problem 10

(a): $\begin{matrix} P (A_{3} | A_{1}) & = P (A_{2} ∣ ∣ A_{1}) P (A_{3} ∣ ∣ A_{2}, A_{1}) + P (A_{2}^{c} ∣ ∣ A_{1}) P (A_{3} ∣ ∣ A_{2}^{c}, A_{1}) = 0.8 * 0.8 + 0.2 * 0.3 = 0.7 \end{matrix}$
(b): $\begin{matrix} P (A_{3} ∣ ∣ A_{1}^{c}) & = P (A_{2} ∣ ∣ A_{1}^{c}) P (A_{3} ∣ ∣ A_{2}, A_{1}^{c}) + P (A_{2}^{c} ∣ ∣ A_{1}^{c}) P (A_{3} ∣ ∣ A_{2}^{c}, A_{1}^{c}) = 0.3 * 0.8 + 0.7 * 0.3 = 0.45 \end{matrix}$

$\begin{matrix} P (A_{3}) & = P (A_{1}) P (A_{3} ∣ ∣ A_{1}) + P (A_{1}^{c}) P (A_{3} ∣ ∣ A_{1}^{c}) = 0.75 * 0.7 + 0.25 * 0.45 = 0.64 \end{matrix}$