Waiting Time and Occurrence Race: HT versus HH

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Markov Chains / Renewal Theory.

Your Task
A fair coin is tossed repeatedly.
(1) Compute the expected number of tosses to see the pattern $HT$ (Heads followed immediately by Tails).
(2) Compute the expected number of tosses to see the pattern $HH$ (two consecutive Heads).
(3) In a two-player game where Player 1 chooses $HT$ and Player 2 chooses $HH$, what is the probability that $HT$ appears before $HH$?

Constraints

• All trials are mutually independent and identically distributed.
• Probabilities lie strictly within $(0, 1)$ unless degenerate.