV)=
Exercise3.11.12
QandRareindependentevents.P(Q) = 0.4;P(QANDR) = 0.1.FindP(R).
Exercise3.11.13
(Solutiononp.162.)
YandZareindependentevents.
a. RewritethebasicAdditionRuleP(YORZ) P(Y) + P(ZP(YANDZ)usingthe
informationthatYandZareindependentevents.
b. UsetherewrittenruletofindP(Z)ifP(YORZ) = 0.71andP(Y) = 0.42.
Exercise3.11.14
GandHaremutuallyexclusiveevents.P(G)=0.5;P(H)=0.3
a. ExplainwhythefollowingstatementMUSTbefalse:P(HjG)=0.4.
b. Find:P(HORG).
c. AreGandHindependentordependentevents?Explaininacompletesentence.
Exercise3.11.15
(Solutiononp.162.)
ThefollowingarerealdatafromSantaClaraCounty,CA.Asofacertaintime,therehadbeen
atotalof3059documentedcasesofAIDSinthecounty. Theyweregroupedintothefollowing
categories(Source:SantaClaraCountyPublicH.D.):
Homosexual/Bisexual
IVDrugUser*
HeterosexualContact
Other
Totals
Female
0
70
136
49
____
Male
2146
463
60
135
____
Totals
____
____
____
____
____
Table3.5
:*includeshomosexual/bisexualIVdrugusers
SupposeoneofthepersonswithAIDSinSantaClaraCountyisrandomlyselected.Computethe
following:
a. P(personisfemale)=
b. P(personhasariskfactorHeterosexualContact)=
c. P(personisfemaleORhasariskfactorofIVDrugUser)=
d. P(personisfemaleANDhasariskfactorofHomosexual/Bisexual)=
e. P(personismaleANDhasariskfactorofIVDrugUser)=
f. P(femaleGIVENpersongotthediseasefromheterosexualcontact)=
g. ConstructaVennDiagram.Makeonegroupfemalesandtheothergroupheterosexualcontact.
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147
Exercise3.11.16
Solvethesequestionsusingprobabilityrules. DoNOTusethecontingencytableabove. . 3059
casesofAIDShadbeenreportedinSantaClaraCounty,CA,throughacertaindate. Thosecases
willbeourpopulation. Ofthosecases,6.4%obtainedthediseasethroughheterosexualcontact
and7.4%arefemale.Outofthefemaleswiththedisease,53.3%gotthediseasefromheterosexual
contact.
a. P(personisfemale)=
b. P(personobtainedthediseasethroughheterosexualcontact)=
c. P(femaleGIVENpersongotthediseasefromheterosexualcontact)=
d. ConstructaVennDiagram.Makeonegroupfemalesandtheothergroupheterosexualcontact.
Fillinallvaluesasprobabilities.
Exercise3.11.17
(Solutiononp.163.)
Thefollowingtableidentifiesagroupofchildrenbyoneoffourhaircolors,andbytypeofhair.
HairType
Brown
Blond
Black
Red
Totals
Wavy
20
15
3
43
Straight
80
15
12
Totals
20
215
Table3.6
a. Completethetableabove.
b. Whatistheprobabilitythatarandomlyselectedchildwillhavewavyhair?
c. Whatistheprobabilitythatarandomlyselectedchildwillhaveeitherbrownorblondhair?
d. Whatistheprobabilitythatarandomlyselectedchildwillhavewavybrownhair?
e. Whatistheprobabilitythat arandomlyselectedchildwillhavered hair, giventhathehas
straighthair?
f. IfBistheeventofachildhavingbrownhair,findtheprobabilityofthecomplementofB.
g. Inwords,whatdoesthecomplementofBrepresent?
Exercise3.11.18
Apreviousyear,theweightsofthemembersoftheSanFrancisco49ersandtheDallasCowboys
werepublishedintheSanJoseMercuryNews. Thefactualdataarecompiledintothefollowing
table.
Shirt#
210
211-250
251-290
290
1-33
21
5
0
0
34-66
6
18
7
4
66-99
6
12
22
5
Table3.7
Forthefollowing,supposethatyourandomlyselectoneplayerfromthe49ersorCowboys.
a. Findtheprobabilitythathisshirtnumberisfrom1to33.
b. Findtheprobabilitythatheweighsatmost210pounds.
