Let the mean and variance of $7$ observations $2, 4, 10, x, 12, 14, y$ (where $x > y$) be $8$ and $16$ respectively. Two numbers are chosen from the set $\{1, 2, 3, x-4, y, 5\}$ one after another without replacement. Then the probability that the smaller number among the two chosen numbers is less than $4$ is:

$A$ and $B$ are two candidates seeking admission in a college. The probability that $A$ is selected is $0.7$ and the probability that exactly one of them is selected is $0.6$. Find the probability that $B$ is selected.

An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is $2/3$,then the eccentricity of the ellipse is:

In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of $52$ playing cards by a person $B$. They win the game if $A$ gets a prime score as the sum of the numbers appearing on both the dice and $B$ gets a face card and a card having a prime number. Then the probability that both $A$ and $B$ win is:

Three numbers are chosen at random,one after another with replacement,from the set $S = \{1, 2, 3, \ldots, 100\}$. Let $p_1$ be the probability that the maximum of the chosen numbers is at least $81$ and $p_2$ be the probability that the minimum of the chosen numbers is at most $40$.
$(1)$ The value of $\frac{625}{4} p_1$ is
$(2)$ The value of $\frac{125}{4} p_2$ is

The probability that the product of the outcomes when three dice are rolled simultaneously is divisible by $4$ is equal to