If a variable takes values $0, 1, 2, \dots, n$ with frequencies ${q^n}, \frac{n}{1}{q^{n - 1}}p, \frac{n(n - 1)}{1 \times 2}{q^{n - 2}}{p^2}, \dots, {p^n}$,where $p + q = 1$,then the mean is

The mean and standard deviation of a binomial variate $X$ are $4$ and $\sqrt{3}$ respectively. Then $P(X \geq 1)$ is equal to

In an experiment, a person gets success $\alpha$ times out of $\beta$ trials. If the experiment consists of $n$ trials, then the probability that he fails at least $(n-1)$ times is

An urn contains $25$ balls of which $10$ balls bear a mark $'X'$ and the remaining $15$ bear a mark $'Y'$. $A$ ball is drawn at random from the urn,its mark is noted down and it is replaced. If $6$ balls are drawn in this way,find the probability that not more than $2$ will bear $'Y'$ mark.

Find the probability of throwing at most $2$ sixes in $6$ throws of a single die.

The probability of a man hitting a target is $\frac{2}{3}$. What is the minimum number of times he must fire so that the probability of hitting the target at least once is more than $90 \%$?