$A$ 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

If the difference between the mean and variance of a binomial variate is $\frac{5}{9}$,then the probability of getting $2$ successes for an event when the experiment is conducted $5$ times,is

In a binomial distribution,the probability of success is $\frac{1}{4}$ and the standard deviation is $3$. Then,its mean is

If the mean and the variance of a Binomial variate $X$ are $2$ and $1$ respectively,then the probability that $X$ takes a value greater than or equal to $1$ is

The probability of success $p$ for the Binomial distribution satisfying the relation $4 P(X=4) = P(X=2)$ with parameter $n=6$ is

$X$ denotes the number of times heads occur in $n$ tosses of a fair coin. If $P(X=4)$,$P(X=5)$,and $P(X=6)$ are in arithmetic progression,the largest value of $n$ is