problem 12

(a): Let $A_{i}$ be the event that Alice sends bit $i$ . Let $B_{j}$ be the event that Bob recieves bit $j$ .

$\begin{matrix} P (A_{1} | B_{1}) & = \frac{P (A_{1}) P (B_{1} | A_{1})}{P (A_{1}) P (B_{1} | A_{1}) + P (A_{0}) P (B_{1} | A_{0})} = \frac{0.5 * 0.9}{0.5 * 0.9 + 0.5 * 0.05} = 0.95 \end{matrix}$
(b): Let $B_{j, k, l}$ be the event that Bob recieves bit tuple $j, k, l$ .

$\begin{matrix} P (A_{1} | B_{110}) & = \frac{P (A_{1}) P (B_{110} | A_{1})}{P (A_{1}) P (B_{110} | A_{1}) + P (A_{0}) P (B_{110} | A_{0})} = \frac{0.5 * 0 . 9^{2} * 0.1}{0.5 * 0 . 9^{2} * 0.1 + 0.5 * 0.0 5^{2} * 0.95} = 0.97 \end{matrix}$