If two cards are drawn at random simultaneously from a pack of $52$ playing cards, then the probability of getting a face card and a spade card other than the face card is:

If $P(B) = \frac{3}{4}$,$P(A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P(\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$,then $P(B \cap C)$ is

The probability that $A$ speaks truth is $75 \%$ and the probability that $B$ speaks truth is $80 \%$. The probability that they contradict each other when asked to speak on a fact is

Consider the set $A_n$ of points $(x, y)$ such that $0 \leq x \leq n, 0 \leq y \leq n$,where $n, x, y$ are integers. Let $S_n$ be the set of all lines passing through at least two distinct points from $A_n$. Suppose we choose a line $l$ at random from $S_n$. Let $P_n$ be the probability that $l$ is tangent to the circle $x^2+y^2=n^2\left(1+\left(1-\frac{1}{\sqrt{n}}\right)^2\right)$. Then,the limit $\lim _{n \rightarrow \infty} P_n$ is

If a year is chosen at random from the years having $53$ Sundays and $53$ Mondays,what is the probability that it is a leap year?

If three numbers are randomly selected from the set $\{1, 2, 3, \ldots, 50\}$, then the probability that they are in arithmetic progression is