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CONDITIONAL PROBABILITY &

INDEPENDENCE

AppliedSta+s+csandCompu+ngLab

IndianSchoolofBusiness

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LearningGoals

Condi6onalProbabili6es TheMul6plica6onRule IndependenceofEvents

2

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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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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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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