Exercise 2.10

Answers

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

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

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2021-12-05 00:00