Question

If X and Y are two events such that `PX//Y)=(1)/(2), P(Y//X)=(1)/(3)` and `P(XcapY)(1)/(6)`. Then, which of the following is/are correct ? A. `P(XcupY)=2//3` B. X and Y are independent C. X and Y are not independent D. `P(X^(c )capY)=1//3` Select the correct answer from above options

Answer 1

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)`