Noisy Signal Detection and Evidence Threshold

A hidden signal $S$ is equally likely to be $+1$ or $-1$. At each time step you receive a noisy reading: if $S = +1$ the reading is $+1$ with probability $\frac{2}{3}$ and $-1$ with probability $\frac{1}{3}$; if $S = -1$ the reading is $-1$ with probability $\frac{2}{3}$ and $+1$ with probability $\frac{1}{3}$. Readings are conditionally independent given $S$.

(a) You observe the sequence $(+1, +1, -1)$. Find the posterior probability $P(S = +1 \mid \text{observations})$.

(b) Starting from the uniform prior, what is the minimum number $n$ of consecutive $+1$ readings required so that $P(S = +1 \mid n \text{ consecutive } +1) > 0.95$?