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

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

$S=\left\{\right(B,B,B),(G,G,G),(B,G,G),(B,B,G),(B,G,B\left)\right(G,B,B),(G,B,G),(G,G,B\left)\right\}$

$E_1=eventthatafamilyhasatleastonegirl$

Then $E_1$ $=\left\{\right(G,B,B),(B,G,B),(B,B,G),(G,G,B),(B,B,G),(G,B,G\left)\right(G,G,G)$

$E_2=Eventthattheelderchildisagirl$

Then $E_2$ $=\left\{\right(G,B,B),(G,G,B\left)\right(G,B,G),(G,G,G)$

$E_1\cap E_2=(G,B,B),(G,G,B),(G,B,G),(G,G,G)$

$P\left(\frac{E_2}{E_1}\right)=\frac{P(E_2\cap E_1)}{P\left(E_1\right)}$

$$\begin{gathered} P\left(\frac{E_2}{E_1}\right)=\frac{{\displaystyle \frac{4}{8}}}{{\displaystyle \frac{7}{8}}} \\ =\frac{4}{7} \end{gathered}$$

Hence, the probability that the eldest child is a girl given that the family has at least one girl is $\frac{4}{7}$.