2.6.3 problem 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)

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

(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)

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