$A$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B|A)$,if $A$ is a subset of $B$.

Let $X$ and $Y$ be two events of a sample space such that $P(X)=\frac{1}{3}$, $P(X|Y)=\frac{1}{2}$ and $P(Y|X)=\frac{2}{5}$, then which of the following is true?

If $A$ and $B$ are two events such that $P(A)=\frac{1}{3}$,$P(B)=\frac{1}{2}$ and $P(A \cap B)=\frac{1}{6}$,then $P(A^{\prime} | B)$ is

$A$ coin is tossed until a head appears or it has been tossed thrice. Given that a head does not appear on the first toss,what is the probability that the coin is tossed thrice?

Two integers $r$ and $s$ are drawn one at a time without replacement from the set $\{1, 2, \ldots, n\}$. Then $P(r \leq k \mid s \leq k) =$

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.