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?

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Solution

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.

$Clearly, S = \{B_{1} B_{2}, B_{1} G_{2}, G_{1} B_{2}, G_{1} G_{2}\} A = \{G_{1} G_{2}\} B = \{B_{1} G_{2}, G_{1} G_{2}\} C = \{B_{1} G_{2}, G_{1} B_{2}, G_{1} G_{2}\}$

$A \cap B = \{G_{1} G_{2}\} and A \cap C = \{G_{1} G_{2}\} (i) Required probability = P (A / B) = \frac{n (A \cap B)}{n (B)} = \frac{1}{2} (ii) Required probability = P (A / C) = \frac{n (A \cap B)}{n (C)} = \frac{1}{3}$

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