Let X any Y be two events, such that `P(X//Y)=(1)/(2), P(Y//X)=(1)/(3) and P(X cap Y)=(1)/(6)`. Which of the following is (are) correct ?

Options

A. `P(X cup Y)=(2)/(3)`
B. X and Y are independent
C. `P(X^(c ) cap Y)=(1)/(6)`
D. X and Y are not independent

Answer

D

Explanation

We have,
`P(X//Y)=(1)/(2),P(Y//X)=(1)/(3) " and ", P(X cap Y)=(1)/(6)`
`therefore P(X)=(P(X cap Y))/(P(Y//X))=((1)/(6))/((1)/(3))=(1)/(2),P(Y)=(P(X capY))/(P(X//Y))=((1)/(6))/((1)/(2))=(1)/(3)`
Clearly,`P(X cup Y)=P(X)P(Y)`.
So, X and Y are independent events.
`therefore P(X cup Y)=1-P(overline(X))P(overline(Y))=1-(1)/(2)xx(2)/(3)=(2)/(3)`
and, `P(X^(c ) cap Y)=P(Y)-P(X cap Y)=(1)/(3)-(1)/(6)=(1)/(6)`
Hence, option (d) in incorrect.