$A$ and $B$ are independent events. If $P(A \cup B)=0.5$ and $P(A)=0.2$,then $P(B) = $ . . . . . . . (in $/8$)

Let $\alpha$ be a root of $x^2+x+1=0$ and suppose that a fair die is thrown $3$ times. If $a, b,$ and $c$ are the numbers shown on the die,then the probability that $\alpha^a+\alpha^b+\alpha^c=0$ is

All face cards from a pack of $52$ playing cards are removed. From the remaining $40$ cards,two are drawn randomly without replacement. The probability of drawing a pair (cards of the same denomination) is:

Three students $S_1, S_2$ and $S_3$ are given a problem to solve. Consider the following events:
$U:$ At least one of $S_1, S_2$ and $S_3$ can solve the problem,
$V: S_1$ can solve the problem,given that neither $S_2$ nor $S_3$ can solve the problem,
$W: S_2$ can solve the problem and $S_3$ cannot solve the problem,
$T: S_3$ can solve the problem.
For any event $E$,let $P(E)$ denote the probability of $E$.
If $P(U)=\frac{1}{2}, P(V)=\frac{1}{10}$ and $P(W)=\frac{1}{12}$,then $P(T)$ is equal to

Let $S = \{w_1, w_2, \ldots\}$ be the sample space associated with a random experiment. Let $P(w_n) = \frac{P(w_{n-1})}{2}$ for $n \geq 2$. Let $A = \{2k + 3\ell : k, \ell \in \mathbb{N}\}$ and $B = \{w_n : n \in A\}$. Then $P(B)$ is equal to:

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C}) = \frac{1}{2}$,$P(A) > 0$,$P(B) = b$,and $P(C) = c$,then $P((\bar{B} \cap \bar{C}) \mid A) = $