Flips until three heads in a row

Question

You flip a fair coin repeatedly. The moment you see $3$ heads in a row you stop. A single tail at any point resets your current streak back to zero.

On average, how many flips does it take to get three heads in a row?

Accepted Answer

Track your progress by how many heads you currently have in a row: state 0, 1, or 2, with state 3 meaning "done." Let ej be the expected number of remaining flips when you are sitting on a streak of j heads. From any state j < 3 you flip once. With probability 1/2 you get a head and advance to j+1; with probability 1/2 you get a tail and crash back to 0: e0 = 1 + 	frac12 e1 + 	frac12 e0, \qquad e1 = 1 + 	frac12 e2 + 	frac12 e0, \qquad e2 = 1 + 	frac12 e3 + 	frac12 e0, with e3 = 0. From the first equation, 	frac12 e0 = 1 + 	frac12 e1, so e0 = 2 + e1. Similarly e1 = 2 + e2 and e2 = 2 + …

Flips until three heads in a row

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