If $A$ and $B$ are two events such that $P(A) = \frac{1}{3}$,$P(B) = \frac{1}{5}$,and $P(A \cup B) = \frac{1}{3}$,then the value of $P(A^{\prime} | B^{\prime}) + P(B^{\prime} | A^{\prime})$ is:

If $P(A \cap B) = \frac{7}{10}$ and $P(B) = \frac{17}{20}$,where $P$ stands for probability,then $P(A \mid B)$ is equal to:

Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{0, 1\}$. Let the events $E_1$ and $E_2$ be given by $E_1 = \{A \in S : \operatorname{det} A = 0\}$ and $E_2 = \{A \in S : \text{sum of entries of } A \text{ is } 7\}$. If a matrix is chosen at random from $S$,then the conditional probability $P(E_1 \mid E_2)$ equals:

If a die is rolled twice and the sum of the numbers appearing on them is observed to be $6$, then the probability that the number $1$ appears at least once on them is

Evaluate $P(A \cup B)$,if $2 P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$.

Let $A, B, C$ be pairwise independent events with $P(C) > 0$ and $P(A \cap B \cap C) = 0$. Then $P(A' \cap B' | C) = $