$A$ box contains $100$ balls,numbered from $1$ to $100$. If $3$ balls are selected one after the other at random with replacement from the box,then the probability that the sum of the three numbers on the balls selected is an odd number,is

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C}) = \frac{1}{2}$,$P(A) > 0$,$P(B) = b$,and $P(C) = c$,then $P((\bar{B} \cap \bar{C}) \mid A) = $

Two balls are drawn from an urn containing $7$ white,$6$ red,and $8$ black balls one after the other without replacement. The probability that at least one of them is white is:

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:

In a throw of three dice,the probability that at least one die shows up $1$,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