Waiting times for HHT and for THH

Question

A fair coin is flipped until a given three-flip pattern appears. Find the expected number of flips needed to observe HHT, and separately the expected number needed to observe THH.

Short answer

$8$ in both cases. Neither pattern has a proper prefix equal to a suffix, so there is no overlap to exploit and the waiting time is just $2^3=8$ (confirmable with a small Markov chain on the matched prefix).

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 5.1 Markov Chain, 'Coin triplets A', p.108; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Markov Chains — Transition Matrices and the Markov Property
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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