If $X$ is a Poisson variate such that $\frac{5}{3} k = P(X=2) = P(X=3)$, then $P(X=5) =$

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that both balls are red.

If the range of a random variable $X$ is $\{0, 1, 2, 3, 4, \ldots\}$ with $P(X=k) = \frac{(k+1)a}{3^k}$ for $k \geq 0$,then $a$ is equal to

In a game,$3$ coins are tossed. $A$ person is paid ₹ $7$ if he gets all heads or all tails,and he is supposed to pay ₹ $3$ if he gets one head or two heads. The amount he can expect to win on an average per game is ₹

The probability distribution of a random variable $X$ is given below.

$X = x$	$0$	$1$	$2$	$3$
$P(X = x)$	$\frac{1}{10}$	$\frac{2}{10}$	$\frac{3}{10}$	$\frac{4}{10}$

Then the variance of $X$ is

Two balls are drawn at random from a bag containing $5$ black balls and $3$ white balls. If the random variable $X$ denotes the number of white balls drawn, then the mean of $X$ is