Question

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

Answer 1

It is given that the family has three children. Sample space,
$S=\{\begin{array}{ccc}(B,B,B),(B,B,B),(B,G,B,(B,G,G)& & \\ (G,B,B)(G,B,G)& & \\ (G,G,B),(G,G,G)& & \end{array}$
Let, $E_1$= The event that the family has at least one girl. $E_1=\{\begin{array}{ccc}(B,B,G),(B,G,B),(B,G,G),(G,B,B)& & \\ (G,B,G)(G,G,B),(G,G,G)& & \end{array}$and $E_2$= The event that the eldest child is a girl.
and, $E_2=(G,B,B),(G,B,G),(G,G,B),(G,G,G)$
$\because E_1\cap E_2=(G,B,B),(G,B,G),(G,G,B),(G,G,G)$ $P\left(E_1\right)=\frac{7}{8},P(E_1\cap E_2)=\frac{4}{8}=\frac{1}{2}$
Now,
$P\left(\frac{E_2}{E_1}\right)=\frac{P(E_1\cap E_2)}{P(E_1}=\frac{\frac{1}{2}}{\frac{7}{8}}=\frac{4}{7}$
Hence, the correct option is (𝐷).