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

P(X|Y) = P(X∩Y)/P(Y) = 1/2
→ (1/6)/P(Y) = 1/2
→ P(Y) = 1/3
P(Y|X) = P(X∩Y)/P(X) = 1/3
→ (1/6)/P(X) = 1/3
→ P(X) = 1/2
If X and Y are independent, then P(X∩Y) = P(X)P(Y)
However, P(X)P(Y) = (1/2)(1/3) = 1/6 = P(X∩Y)
Therefore, X and Y are independent.
P(X∪Y) = P(X) + P(Y) - P(X∩Y) = 1/2 + 1/3 - 1/6 = 2/3
P(Xᶜ∩Y) = P(Y) - P(X∩Y) = 1/3 - 1/6 = 1/6