# Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is

Assume that in a family, each child is equally likely to be a boy or a girl.

$$S=\{(B, B, B),(G, G, G),(B, G, G),(G, B, G),(G, G, B),(G, B, B),(B, G, B) (B, B, G)\}\\ E_{1}= \text {Event that a family has atleast one girl, then } E_{1}=\{(G, B, B),(B, G, B),(B, B, G),(G, G, B),(B, G, G),(G, B, G),(G, G, G)\}\\ E_{2}= \text { Event that the eldest child is a girl, then } E_{2}=\{(G, B, B),(G, G, B),(G, B, G),(G, G, G)\}\\ E_{1} \cap E_{2}=\{(G, B, B),(G, G, B),(G, B, G),(G, G, G)\}\\ P\left(E_{2} / E_{1}\right)=\frac{P\left(E_{1} \cap E_{2}\right)}{P\left(E_{1}\right)}=\frac{\frac{4}{8}}{\frac{7}{8}}=\frac{4}{7}$$