Let $A, B, C$ be pairwise independent events such that $P(C) > 0$ and $P(A \cap B \cap C) = 0$. Then $P(A' \cap B'|C)$ is equal to:

Two numbers are selected at random from the set $\{1, 2, 3, \ldots, 13\}$. If the sum of the selected numbers is even, what is the probability that both the numbers are odd?

Let $E_{1}$ and $E_{2}$ be two events such that the conditional probabilities $P(E_{1} \mid E_{2}) = \frac{1}{2}$,$P(E_{2} \mid E_{1}) = \frac{3}{4}$ and $P(E_{1} \cap E_{2}) = \frac{1}{8}$. Then:

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails } | \text{ } B \text{ has failed})$.

If $E_{1}$ denotes the event of getting a sum of $6$ when throwing two dice and $E_{2}$ is the event of getting a $2$ on at least one of the two dice,then $P(E_{2} / E_{1})$ is (in $/ 5$)

Two dice are thrown simultaneously. If $A$ is the event of getting the sum of the numbers on two dice as greater than or equal to $8$ and $B$ is the event of getting a number less than or equal to $3$ on at least one of the dice. Then,$P(B / A) = $