Analyze HT vs HH stopping-time probabilities

We toss a fair coin repeatedly and stop at the first appearance of either HT (A wins) or HH (B wins). A tie cannot happen.

1. Find exact expressions for $P(A \text{ wins})$ and $P(B \text{ wins})$.

2. For $m \ge 2$, derive the probability mass function of the stopping time, giving $P(A \text{ wins at flip } m)$ and $P(B \text{ wins at flip } m)$.

3. Calculate the expected stopping time $E[T]$.

4. Generalize part (1) to a biased coin where $P(H)=p$, and give closed-form formulas for $P(A \text{ wins})$ and $P(B \text{ wins})$.

5. Verify your results by simulation with at least $10^6$ trials, and compare the empirical PMFs to the theoretical PMFs.

Overview: The question assesses understanding of discrete probability, stopping-time analysis, pattern occurrence in Bernoulli sequences, deriving PMFs and expectations, and generalizing to biased coins.