Correct Answer - A::B
Plan
(i) Conditional probability, i.e., P(A/B) `= (P(A nn B))/(P(B))`
(ii) P (A uu B) = P(A) + P(B) - P(A nn B)`
`(iii) Independent event, then `P(A nn B) = P(A) * P(B) `
Here, `P(X//Y) = (1)/(2) , P ((Y)/(X)) = (1)/(3)`
and `P (X nn Y) = 6`
`therefore P ((X)/(Y)) = (P (X nn Y))/(P(Y))`
`rArr (1)/(2) = (1//6)/(P (Y)) rArr P(Y)) = (1)/(3) " "... (i)`
`P ((X)/(Y)) = (1)/(3) rArr (P(X nn Y))/(P(X)) = (1)/(3)`
`rArr (1)/(6) = (1)/(3) P(X)`
`therefore P(X) = (1)/(2) " "...(ii)`
`P(X nn Y) = P(X) + P(Y) - P(X nn Y)`
` = (1)/(2) + (1)/(3) - (1)/(6) = (2)/(3) " " .... (iii)`
`P(X nn Y) = (1)/(6) "and " P(X) * P(Y) = (1)/(2) * (1)/(3) = (1)/(6)`
`rArr P(X nn Y) = P(X) * P(Y)`
i.e. independent events
`therefore P(X^(e) nn Y) = P(Y) - P(X nn Y)`
` = (1)/(3) - (1)/(6) = (1)/(6)`