Let $X$ and $Y$ be two events such that $P(X)=\frac{1}{3}$,$P(X \mid Y)=\frac{1}{2}$ and $P(Y \mid X)=\frac{2}{5}$. Then:
$A) P(X^{\prime} \mid Y)=\frac{1}{2}$
$B) P(X \cap Y)=\frac{1}{5}$
$C) P(X \cup Y)=\frac{2}{5}$
$D) P(Y)=\frac{4}{15}$

Assume that each born child is equally likely to be a boy or a girl. If a family has two children,what is the conditional probability that both are girls given that at least one is a girl?

If $A$ and $B$ are two events such that $P(B) \neq 0$ and $P(B) \neq 1$,then $P(\bar{A} \mid \bar{B})$ is equal to:

One card is drawn at random from a well-shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent?
$E:$ 'the card drawn is black'
$F:$ 'the card drawn is a king'

Let $B(\alpha, \beta, \gamma)$ represent that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls, and $\gamma$ blue balls. Given $B_1(2, 3, 2)$, $B_2(3, 2, 2)$, $B_3(2, 2, 3)$. $A$ die is rolled. If the die shows $2, 3,$ or $5$, a ball is drawn from bag $B_1$. If the die shows $4$ or $6$, a ball is drawn from bag $B_2$. If the die shows $1$, a ball is drawn from bag $B_3$. Find the probability of drawing a green ball.

When $2$ dice are thrown, it is observed that the sum of the numbers appearing on the top faces of both dice is a prime number. What is the probability that at least one of the numbers in the pair is a multiple of $3$?