Sequential Signal Updating and the Tower Property

A quant researcher believes a directional signal has accuracy $p$ that is either $\frac{1}{3}$ or $\frac{2}{3}$, each equally likely a priori. On each day the signal independently (given $p$) predicts the market direction, and is correct with probability $p$.

(a) On Day 1 the signal is correct. What is the posterior $P\!\left(p = \tfrac{2}{3} \mid C_1\right)$?

(b) On Day 2 the signal is wrong. Starting from the Day-1 posterior, compute the updated $P\!\left(p = \tfrac{2}{3} \mid C_1, W_2\right)$.

(c) Verify the tower property of conditional expectation: show that $E[p] = E\!\left[\,E[p \mid D_1]\,\right]$, where $D_1 \in \{C_1, W_1\}$ denotes the Day-1 outcome. Compute all quantities explicitly.