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?

$A$ die is thrown four times. The probability of getting a perfect square in at least one throw is

$A$ die is thrown $5$ times. If getting an odd number is considered a success,what is the variance of the probability distribution?

If the mean and variance of a binomial variable $X$ are $2$ and $1$ respectively,then what is the probability that $X \geq 1$?

$A$ fair die with faces $1, 2, 3, 4, 5,$ and $6$ is tossed $4$ times. What is the probability that in all $4$ tosses,the value obtained is not less than $2$ and not greater than $5$ (in $/81$)?

There are $6$ boxes labelled $B_1, B_2, \ldots, B_6$. In each trial,two fair dice $D_1, D_2$ are thrown. If $D_1$ shows $j$ and $D_2$ shows $k$,then $j$ balls are put into the box $B_k$. After $n$ trials,what is the probability that $B_1$ contains at most one ball?