Let X and Y be two events such that $$P(X \mid Y) = \frac{1}{2}, P(Y \mid X) = \frac{1}{3}$$ and $$P(X \cap Y) = \frac{1}{6}$$. Which of the following is (are) correct?

A

$$P(X \cup Y) = \frac{2}{3}$$

B

X and Y are independent

C

X and Y are not independent

D

$$P(X^c \cap Y) = \frac{1}{3}$$