Exercise 2.62

Let $D$ be the event that the mother has the disease. Let $C_i$ be the event that the $i$-th child has the disease.

(a)

$$\begin{array}{llll}\hfill P(C_1^c\cap C_2^c)& =P(D)P(C_1^c\cap C_2^c|D)+P(D^c)P(C_1^c\cap C_2^c|D^c)& \hfill & \\ \hfill & =\frac{1}{3}\ast \frac{1}{4}+\frac{2}{3}& \hfill & \\ \hfill & =\frac{9}{12}& \hfill & \end{array}$$

(b)

The two events are not independent. If the elder child has the disease, the mother has the disease, which means the younger child has probability $\frac{1}{2}$ of having the disease. Unconditionally, the younger child has probability $\frac{1}{6}$ of having the disease.

(c)

$$\begin{array}{llll}\hfill P(D|C_1^c\cap C_2^c)& =\frac{P(D)P(C_1^c\cap C_2^c|D)}{P(C_1^c\cap C_2^c)}& \hfill & \\ \hfill & =\frac{\frac{1}{3}\ast \frac{1}{4}}{\frac{1}{3}\ast \frac{1}{4}+\frac{2}{3}}& \hfill & \\ \hfill & =\frac{1}{9}& \hfill & \end{array}$$