$A$ pair of fair dice is rolled together till a sum of either $5$ or $7$ is obtained. Then the probability that $5$ comes before $7$ is

In a single throw of three dice,the probability of getting a sum at least $5$ is

Two players,$P_1$ and $P_2$,play a game against each other. In every round,each player rolls a fair die once. Let $x$ and $y$ denote the outcomes for $P_1$ and $P_2$. If $x > y$,$P_1$ scores $5$ points and $P_2$ scores $0$. If $x = y$,each scores $2$ points. If $x < y$,$P_1$ scores $0$ and $P_2$ scores $5$. Let $X_n$ and $Y_n$ be the total scores of $P_1$ and $P_2$ after $n$ rounds. Match the following:

List-$I$	List-$II$
$(I)$ Probability of $(X_2 \geq Y_2)$ is	$(P)$ $\frac{3}{8}$
$(II)$ Probability of $(X_2 > Y_2)$ is	$(Q)$ $\frac{11}{16}$
$(III)$ Probability of $(X_3 = Y_3)$ is	$(R)$ $\frac{5}{16}$
$(IV)$ Probability of $(X_3 > Y_3)$ is	$(S)$ $\frac{355}{864}$
	$(T)$ $\frac{77}{432}$

Two friends $A$ and $B$ meet every weekend either at a party or at a Sports Club. The probability that they meet at the Sports Club is $\frac{4}{9}$. The probability that they dine together at a party and at the Club are respectively $\frac{1}{3}$ and $\frac{2}{5}$. On a certain weekend, the probability that they disperse without dining together is:

Two fair dice,each with faces numbered $1, 2, 3, 4, 5$ and $6$,are rolled together and the sum of the numbers on the faces is observed. This process is repeated until the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number,then the value of $14p$ is . . . . .

$A$ and $B$ toss a fair coin each simultaneously $50$ times. The probability that both of them will not get tail at the same toss is