Lesson 3 Conditional Probability and Compound Events … Lesson 3 Conditional Probability... · Unit #9: Probability and Statistics Lesson #3: Conditional Probability and Compound

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



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 .



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



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



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




(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


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



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%



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



Recall: Compound EventsIndependent EventsIftheresultofthesecondeventisnotaffectedbytheresultofthefirstevent.

·

Dependent EventsIftheresultofoneeventIS affectedbytheresultofanotherevent,theeventsaresaidtobedependent.

· , ·

Compound EventsExample1:Findtheprobabilityofrollinga5onastandard6‐sideddieandthenrollinganumbergreaterthan2.

Example2:Aboxcontains5bluepensand3redpens.FindtheprobabilitythatMackenziewillpicktwobluepens.

*IndependentEvents!

⟹ ·

, , , , ,

·

*DependentEvents!

⟹ · ·



Example3:Aspinnerisgivenbelow,andisspuntwice.

(a)Probabilityyouwillspinanevennumberonthefirstspinandanumberlessthanfiveonthesecondspin.

(b)Probabilityyouwillspinaprimenumberandaperfectsquare.

*IndependentEvents!

⟹ ·

·

*Ineitherorder!

⇒ · ·

· ·

Example4:Aclassconsistsof12girlsand8boys.Agroupof3ispickedoutatrandomtogiveaspeech.Findtheprobabilitythattheywillallbeboys.

*DependentEvents!

⟹ · ·

· ·



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%

⇒

· ·

. . . .

. . .



Example7:(a)Adrawercontains3redpaperclips,4greenpaperclips,and5bluepaperclips. Onepaperclipistakenfromthedrawerandthenreplaced. Anotherpaperclipistakenfromthedrawer. Whatistheprobabilitythatthefirstpaperclipisred&thesecondpaperclipisblue?

*IndependentEvents!

⟹ ·

·

Example7:(b)Adrawercontains3redpaperclips,4greenpaperclips,and5bluepaperclips. Onepaperclipistakenfromthedrawerandnotreplaced. Anotherpaperclipistakenfromthedrawer.Whatistheprobabilitythatthefirstpaperclipisredandthesecondpaperclipisblue?

*DependentEvents!

⟹ ·

·



(c)Showthattheeventsinpart(b) aredependentevents.

: ·

: ·

: · ·

·

· ·

? ·

?

. .∴ !

Example8:Abagcontainsmarblesofthreedifferentcolors:8black,6white,and4red.Threemarblesareselectedatrandom,withoutreplacement.Findtheprobabilitythattheselectioncontainseachoftheoutcomeslistedbelow.Expressyouranswersasadecimaltothenearesthundredth.(a) Threeblackmarbles (b) Threewhitemarbles

*DependentEvents!

⟹ · ·

.

· ·

⟹ · ·

.

· ·



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!)

& &

· ·

& &

& &

· · · ·

.



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

Documents

Lesson 3 Conditional Probability and Compound Events … Lesson 3 Conditional Probability... · Unit #9: Probability and Statistics Lesson #3: Conditional Probability and Compound