Sushma has $2$ children. Let $A$ be the event that the oldest child is a girl, $B$ be the event that at least one child is a girl, and $C$ be the event that both children are girls. Which of the following are correct statements:
(a) $P(C\mid A)=\frac{1}{2}.$
(b) $P(C\mid B)=\frac{1}{2}.$
(c) $P(C\mid A)=\frac{1}{3}.$
(d) $P(C\mid B)=\frac{1}{3}.$
I tried doing it by venn diagram but couldn't get the result.
By definition, $P(C|A) = \frac{P(C \cap A)}{P(A)}$ and $P(C|B) = \frac{P(C \cap B)}{P(B)}$.
If we assume $P(A) = \frac{1}{2}$, then $P(C|A) = \frac{0.5 \times 0.5}{0.5} = \frac{1}{2}$.
Clearly $P(B) =\frac{3}{4}$ and so $P(C|B) = \frac{0.5 \times 0.5}{0.75} = \frac{1}{3}$.
So (a) and (d) are the only true statements here.
Alternatively, we can use some quick intuition. $P(C|A)$ means the probability that both children are girls, given that the oldest child is a girl. Clearly, this is just the probability of the youngest child being a girl: $\frac{1}{2}$. $P(C|B)$ means the probability that both children are girls, given that at least one of them is a girl. In this case, we see that there are three possibilities: the oldest child being a girl, with the youngest being a boy, the oldest being a boy with the youngest being a girl, and both children being girls. Of these three possibilities, only one of them meets our criteria, so clearly the probability is $\frac{1}{3}$.