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

$A$ ship is fitted with three engines $E_1, E_2$,and $E_3$. The engines function independently of each other with respective probabilities $\frac{1}{2}, \frac{1}{4}$,and $\frac{1}{4}$. For the ship to be operational,at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X_1, X_2$,and $X_3$ denote respectively the events that the engines $E_1, E_2$,and $E_3$ are functioning. Which of the following is (are) true?
$(A) P(X_1^c \mid X) = \frac{3}{16}$
$(B) P(\text{Exactly two engines are functioning} \mid X) = \frac{7}{8}$
$(C) P(X \mid X_2) = \frac{5}{16}$
$(D) P(X \mid X_1) = \frac{7}{16}$

If $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then

There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order until both the faulty machines are identified. The probability that only two tests are needed is:

First bag contains $3$ red and $5$ black balls and second bag contains $6$ red and $4$ black balls. $A$ ball is drawn from each bag. The probability that one ball is red and the other is black,is

Two persons $A$ and $B$ are throwing an unbiased six-faced dice alternatively,with the condition that the person who throws $3$ first wins the game. If $A$ starts the game,then the probabilities of $A$ and $B$ winning the game are,respectively: