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Transcript of Lesson 3 Conditional Probability and Compound Events … Lesson 3 Conditional Probability... ·...
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 1
Lesson #3: Conditional Probability
and Compound
Events
Conditional ProbabilityConditionalProbability ofaneventB,inrelationtoeventA,istheprobabilitythateventBwilloccurgiventheknowledgethataneventhasalreadyoccurred.
Notation:P B A isprobabilityofeventBgiveneventA
2Types:1.IndependentEvents2.DependentEvents
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 2
Conditional ProbabilityIndependentEventsGiven:Youtosstwopennies.Thefirstpennylandsonheads.Thesecondpennyfallsoffofthetableandyoucannotseeit.Whatistheprobabilitythatbothpennieslandonheads?
P Heads Heads12
Sincethefirstcoinlandedonheads,whichisindependent ofthecoinlandingon
headsforthesecondtoss,theprobabilitythatthesecondcoinlandsonheadsis
ConditionalProbabilityofIndependentEventsIfeventsAandBareindependent,theconditionalprobabilityofeventBgiveneventAissimplythe
probabilityofeventB.
Conditional ProbabilityExample1:Youtosstwodice.Thefirstdielandsona5.Whatistheprobabilitythattheseconddielandsona4?
P 4 5
16
WHY? Sincethefirstdielandedona5,whichisindependent ofthedielandingona4forthesecondtoss,theprobabilitythattheseconddielandsona4is .
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 3
Conditional ProbabilityDependentEventsGiven:Abagcontains12redM&Ms,12blueM&Ms,and12greenM&Ms. WhatistheprobabilityofdrawingasecondM&Mofthesamecolorasthefirstcolor?
P red redSincetherearethesame
numberofeachcolor,itdoesnotmatterifwefindtheprobability
ofred,blue,orgreen!
Recall:IftheresultofoneeventIS affectedbytheresultofanotherevent,theeventsaresaidtobedependent.
· , assumingAoccured ·*usingnewnotation*
Conditional ProbabilityDependentEvents
ConditionalProbabilityofDependentEvents
IfeventsAandBaredependent,theconditionalprobabilityofeventBgiveneventAis
Solvefor :
·
B A
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 4
Conditional ProbabilityDependentEventsExample1:Abagcontains12redM&Ms,12blueM&Ms,and12greenM&Ms. WhatistheprobabilityofdrawingasecondM&Mofthesamecolorasthefirstcolor?
P B A red red
P redandredP red
1236 ·
1135
1236
1135
Recall: · , assumingAoccured
Conditional ProbabilityExample2:Youselect2cardsfromastandarddeckofcards.Whatistheprobabilityofdrawingtwocardsfromthesamesuit,ifthefirstcardisnot replaced?
P B A
P heartsandheartsP hearts
1352 ·
1251
1352
1251
Recall: · , assumingAoccured
P hearts hearts
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 5
Example3:WVHSstatisticsstudentssurveyed52seniorsabouttheirplansafterhighschoolandthedataispresentedinthetablebelow.
Male (M) Female (F) Total
Going to College (C) 16 13 29
Not Going to College (N)
14 9 23
Total 30 22 52
Findthefollowingprobabilities:(a)P F (b)P C (c)P C F (d)P M C
2252
2952
P CandFP F
1322
P MandCP C
1629
Example4:Resultsfromamiddleschoolsurveyaresummarizedbelow.Usingthese545studentsasthesamplespace,astudentfromthestudyisrandomlyselected:
(a) Ifthestudentselectedisa7thgradestudent,whatistheprobabilitythatthestudentprefersice‐skating?
GradeSnow
boardingSkiing
Ice Skating TOTAL
6th 68 41 46 155
7th 84 56 70 210
8th 59 74 47 180
TOTAL 211 171 163 545
P skatingand7thgradeP 7thgrade
70210
P iceskating 7thgrade
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 6
Example4:Resultsfromamiddleschoolsurveyaresummarizedbelow.Usingthese545studentsasthesamplespace,astudentfromthestudyisrandomlyselected:
(b)Ifthestudentselectedpreferssnowboarding,whatistheprobabilitythatthestudentselectedisa6th gradestudent?
GradeSnow
boardingSkiing
Ice Skating TOTAL
6th 68 41 46 155
7th 84 56 70 210
8th 59 74 47 180
TOTAL 211 171 163 545
P 6thgradeandsnowboardP snowboard
68211
P 6thgrade snowboard
Example4:Resultsfromamiddleschoolsurveyaresummarizedbelow.Usingthese545studentsasthesamplespace,astudentfromthestudyisrandomlyselected:
GradeSnow
boardingSkiing
Ice Skating TOTAL
6th 68 41 46 155
7th 84 56 70 210
8th 59 74 47 180
TOTAL 211 171 163 545
(c)Ifthestudentselectedisan8th gradestudent,whatistheprobabilitythatthestudentprefersskiingoriceskating?
P skiingand8th P skatingand8thP 8thgrade
74 47180
121180
P skiingORiceskating 8thgrade
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 7
Example5:Asurveywasdoneofcommutersinthreemajorcitiesabouthowtheyprimarilygottowork.Theresultsareshownasprobabilitiesinthetablebelow:
Findthefollowingprobabilities:(a)P train LA (b)P NYC Car (c)P walk Chicago
Car Train Walk Total
NYC .05 .25 .10 .40
LA .18 .12 .05 .35
Chicago .08 .14 .03 .25
Total .31 .51 .18 1.00
P trainandLA
P LA
.12
.35.3429
P NYCandcar
P car
.05
.31.1613
P walkandChicago
P Chicago
.03
.25 .12
Example6:Amarketingcompanyasked1,000customershowtheyprefertoviewmovies.Theresultaresummarizedinthetablebelow.
