In a game,a person wins $5$ rupees for getting a number greater than $4$ and loses $1$ rupee otherwise,when a fair die is thrown. $A$ man participates in the game and decides to quit as soon as he gets a number greater than $4$. If he plays for a maximum of $3$ throws,what is the expected value (mean value) of the amount he wins or loses?

In a tournament,there are $12$ players $P_1, P_2, P_3, \dots, P_{12}$ divided into $6$ pairs at random. From each game,a winner is decided based on the game played between the two players of the pair. Assuming each player is of equal strength,what is the probability that exactly one out of $P_1$ and $P_2$ is among the losers?

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

For the three events $A, B$ and $C$,$P$ (exactly one of the events $A$ or $B$ occurs) = $P$ (exactly one of the events $B$ or $C$ occurs) = $P$ (exactly one of the events $C$ or $A$ occurs) = $p$ and $P$ (all the three events occur simultaneously) = $p^2$,where $0 < p < 1/2$. Then the probability of at least one of the three events $A, B$ and $C$ occurring is

Two distinct numbers $a$ and $b$ are chosen randomly from the set $S = \{2^1, 2^2, 2^3, \dots, 2^{25}\}$. What is the probability that $\log_2(ab)$ is an integer?

Two players $A$ and $B$ are alternately throwing a coin and a die together. $A$ player who first throws a head and a $6$ wins the game. If $A$ starts the game,then the probability that $B$ wins the game is: