Waiting time for the pattern HTH

Question

A fair coin is tossed repeatedly. What is the expected number of tosses needed until the sequence heads, tails, heads appears for the first time as three consecutive outcomes?

Short answer

$10$ . Set up states for progress (nothing, H, HT) and solve the first-step equations; equivalently, by the overlap (martingale/gambler) argument, the expected wait is $2^3$ plus $2^1$ for the self-overlap of HTH with its own prefix H, giving $8+2=10$ .

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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