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148
CHAPTER3. PROBABILITYTOPICS
c. Findtheprobabilitythathisshirtnumberisfrom1to33ANDheweighsatmost210pounds.
d. Findtheprobabilitythathisshirtnumberisfrom1to33ORheweighsatmost210pounds.
e. Findtheprobabilitythathisshirtnumberisfrom1to33GIVENthatheweighsatmost210
pounds.
f. Ifhavingashirtnumberfrom1to33andweighingatmost210poundswere independent
events,thenwhatshouldbetrueaboutP(Shirt#1-33|210pounds)?
Exercise3.11.19
(Solutiononp.163.)
Approximately 281,000,000 0 people over r age 5 5 live in n the e United d States.
Of
these people, , 55,000,000 0 speak a a language other than n English at t home.
Of
those who speak k another r language at home, , 62.3%
speak Spanish.
(Source:
http://www.census.gov/hhes/socdemo/language/data/acs/ACS-12.pdf)
Let:E=speakEnglishathome;E’=speakanotherlanguageathome;S=speakSpanish;
Finisheachprobabilitystatementbymatchingthecorrectanswer.
ProbabilityStatements
Answers
a.P(E’)=
i.0.8043
b.P(E)=
ii.0.623
c.P(SandE’)=
iii.0.1957
d.P(S|E’)=
iv.0.1219
Table3.8
Exercise3.11.20
Theprobabilitythatamaledevelopssomeformofcancerinhislifetimeis0.4567(Source:Ameri-
canCancerSociety).Theprobabilitythatamalehasatleastonefalsepositivetestresult(meaning
thetestcomesbackforcancerwhenthemandoesnothaveit)is0.51(Source:USAToday).Someof
thequestionsbelowdonothaveenoughinformationforyoutoanswerthem.Write“notenough
information”forthoseanswers.
Let:C=amandevelopscancerinhislifetime;P=manhasatleastonefalsepositive
a. Constructatreediagramofthesituation.
b. P(C)=
c. P(PjC)=
d. P(PjC)=
e. Ifatestcomesuppositive,baseduponnumericalvalues,canyouassumethatmanhascancer?
Justifynumericallyandexplainwhyorwhynot.
Exercise3.11.21
(Solutiononp.163.)
In1994,theU.S.governmentheldalotterytoissue55,000GreenCards(permitsfornon-citizens
toworklegallyintheU.S.).RenateDeutsch,fromGermany,wasoneofapproximately6.5million
peoplewhoenteredthislottery.LetG=wonGreenCard.
a. WhatwasRenate’schanceofwinningaGreenCard?Writeyouranswerasaprobabilitystate-
ment.
b. Inthesummerof1994,Renatereceivedaletterstatingshewasoneof110,000finalistschosen.
Oncethefinalistswerechosen,assumingthateachfinalisthadanequalchancetowin,what
wasRenate’schanceofwinningaGreenCard?LetF=wasafinalist. Writeyouransweras
aconditionalprobabilitystatement.
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149
c. Are Gand Findependent or dependentevents? ? Justifyyouranswernumericallyandalso
explainwhy.
d. AreGandFmutuallyexclusiveevents?Justifyyouranswernumericallyandalsoexplainwhy.
NOTE
: P.S.Amazingly,on2/1/95,RenatelearnedthatshewouldreceiveherGreenCard–true
story!
Exercise3.11.22
ThreeprofessorsatGeorgeWashingtonUniversitydidanexperimenttodetermineifeconomists
aremoreselfishthanotherpeople.Theydropped64stamped,addressedenvelopeswith$10cash
indifferentclassroomsontheGeorgeWashingtoncampus.44%werereturnedoverall.Fromthe
economicsclasses56%oftheenvelopeswerereturned.Fromthebusiness,psychology,andhistory
classes31%werereturned.(Source:WallStreetJournal)
Let:R=moneyreturned;E=economicsclasses;O=otherclasses
a. Writeaprobabilitystatementfortheoverallpercentofmoneyreturned.
b. Writeaprobabilitystatementforthepercentofmoneyreturnedoutoftheeconomicsclasses.
c. Writeaprobabilitystatementforthepercentofmoneyreturnedoutoftheotherclasses.
d. Ismoneybeingreturnedindependentoftheclass?Justifyyouranswernumericallyandexplain
it.
e. Baseduponthisstudy,doyouthinkthateconomistsaremoreselfishthanotherpeople?Explain
whyorwhynot.Includenumberstojustifyyouranswer.
