Mystery Coin Identification

A box contains three coins, each equally likely to be selected:

• Coin F (fair): $P(H) = \frac{1}{2}$
• Coin H (heads-biased): $P(H) = \frac{3}{4}$
• Coin T (tails-biased): $P(H) = \frac{1}{4}$

You pick one coin uniformly at random and flip it three times, obtaining the sequence H, T, H (flips are conditionally independent given the coin).

(a) Find the posterior probability that the coin is each type.

(b) What is the conditional probability that the fourth flip is heads?