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 atleast one girl is

Here, 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)

${E_1} =$ 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)}

${E_2} =$ Event that the eldest child is a girl, then

${E_2} =${(G, B, B),(G, G, B),(G, B, G),(G, G, G)}
$\therefore$ ${E_1} \cap {E_2} = \{ (G,B,B),(G,G,B),(G,B,G),(G,G,G)\}$

$\therefore$ $P\left( {{E_2}/{E_1}} \right) = \frac{{P\left( {{E_1} \cap {E_2}} \right)}}{{P\left( {{E_1}} \right)}} = \frac{{4/8}}{{7/8}} = \frac{4}{7}$