Exercise 2.7

Answers

(a): $\begin{array}{l} P (D | H) & = \frac{P (D) P (H | D)}{P (H)} \\ = \frac{P (D) P (H | D)}{P (D) P (H | D) + P (D^{c}) P (H | D^{c})} \\ = \frac{\frac{1}{2} (\frac{1}{100} + \frac{99}{100} {(\frac{1}{2})}^{7})}{\frac{1}{2} (\frac{1}{100} + \frac{99}{100} {(\frac{1}{2})}^{7}) + \frac{1}{2} {(\frac{1}{2})}^{7}} \\ = 0.69 \end{array}$
(b): Let $C$ be the event that the chosen coin is double-headed. $\begin{array}{l} P (C | H) & = P (D | H) P (C | D, H) + P (D^{c} | H) P (C | D^{c}, H) \\ = 0.69 * 0.56 + 0 \\ = 0.39 \end{array}$

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2021-12-05 00:00