Which comes first, HHT or THH

Question

A fair coin is tossed repeatedly until either the run HHT or the run THH has appeared. What is the probability that HHT appears first?

Short answer

$1/4$ . If the first two tosses are HH, HHT must come first (the next T completes it). Otherwise a T appears before any HH, and from then on the first HH completes THH before HHT is possible. So HHT wins only with probability $\tfrac14$ .

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 5.1 Markov Chain, 'Coin triplets B', p.110; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Markov Chains — Transition Matrices and the Markov Property
Exit Distributions and Exit Times
Sample Spaces, Events, and the Probability Axioms

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