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 $P(0 < X \le 2)$.

$A$ fair coin is tossed $4$ times. If $X$ is a random variable which indicates the number of heads,then $P[X < 3] = $

$A$ boy rolls a die once. If an even number appears, the number of chocolates the boy gets is equal to two more than the number that appeared. If an odd number appears on the die, the number of chocolates he gets is equal to three more than the number that appeared. If a random variable $X$ represents the number of chocolates the boy receives, then the range of $X$ is:

The range of a discrete random variable $X$ is $\{1, 2, 3\}$ and the probabilities of its elements are given by $P(X=1) = 3k^3$, $P(X=2) = 2k^2$, and $P(X=3) = 7 - 19k$. Then $P(X=3) = $

If a die is thrown at random,then the expectation of the number on it is

Which of the following can not be a valid assignment of probabilities for outcomes of sample space $S = \{\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}, \omega_{5}, \omega_{6}, \omega_{7}\}$?

Outcome	Probability
$\omega_{1}$	$0.1$
$\omega_{2}$	$0.01$
$\omega_{3}$	$0.05$
$\omega_{4}$	$0.03$
$\omega_{5}$	$0.01$
$\omega_{6}$	$0.2$
$\omega_{7}$	$0.6$