In a game,$3$ coins are tossed. $A$ person is paid $₹150$ if he gets all heads or all tails,and he is supposed to pay $₹50$ if he gets one head or two heads. The amount he can expect to win or lose on an average per game in $₹$ is:

If a discrete random variable $X$ has the probability distribution as follows:

$X = x$	$0$	$1$	$2$	$3$
$P(X = x)$	$k$	$3k$	$3k$	$k$

Then $Var(X) = $

If $3$ is the variance of a Poisson distribution,then $P(1 < x < 4) = $

$A$ person throws two fair dice. He wins $Rs.\, 15$ for throwing a doublet (same numbers on the two dice),wins $Rs.\, 12$ when the throw results in the sum of $9$,and loses $Rs.\, 6$ for any other outcome on the throw. Then the expected gain/loss (in $Rs.$) of the person is

$A$ 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 $\sigma^2 + \mu^2 =$

If a fair coin is tossed $5$ times,the probability that heads does not occur two or more times in a row is