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AppliedSta+s+csandCompu+ngLab
CONDITIONAL PROBABILITY &
INDEPENDENCE
AppliedSta+s+csandCompu+ngLab
IndianSchoolofBusiness
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AppliedSta+s+csandCompu+ngLab
LearningGoals
Condi6onalProbabili6es TheMul6plica6onRule IndependenceofEvents
2
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AppliedSta+s+csandCompu+ngLab
AnExample
10candidates,goforaninterviewatfirmXforthesamejob.Hereissomeinforma6ononthecandidates:
ABCDEFGHIJ
OurSampleSpace(fortheexperimentthat10candidatesgoforaninterviewandonegetsselected)is:
Theprobabili6esofeachcandidategengthejobarerespec6vely
3
Boys Girls
Engineers
1021..,........., ppp
},,,,,,,,,{ JIHGFEDCBAS =
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AnExample(Contd) Letusdefinefewevent:B:Aboyisselected
G:Agirlisselected
E:Anengineerisselected
B={A,B,C,D}andP(B)=
G={E,F,G,H,I,J}andP(G)=
E={C,D,E,F}andP(E)=
Theresultsareexpectedwithinaweekoftheinterview.For,somereasonthe
firmhasntgo]enbacktoanyofthecandidateswiththeresults.So,the
anxiouscandidatesspeakwiththeHRmanager.Themanagerassures
themthatoneofthemhasbeenselectedbutsheisnotsurewhoitis.All
sheknowsisthatthecandidateisanengineer.
Now,giventhisinforma6onwhatistheprobabilitythatagirlgetsthejob?
4
4321pppp +++
1098765pppppp +++++
6543 pppp +++
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AnExample(Contd) Wenowhaveanewandsmallersamplespace,
Intui6velyweknowthattherespec6veprobabili6esinS1herearepropor6onaltotheirpreviousprobabili6es
Letussaytheprobabili6esaresuchthat Wealsoknowthat
5
},,,{1 FEDCS =
'
4
'
3
'
2
'
1 ,,, pppp
1
'
11
'
1cpppp =
=
===
====4
1
4
1
4
1
4
1
' 1111
i
i
i
i
i
i
i
i
p
cpccpp
6543
6'
4
6543
5'
3
6543
4'
2
6543
3'
1
'
&'
pppp
pp
pppp
pp
pppp
pp
pppp
pp
+++
=
+++
=
+++
=
+++
=
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Wewanttoobtaintheprobabilitythatagirlwillbeselectedgiventhatanengineerhasgotthejob,i.e.P(GgivenEhasoccurred)
ThisisdenotedbyP(G|E)andreadasProbabilityofGgivenE,wecallsuchaprobabilityaCondi+onalProbability
Now,thereareonlytwogirlsintheGandEcategory,theprobabilitythatoneofthemisselectedis:
LookattheeventGE={E,F}: Similarly,theeventE={C,D,E,F}: Sowehave:
AnExample(Contd)
6543
65
6543
6
6543
5)|(pppp
pppppp
ppppp
pEGP+++
+
=
+++
+
+++
=
65)( ppEGP +=
6543)( ppppEP +++=
)()()|(
EP
EGPEGP
=
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AppliedSta+s+csandCompu+ngLab
Condi6onalProbability:Defini6onLetEandFbetwoevents,then:
Anotherexample:
AfarmerapproachesaRuralBankforaloan.Frompastexperiencethebankes6matesthattheprobabilitythatthisfarmerwilldefaultontheloanisaround4%.Thebanktellsthe
farmerthattheywouldgetbacktohiminaweekwithaloanproposal.Intheweekthatfollows,anewresearchclaimsthatfarmersaremorelikelytodefaultonloanswhentheeconomyislowthanwhenitishigh.Withthisnewinforma6on,thebankisnolongerabletoassumethattheprobabilitythatthefarmerwoulddefaultontheloaniss6ll4%
LetDbetheeventthatthefarmerdefaults Listheeventthateconomyislow Theresearchshowedthat3%ofthefarmersthatwerescheduledtorepayloansduring
thelasteconomiclowdefaulted
Theprobabilitythattheeconomywillbelowatanygiven6mepointis7% Sowehave:
Whichmeansthattheprobabilitythatthefarmerwoulddefaultinaloweconomyisawhooping43%!
7
0)(,)(
)()|( >
= FP
FP
FEP
FEP
P(D | L)=P(DL)
P(L)=
0.03
0.07 0.43
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TheMul6plica6onRule
Letuslookat:
Similarly, Example:
8
P(E| F)=P(EF)
P(F)
P(EF)= P(E| F)P(F) (MultiplicationRule)
P(EF)= P(F | E)P(E)
Examplefrom:AczelA.,SounderpandianJ.Completebusinesssta6s6cs
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Independence
Anotherresearchoffarmerandloandefaults,showedthattheprobabilityofdefaultdoesn'tactuallychangeastheeconomychanges
ThatisP(D|L)isnotdifferentfroP(D),i.e.Economicsitua6ondoesnthaveanyeffectonthedefaultratesoffarmers
Wethensaythattheeventthatthefarmerdefaultsisindependentfromtheeventthattheeconomyislow.
Aformaldefini6on: TwoeventsE&Faresaidtobeindependentif:
Forseveralevents:
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P(E | F)= P(E)
P(EF)= P(E)P(F)
E1,E2................,En
P( Ei
i=1
n
)= P(Eii=1
n
)
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AppliedSta+s+csandCompu+ngLab
IndependentEventsvs.DisjointEvents
Thesearedifferentconcepts! TocheckforindependenceofeventsEandFwecheckif:P(E|F)=P(E) TocheckiftheyareDisjointorMutuallyExclusivewecheck:P(EF)=0 Example:
Consideradeckofcards C:AcardisaClub R:AcardisRedConsiderP(R|C)=P(R)
0NotIndependent!
ConsiderP(CR)
=0..Disjoint!
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Thankyou
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