$A$ die is loaded such that the probability of throwing the number $i$ is proportional to its reciprocal. Then the probability that $3$ appears in a single throw is

Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$ or $B$ occurs is $\frac{1}{2}$,then the probability of both of them occurring together is

$A, B, C, D$ cut a pack of $52$ well-shuffled playing cards successively in the same order. If the person who cuts a spade first wins the game and the game continues until this happens,then the probability that $A$ wins the game is

If $A$ and $B$ are any two events such that $P(\overline{A \cup B}) = \frac{1}{6}$,$P(A \cap B) = \frac{1}{4}$,and $P(\bar{A}) = \frac{1}{4}$,then the events $A$ and $B$ are

In a single throw of two dice,the probability of getting a sum of more than $7$ is

If two dice are thrown together,then the probability that the sum of the numbers appearing on them is a prime number is: