Question

Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?

Answer 1

Consider the given events.
A = Both the children are girls.
B = The youngest child is a girl.
C = At least one child is a girl.

\[\text{ Clearly } , \]
\[S = \left\{ B_1 B_2 , B_1 G_2 , G_1 B_2 , G_1 G_2 \right\}\]
\[A = \left\{ G_1 G_2 \right\}\]
\[B = \left\{ B_1 G_2 , G_1 G_2 \right\} \]
\[C = \left\{ B_1 G_2 , G_1 B_2 , G_1 G_2 \right\}\]

\[A \cap B = \left\{ G_1 G_2 \right\} \text{ and }  A \cap C = \left\{ G_1 G_2 \right\}\]
\[\left( i \right) \text { Required probability } = P\left( A/B \right) = \frac{n\left( A \cap B \right)}{n\left( B \right)} = \frac{1}{2}\]
\[\left( ii \right) \text{ Required probability }  = P\left( A/C \right) = \frac{n\left( A \cap B \right)}{n\left( C \right)} = \frac{1}{3}\]