Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2, \sqrt{3N}, N + 2$ are in geometric progression be $\frac{k}{48}$. Then the value of $k$ is

$A$ bag contains $3$ white and $5$ black balls. If one ball is drawn,then the probability that it is black,is

The probabilities that $A$ and $B$ will die within a year are $p$ and $q$ respectively. Then,the probability that only one of them will be alive at the end of the year is

Let $S$ be the sample space of the random experiment of throwing simultaneously two unbiased dice with six faces (numbered $1$ to $6$) and let $E_k = \{(a, b) \in S : ab = k\}$ for $k \geq 1$. If $p_k = P(E_k)$ for $k \geq 1$,then which of the following is correct?

$A, B, C$ are any three events. If $P(S)$ denotes the probability of $S$ happening,then $P(A \cap (B \cup C)) = $

$A$ problem in Algebra is given to two students $A$ and $B$ whose chances of solving it are $\frac{2}{5}$ and $\frac{3}{4}$ respectively. The probability that the problem is solved, if both of them try independently, is