If a die is rolled twice and the sum of the numbers appearing on them is observed to be $6$, then the probability that the number $1$ appears at least once on them is

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of $4$. If $p$ is the conditional probability that number $4$ has appeared at least once, then $3p + 2 =$

$A$ pair of dice is thrown. Then the probability that either of the dice shows $2$ when their sum is $6$ is

If $A$ and $B$ are two events such that $P(A | B) = 0.6$, $P(B | A) = 0.3$, and $P(A) = 0.1$, then $P(\bar{A} \cap \bar{B})$ equals:

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:
$A) P(X^{\prime} \mid Y)=\frac{1}{2}$
$B) P(X \cap Y)=\frac{1}{5}$
$C) P(X \cup Y)=\frac{2}{5}$
$D) P(Y)=\frac{4}{15}$

If $A$ and $B$ are any two events such that $P(A)+P(B)-P(A \cap B)=P(A)$,then