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 ?

Question

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

Answer 1

Correct Answer - D We have, P(X/Y)= 1 2 ,P(Y/X)= 1 3 and ,P(X∩Y)= 1 6 ∴P(X)= P(X∩Y) P(Y/X) = 1 6 1 3 = 1 2 ,P(Y)= P(X∩Y) P(X/Y) = 1 6 1 2 = 1 3 Clearly,P(X∪Y)=P(X)P(Y) . So, X and Y are independent events. ∴P(X∪Y)=1-P( ˉ X )P( ˉ Y )=1- 1 2 × 2 3 = 2 3 and, P(Xc∩Y)=P(Y)-P(X∩Y)= 1 3 - 1 6 = 1 6 Hence, option (d) in incorrect.