Exercise3.11.23
(Solutiononp.163.)
ThechartbelowgivesthenumberofsuicidesestimatedintheU.S.forarecentyearbyage,race
(blackandwhite),andsex.Weareinterestedinpossiblerelationshipsbetweenage,race,andsex.
Wewillletsuicidevictimsbeourpopulation.(Source: TheNationalCenterforHealthStatistics,
U.S.Dept.ofHealthandHumanServices)
RaceandSex
1-14
15-24
25-64
over64
TOTALS
white,male
210
3360
13,610
22,050
white,female
80
580
3380
4930
black,male
10
460
1060
1670
black,female
0
40
270
330
allothers
TOTALS
310
4650
18,780
29,760
Table3.9
NOTE
:Donotinclude"allothers"forparts(f),(g),and(i).
a. Fillinthecolumnforthesuicidesforindividualsoverage64.
b. Fillintherowforallotherraces.
c. Findtheprobabilitythatarandomlyselectedindividualwasawhitemale.
d. Findtheprobabilitythatarandomlyselectedindividualwasablackfemale.
e. Findtheprobabilitythatarandomlyselectedindividualwasblack
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150
CHAPTER3. PROBABILITYTOPICS
f. Findtheprobabilitythatarandomlyselectedindividualwasmale.
g. Outoftheindividualsoverage64,findtheprobabilitythatarandomlyselectedindividualwas
ablackorwhitemale.
h. Comparing“RaceandSex”to“Age,”whichtwogroupsaremutuallyexclusive?Howdoyou
know?
i. Arebeingmaleandcommittingsuicideoverage64independentevents?Howdoyouknow?
Thenexttwoquestionsrefertothefollowing: ThepercentoflicensedU.S.drivers(fromarecentyear)
thatarefemaleis48.60.Ofthefemales,5.03%areage19andunder;81.36%areage20-64;13.61%areage
65orover.OfthelicensedU.S.maledrivers,5.04%areage19andunder;81.43%areage20-64;13.53%are
age65orover.(Source:FederalHighwayAdministration,U.S.Dept.ofTransportation)
Exercise3.11.24
Completethefollowing:
a. Constructatableoratreediagramofthesituation.
b. P(driverisfemale)=
c. P(driverisage65orover|driverisfemale)=
d. P(driverisage65oroverANDfemale)=
e. Inwords,explainthedifferencebetweentheprobabilitiesinpart(c)andpart(d).
f. P(driverisage65orover)=
g. Arebeingage65oroverandbeingfemalemutuallyexclusiveevents?Howdoyouknow
Exercise3.11.25
(Solutiononp.163.)
Supposethat10,000U.S.licenseddriversarerandomlyselected.
a. Howmanywouldyouexpecttobemale?
b. Usingthetableortree diagramfromthepreviousexercise,constructacontingencytableof
genderversusagegroup.
c. Usingthecontingencytable,findtheprobabilitythatoutoftheage20-64group,arandomly
selecteddriverisfemale.
Exercise3.11.26
Approximately86.5%ofAmericanscommutetoworkbycar,truckorvan. Outofthatgroup,
84.6%drivealoneand15.4%driveinacarpool.Approximately3.9%walktoworkandapproxi-
mately5.3%takepublictransportation. (Source:BureauoftheCensus,U.S.Dept. ofCommerce.
Disregardroundingapproximations.)
a. Constructatableoratreediagramofthesituation. Includeabranchforallothermodesof
transportationtowork.
b. Assumingthatthewalkerswalkalone,whatpercentofallcommuterstravelalonetowork?
c. Supposethat1000workersarerandomlyselected.Howmanywouldyouexpecttotravelalone
towork?
d. Supposethat1000workersarerandomlyselected. Howmanywouldyouexpecttodriveina
carpool?
Exercise3.11.27
Explainwhatiswrongwiththefollowingstatements.Usecompletesentences.
a. Ifthere’sa60%chanceofrainonSaturdayanda70%chanceofrainonSunday,thenthere’sa
130%chanceofrainovertheweekend.
b. Theprobabilitythatabaseballplayerhitsahomerunisgreaterthantheprobabilitythathe
getsasuccessfulhit.
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