How many flips until you see H-T-H?

Question

Flip a fair coin repeatedly and stop the first time you see the pattern H, T, H in three consecutive flips. What is the expected number of flips?

Accepted Answer

Track how much of the target pattern you have built up. The states are: start (no progress), "H" (matched the first H), "HT" (matched H then T), and done. Let e0, eH, e{HT} be the expected additional flips from each. From start. Flip H (prob 	frac12) 	o state H; flip T (prob 	frac12) 	o still start (a leading T is useless): e0 = 1 + 	frac12 eH + 	frac12 e0 \implies e0 = 2 + eH. From state H. Flip H 	o still state H (the new H restarts the pattern); flip T 	o state HT: eH = 1 + 	frac12 eH + 	frac12 e{HT} \implies eH = 2 + e{HT}. From state HT. Flip H 	o done (we have H-T-H). Flip T …

How many flips until you see H-T-H?

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