Compute posterior and predictive coin probabilities

There are two coins: F is fair, and B is biased with $P(H)=0.7$. You choose one coin uniformly at random and toss it three times. The first two tosses come up Heads, while the third toss has not yet been observed. a) Compute $P(\text{picked } B \mid \text{first two flips are Heads})$. b) Compute $P(\text{next flip is Heads} \mid \text{first two flips are Heads})$. c) Continuing with the same coin, let $T$ be the total number of flips until the first Tail occurs, including that flip. Compute $E[T \mid \text{first two flips are Heads}]$, showing your derivation.

Overview: This problem tests Bayesian inference and probabilistic reasoning, specifically posterior updating, posterior predictive probability, and the conditional expectation of a stopping time after some outcomes have been observed.