Two dice are rolled. If $A$ denotes the event that the same number shows on each die and $B$ denotes the event that the sum of the numbers on both dice is greater than $7$,then $P(A \mid B)$ and $P(B \mid A)$ respectively are

Given that $A$ and $B$ are two events such that $P(B) = \frac{3}{5}$,$P\left(\frac{A}{B}\right) = \frac{1}{2}$,and $P(A \cup B) = \frac{4}{5}$,then $P(A)$ is equal to:

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

The probabilities of Virat becoming the Man of the Match in the $1^{st}$,$2^{nd}$,and $3^{rd}$ matches of India in the World Cup $2015$ are $\frac{3}{7}$,$\frac{2}{7}$,and $\frac{1}{7}$ respectively. If Virat got the Man of the Match in exactly one match,what is the probability that he got it in the $3^{rd}$ match?

Let $E^c$ denote the complement of an event $E$. Let $E, F, G$ be pairwise independent events with $P(G)>0$ and $P(E \cap F \cap G) = 0$. Then $P(E^c \cap F^c \mid G)$ equals

$A$ candidate takes three tests in succession and the probability of passing the first test is $p$. The probability of passing each succeeding test is $p$ if he passes the preceding one, and $\frac{p}{2}$ if he fails the preceding one. The candidate is selected if he passes at least two tests. The probability that the candidate is selected is: