The mean of the numbers obtained on throwing a die having written $1$ on three faces,$2$ on two faces,and $5$ on one face is

If $X$ is a random variable with probability distribution $P(X=k) = \frac{(2k+3)c}{3^k}$, $k=0, 1, 2, \ldots, \infty$, then $P(X=3) =$

Two persons $A$ and $B$ play a game by throwing two dice. If the sum of the numbers appeared on the two dice is even, $A$ will get $\frac{1}{2}$ point and $B$ will get $\frac{1}{2}$ point. If the sum is odd, $A$ will get one point and $B$ will get no point. The arithmetic mean of the random variable of the number of points of $A$ is

$A$ fair $n$-faced die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$,then $n=$ (where $n \in N$ ).

$A$ random variable $X$ takes the values $1, 2, 3$ and $4$ such that $2 P(X=1) = 3 P(X=2) = P(X=3) = 5 P(X=4)$. If $\sigma^2$ is the variance and $\mu$ is the mean of $X$, then $\sigma^2 + \mu^2 =$

In a book of $250$ pages,there are $200$ typographical errors. Assuming that the number of errors per page follows the Poisson distribution,the probability that a random sample of $5$ pages will contain no typographical error is