Let X and Y be two events such that `P(X)=1/3, P(X|Y)=1/2and P(Y|X)=2/5.` Then

Options

A. `P(X//Y)=(1)/(2)`
B. `P(X cap Y)=(1)/(5)`
C. `P(X cup Y)=(2)/(5)`
D. `P(Y)=(4)/(15)`

Answer

A, D

Explanation

We have,
`P(Y//X)=(2)/(5) implies (P(X cap Y))/(P(X))=(2)/(5) implies P(X cap Y)=(2)/(5)P(X)=(2)/(5)xx(1)/(3)=(2)/(15)`
It is given that
`P(X//Y)=(1)/(2) implies (P(X cap Y))/(P(Y))=(1)/(2) implies P(Y)=2P(X cap Y)=2xx(2)/(15)=(4)/(15)`
So, option (d) is correct.
Thus, we have
`P(X)=(1)/(3),P(Y)=(4)/(15) " and " P(X cap Y)=(2)/(15)`
`therefore P(X cup Y)=P(X)+P(Y)-P(X cap Y)`
`=(1)/(3)+(4)/(15)-(2)/(15)=(7)/(15)`
Now, `P(X'//Y)=1-P(X//Y)=1-(1)/(2)=(1)/(2)`
So, option (a) is correct.