If four dice are thrown simultaneously,then the probability that none of the dice shows the number $1$ on its face is:

For $k > 0$,calculate the value of $\sum_{x=0}^{\infty} \frac{k^x}{x !} \lim _{n}$ ${\rightarrow \infty} \frac{n !}{(n-x) !}\left(1-\frac{k}{n}\right)^{n-x}\left(\frac{1}{n}\right)^x$.

If $X \sim B(9, p)$ is a binomial variate satisfying the equation $P(X=3)=P(X=6)$,then $P(X < 3)=$

In a Binomial distribution $B(n, p)$,the sum and product of the mean and the variance are $5$ and $6$ respectively,then $6(n+p-q)=$

$A$ die is thrown thrice. If getting $1$ or $6$ in a single throw is considered as success, then the variance of the number of successes is

Let in a Binomial distribution,consisting of $5$ independent trials,probabilities of exactly $1$ and $2$ successes be $0.4096$ and $0.2048$ respectively. Then the probability of getting exactly $3$ successes is equal to ....... .