Consider $5$ independent Bernoulli trials,each with a probability of success $p$. If the probability of at least one failure is greater than or equal to $31/32$,then in which interval does $p$ lie?

Suppose $X$ follows a binomial distribution with parameters $n$ and $p$,where $0 < p < 1.$ If $\frac{P(X = r)}{P(X = n - r)}$ is independent of $n$ and $r$,then

The items produced by a firm are supposed to contain $5\%$ defective items. The probability that a sample of $8$ items will contain less than $2$ defective items is:

$A$ and $B$ are playing a chess game with each other. The probability that $A$ wins the game is $0.6$,the probability that he loses is $0.3$,and the probability that it is a draw is $0.1$. If they play three games,what is the probability that $A$ wins at least two games?

The probability of a bomb hitting a bridge is $\frac{1}{2}$ and two direct hits are needed to destroy it. The least number of bombs required so that the probability of the bridge being destroyed is greater than $0.9$ is:

Find the binomial probability distribution whose mean is $3$ and variance is $2$.