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

We have, 
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)} 
E₁ = Event that a family has at least one girl, then
E₁ = {(G, B, B),(B, G, B),(B, B, G),(G, G, B),(B, G, G),(G, B, G),(G, G, G)}
E₂ = Event that the eldest child is a girl, then
E₂ = {(G, B, B),(G, G, B),(G, B, G),(G, G, G)}
$\begin{array}{l}\therefore {\mathrm{E}}_1\cap {\mathrm{E}}_2=\left\{\right(\mathrm{G},\mathrm{B},\mathrm{B}),(\mathrm{G},\mathrm{G},\mathrm{B}),(\mathrm{G},\mathrm{B},\mathrm{G}),\\ \therefore \mathrm{P}\left({\mathrm{E}}_2\mid {\mathrm{E}}_1\right)=\frac{\mathrm{P}\left({\mathrm{E}}_1\cap {\mathrm{E}}_2\right)}{\mathrm{P}\left({\mathrm{E}}_1\right)}\\ =\frac{\frac{4}{8}}{\frac{7}{8}}\\ =\frac{4}{7}\end{array}$