The range of a random variable $X$ is $\{0, 1, 2\}$. If $P(X = 0) = 3c^3$, $P(X = 1) = 4c - 10c^2$, and $P(X = 2) = 5c - 1$, then find $P(0 < X \le 2)$.

If the probability function of a random variable $X$ is given by $P(X=k) = \frac{3^{ck}}{k!}$ for $k = 1, 2, 3, \ldots$ (where $c$ is a constant),then $c =$

India plays two matches each against Australia and West Indies. The probabilities of India scoring $0, 1,$ and $2$ points in a match are $0.45, 0.05,$ and $0.50$ respectively. Assuming the results are independent,what is the probability that India scores at least $7$ points?

For a random variable $X$,if $P(X=k) = \frac{(k+1)a}{3^k}$ for $k=0, 1, 2, \ldots$,then $a = $

$A$ fair die is tossed twice in succession. If $X$ denotes the number of sixes in two tosses,then the probability distribution of $X$ is given by

$A$ box contains $8$ red and $N$ green balls. Two balls are drawn at random from it. If $X$ is the random variable representing the number of green balls drawn and $E(X) = 1.2$, then $N = ...$