Let $A$ and $B$ be two events such that $P(A) = \frac{5}{11}$, $P(B) = \frac{2}{11}$, and $P(A \cup B) = \frac{3}{11}$, then $P(A'|B')$ is . . . . . . .

$A$ coin is tossed until a head appears or it has been tossed thrice. Given that a head does not appear on the first toss,what is the probability that the coin is tossed thrice?

Let $B(\alpha, \beta, \gamma)$ represent that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls, and $\gamma$ blue balls. Given $B_1(2, 3, 2)$, $B_2(3, 2, 2)$, $B_3(2, 2, 3)$. $A$ die is rolled. If the die shows $2, 3,$ or $5$, a ball is drawn from bag $B_1$. If the die shows $4$ or $6$, a ball is drawn from bag $B_2$. If the die shows $1$, a ball is drawn from bag $B_3$. Find the probability of drawing a green ball.

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of $4$. If $p$ is the conditional probability that number $4$ has appeared at least once, then $3p + 2 =$

Let $A$ and $B$ be two independent events such that $P(A) + P(B) = \frac{3}{4}$ and $P(\overline{A} | B) = \frac{2}{5}$. Then,$P(A \cap B)$ is -

If $E_1$ and $E_2$ are two events of a random experiment such that $P(E_1) = \frac{1}{8}$,$P(E_1 \mid E_2) = \frac{1}{3}$,and $P(E_2 \mid E_1) = \frac{1}{4}$,then match the items of List-$I$ with the items of List-$II$.

List-$I$	List-$II$
$A. P(E_1 \cup E_2)$	$I. \frac{3}{29}$
$B. P(E_2)$	$II. \frac{26}{29}$
$C. P(E_1 \mid \bar{E}_2)$	$III. \frac{3}{16}$
$D. P(\bar{E}_1 \mid \bar{E}_2)$	$IV. \frac{3}{32}$