The minimum value of $Z = 3x + y$, subject to the constraints $2x + 3y \leq 6, x + y \geq 1, x \geq 0, y \geq 0$ is....

The centers for disease control have determined that when a person is given a vaccine, the probability that the person will develop immunity to a virus is $0.8$. If $8$ people are given this vaccine, then the probability that exactly $4$ will develop immunity is...

The Variance of the following probability distribution of X is

X	$0$	$1$	$2$	$3$
P(X)	$q^3$	$3q^2 p$	$3qp^2$	$p^3$

where $0 < p < 1, q = 1 - p$

If A and B are two events such that $P(A) = \dfrac{2}{3}$, $P(B) = \dfrac{1}{2}$ and $P(A \mid B) = \dfrac{2}{3}$, then $P(A' \cup B) + P(A \cup B') = $