Question

In a family with two children,find the mean of the number of children who will be girls

Answer 1

Let $X=$  the number of girls in a family with two children.

$$P(X=0)=P(BB)=(\dfrac{1}{2})^2=\dfrac{1}{4}$$

$$P(X=1)=P(GB)+P(BG)=(\dfrac{1}{2})^2+(\dfrac{1}{2})^2=\dfrac{1}{2}$$

$$P(X=2)=P(GG)=(\dfrac{1}{2})^2=\dfrac{1}{4}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 \\ \hline p(x) & 1/4 & 1/2 & 1/4 \\ \end{array}$$

$$mean=\mu=E(X)=\sum _ix_ip(x_i)$$

$$=0(\dfrac{1}{4})+1(\dfrac{1}{2})+2(\dfrac{1}{4})=1$$