$A$ die is thrown twice and the sum of the numbers appearing is observed to be $6$. What is the conditional probability that the number $4$ has appeared at least once?

$A$ family has two children. What is the probability that both the children are boys given that at least one of them is a boy?

If $P(A | B) > P(A)$,then which of the following is correct?

Two planes $I$ and $II$ drop bombs on a target. The probabilities of hitting the target by $I$ and $II$ are $0.3$ and $0.2$ respectively. The second plane drops the bomb only if the first plane fails to hit the target. What is the probability that the second plane hits the target?

If $E_1$ and $E_2$ are two events of the sample space such that $P(E_1) = \frac{1}{4}$, $P(E_1 | E_2) = \frac{1}{2}$ and $P(E_2 | E_1) = \frac{1}{3}$, then $P(E_1 | \bar{E}_2) = $

If $A$ and $B$ are any two events such that $P(A)+P(B)-P(A \cap B)=P(A)$,then