class 11th maths chapter probability cbse

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UNIT IV
PROBABILITY

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Definition of Probability:
Ifthereare‘n’exhaustive,mutuallyexclusiveandequallylikely
outcomesofarandomexperiment,and‘m’ofthemare
favourabletoanevent‘A’,thentheprobabilityofhappeningof‘A’
is:
wheremisno.offavourableevents
nisno.ofunfavourableevents

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Terminology Used in Definition:
RandomExperiment:
Anoccurrencewhichcanberepeatedanumberoftimes
essentiallyunderthesameconditions&whoseresultcan’t
bepredictedbeforehand,isknownasarandom
experimentorsimplyanexperiment.
SampleSpace&SamplePoint:
Thesetofallpossibleoutcomesofaexperimentiscalleda
samplespace(S)
Theelementsofsamplespacearecalledsamplepoints.
Asamplespaceissaidtobefiniteorinfinite.
ForEg:Ifwethrowadice,itcanresultinanyofthesixnumbers
1,2,3,4,5,6.
Thereforesamplespaceofthisexperimentis
S={1,2,3,4,5,6}and
n(S)=6

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Event:
Anysubsetofsamplespaceiscalledanevent.
IfSisasamplespace,thenitisobviousthatthenullsetØ
andthesamplespaceSitselfareevents.
Foreg:E={2,4,6}andn(A)=3
ExhaustiveOutcomes:
Byexhaustivewemeanthatallthepossibleoutcomeshave
beentakenintoconsiderationandoneofthemmust
happenasaresultofanexperiment.
ForEg(1):Ifwethrowadice,therearesixexhaustiveoutcomes,
namelynumbers1,2,3,4,5,6cominguppermost.
Eg(2):Intossingacointherearetwoexhaustiveoutcomes
namelycomingupofhead&tail.

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MutuallyExclusiveOutcomes:
Outcomesaresaidtobemutuallyexclusiveifthe
happeningofanoutcomeexcludesthepossibilityofthe
happeningofotheroutcomes.
Fore.g.:Intossingacoin,ifheadcomingupthencomingupoftail
isexcludedinthatparticularchance.
EquallyLikelyOutcomes:
Outcomesaresaidtobeequallylikelywhenthe
occurrenceofnoneofthemisexpectedinpreferenceto
others.
Independent&DependentEvent:
Twoeventsaresaidtobeindependentiftheprobabilityof
occurrenceofeitherofthemisnotaffectedbytheoccurrence
ornon–occurrenceoftheother.
Ontheotherhand,iftheoccurrenceofoneeventaffectsthe
probabilityofoccurrenceoftheother,thenthesecondeventis
saidtobedependentonthefirst.

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Illustrations1:
Anunbiaseddiceisthrown.Whatistheprobabilityof
i.gettingasix
ii.gettingeitherfiveorsix
Solution:
Inasinglethrowofdice,therearesixpossibleoutcomesi.e.
1,2,3,4,5,6.
Thusn(S)=6
i.gettingasix
Heren(E)=1
Thereforerequiredprobability:
ii.gettingeitherfiveorsix
Heren(E)=2
Thereforerequiredprobability:

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Illustrations2:
Inasimultaneousthrowoftwodie,findtheprobabilityof
gettingatotalof6.
Solution:
Inasimultaneousthrowoftwodie,wehave6*6i.e.36possible
outcomes.
Thusn(S)=36and
E={(1,5),(2,4),(3,3),(4,2),(5,1)}i.e.n(E)=5
Thereforerequiredprobability:

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THEOREMSOFPROBABILITY:
AdditionTheorem(ORTheorem)
MultiplicationTheorem(ANDTheorem)
AdditionTheorem:
Case1:Wheneventsaremutuallyexclusive:
ItstatethatiftwoeventsA&Baremutuallyexclusivethenthe
probabilityofoccurrenceofeitherAorBisthesumofthe
individualprobabilityofA&B.Symbolically
P(AUB)=P(A)+P(B)
Case2:WheneventsareNOTmutuallyexclusive:
ItstatesthatiftwoeventsA&Barenotmutuallyexclusive,then
probabilityoftheoccurrenceofeitherAorBisthesumofthe
individualprobabilityofA&Bminustheprobabilityofoccurrence
ofbothAandB.Symbolically
P(AUB)=P(A)+P(B)–P(AB)

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ConditionalProbability:
TheprobabilityofoccurrenceofeventA,giventhattheeventB
hasalreadyoccurrediscalledconditionalprobabilityof
occurrenceofAontheconditionthatBhasalreadyoccurred.
ItisdenotedbyP(A/B).
IfAandBareindependentevents,thenP(A/B)=P(A).
MultiplicationTheorem:
TheprobabilityofsimultaneousoccurrenceoftwoeventsA&Bis
theproductofprobabilityofAandtheconditionalprobabilityofB
whenAhasalreadyoccurredorvice–versa.Symbolically
P(AB)=P(A).P(B/A),IfP(A)≠0
P(AB)=P(B).P(A/B),IfP(B)≠0
Itisnotedthatincaseofindependentevents:
P(AB)=P(A).P(B)