Tothenearestpercent,whatistheprobabilitythatacustomerwouldprefertoviewmoviesviaonlinedownloadsgiventhattheyareunder35yearsold?
DVD or BlueRay
Netflix or similar
Online Downloads
Age Under 35
52 145 186
Age 35-55
108 110 56
Age over 55
250 75 18
P onlineandunder35P under35
186383
P Online Under35
383NeedTotal!
49%
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 8
Example7:48.6%oftheNorthDakotapopulationover18yearsoldisfemale.65%ofthetotalpopulationofNorthDakotaover18yearsoldhasenrolledincollegeatsomepointsintheirlives.Ifthesetwoeventsareindependentofeachother,whatistheprobabilitythatarandomlyselectedadultinNorthDakotawillbeafemalewhohasenrolledincollege,tothenearestpercent?
Eventsareindependentofeachother! ·
Female College Female · College
.486 · .65
.320274 · 100
32.0274
32%
NeedPercent!
More Examples
with Compound
Events
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 9
Recall: Compound EventsIndependent EventsIftheresultofthesecondeventisnotaffectedbytheresultofthefirstevent.
·
Dependent EventsIftheresultofoneeventIS affectedbytheresultofanotherevent,theeventsaresaidtobedependent.
· , ·
Compound EventsExample1:Findtheprobabilityofrollinga5onastandard6‐sideddieandthenrollinganumbergreaterthan2.
Example2:Aboxcontains5bluepensand3redpens.FindtheprobabilitythatMackenziewillpicktwobluepens.
*IndependentEvents!
⟹ ·
, , , , ,
·
*DependentEvents!
⟹ · ·
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 10
Example3:Aspinnerisgivenbelow,andisspuntwice.
(a)Probabilityyouwillspinanevennumberonthefirstspinandanumberlessthanfiveonthesecondspin.
(b)Probabilityyouwillspinaprimenumberandaperfectsquare.
*IndependentEvents!
⟹ ·
·
*Ineitherorder!
⇒ · ·
· ·
Example4:Aclassconsistsof12girlsand8boys.Agroupof3ispickedoutatrandomtogiveaspeech.Findtheprobabilitythattheywillallbeboys.
*DependentEvents!
⟹ · ·
· ·
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 11
Example5:Apowergeneratingfacilityhasthreeprimarysafetyswitchesincaseofanemergency.Theprobabilitythatanyoftheseswitcheswouldfailis5%.Whatistheprobabilityallthreewillfailgiventhattheswitchesareindependentofoneanother?
*IndependentEvents!
⟹ · ·
.
. ,
· . · .
.
Example6:Acompanywasdeterminingtheeffectivenessofitswarrantysalesonccomputers.Ifacustomerwaschosenatrandom,whatistheprobabilitytheydidnot purchaseawarranty?
% of Customers Purchasing a computer
% of Customers
who purchased a warranty
Type I 68% 35%
Type II 32% 56%
*WhatpercentofcustomersdidNOTpurchaseawarranty?
65%44%
⇒
· ·
. . . .
. . .
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 12
Example7:(a)Adrawercontains3redpaperclips,4greenpaperclips,and5bluepaperclips. Onepaperclipistakenfromthedrawerandthenreplaced. Anotherpaperclipistakenfromthedrawer. Whatistheprobabilitythatthefirstpaperclipisred&thesecondpaperclipisblue?
*IndependentEvents!
⟹ ·
·
Example7:(b)Adrawercontains3redpaperclips,4greenpaperclips,and5bluepaperclips. Onepaperclipistakenfromthedrawerandnotreplaced. Anotherpaperclipistakenfromthedrawer.Whatistheprobabilitythatthefirstpaperclipisredandthesecondpaperclipisblue?
*DependentEvents!
⟹ ·
·
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 13
(c)Showthattheeventsinpart(b) aredependentevents.
: ·
: ·
: · ·
·
· ·
? ·
?
. .∴ !
Example8:Abagcontainsmarblesofthreedifferentcolors:8black,6white,and4red.Threemarblesareselectedatrandom,withoutreplacement.Findtheprobabilitythattheselectioncontainseachoftheoutcomeslistedbelow.Expressyouranswersasadecimaltothenearesthundredth.(a) Threeblackmarbles (b) Threewhitemarbles
*DependentEvents!
⟹ · ·
.
· ·
⟹ · ·
.
· ·
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 14
Example8:Abagcontainsmarblesofthreedifferentcolors:8black,6white,and4red.Threemarblesareselectedatrandom,withoutreplacement.Findtheprobabilitythattheselectioncontainseachoftheoutcomeslistedbelow.Expressyouranswersasadecimaltothenearesthundredth.(c) Ared,ablack,andawhitemarble(inthat order!)
⟹ · ·
.
· ·
Example8:Abagcontainsmarblesofthreedifferentcolors:8black,6white,and4red.Threemarblesareselectedatrandom,withoutreplacement.Findtheprobabilitythattheselectioncontainseachoftheoutcomeslistedbelow.Expressyouranswersasadecimaltothenearesthundredth.(d) Aredmarbleandtwowhitemarbles(inany order!)
& &
· ·
& &
& &
· · · ·
.
Unit #9: Probability and Statistics
Lesson #3: Conditional Probability and Compound Events 15
Review: WVHSstatisticsstudentssurveyed52seniorsabouttheirplansafterhighschoolandthedataispresentedinthetablebelow.
Male (M) Female (F) Total
Going to College (C) 16 13 29
Not Going to College (N)
14 9 23
Total 30 22 52Findthefollowingprobabilities:(a)P C F (b)P M C (c)P N F (d)P F N
P CandF
P F
1322
P MandC
P C
1629
P NandF
P F
922
P FandN
P N
923