Flips until the pattern heads-heads-tails appears

Question

You flip a fair coin until the last three flips are H, H, T. On average, how many flips does it take for HHT to first appear?

Accepted Answer

Track progress toward HHT: S0, S1 (have H), S2 (have HH), done (S3). Let ei be expected remaining flips, e3 = 0. - From S0: H to S1, T stays S0. - From S1 (have H): H to S2; T gives "HT", ending in T which matches nothing, back to S0. - From S2 (have HH): T completes HHT (done); H gives "HHH", whose last two are HH, so you stay at S2. e0 = 1 + 	frac12 e1 + 	frac12 e0, \quad e1 = 1 + 	frac12 e2 + 	frac12 e0, \quad e2 = 1 + 	frac12 e2 + 	frac12 \cdot 0. From the last equation, 	frac12 e2 = 1, so e2 = 2. From the first, e0 = 2 + e1. Then e1 = 1 + 	frac12(2) + 	frac12 e0 = 2 + 	frac12 e…

Flips until the pattern heads-heads-tails appears

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