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 a dice is thrown $5$ times,then the probability of getting $6$ exactly three times is:

If the mean and variance of a binomial variate $X$ are $\frac{4}{3}$ and $\frac{8}{9}$ respectively,then $P(X=2)=$

$A$ man tosses a coin $10$ times,scoring $1$ point for each head and $2$ points for each tail. Let $P(K)$ be the probability of scoring at least $K$ points. The largest value of $K$ such that $P(K) > \frac{1}{2}$ is

If $X$ has a binomial distribution with mean $np$ and variance $npq$,then $\frac{P(X = k)}{P(X = k - 1)}$ is

For an initial screening of an entrance exam,a candidate is given $50$ problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5}$,then the probability that he is unable to solve less than $2$ problems is: