A fair die is rolled. Consider events `E = {1, 3, 5}`, `F = {2, 3}`and `G = {2, 3, 4, 5}`Find(i) `P(E|F)`and `P(F|E)` (ii) `P(E|G)`and `P(G|E)` (iii) `P((EuuF)|G)`and `P((EnnF)|G)`

Answer

N/a

Explanation

Here sample space `S=(1,2,3,4,5,6)`
Given `E-{1,3,5},F-{2,3}`
and `G={2,3,4,5}`
`impliesEnnF={3},EnnG={3,5}`
`EuuF={1,2,3,5},(EuuF)nnG={2,3,5}`
and `(EnnF)nnG={3}`
`impliesn(S)=6,n(E)=3,n(F)=2,n(G)=4`
`n(EnnF)=1,n(EnnG)=2,n(EuuF)=4`
`n[(EuuF)nnG]=3` and `n[(EnnF)nnG]=1`
Therefore,
`P(E)=3/6=1/2,P(F)=2/6=1/3,P(G)=4/6=2/3`
`P(EnnF)=1/6, P(EnnG)=2/6=1/3`
`P[(EuuF)nnG]=3/6=1/2` and `[P(Ennf)nnG]=1/6`
(i) `P(E/F)=(P(nnF))/(P(F))=(1//6)/(1//3)=1/2`
and `P(F/E)=(P(FnnE))/(P(E))=(1//6)/(1//2)=1/3`
(ii) `P(E/G)=(P(EnnG))/(P(G))=(1//3)/(2//3)=1/2`
`P(G/E)=(P(GnnE))/(P(E))=(1//3)/(1//2)=2/3`
(iii) `P((EuuF)/G)=(P[(EuuF)nnG])/(P(G))=(1//2)/(2//3)=3/4`
and `P(((EnnF))G)=(P[(EnnF)nnG])/(P(G))=(1//6)/(2//3)=1/4`