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.