Expected flips until three heads in a row

Question

You flip a fair coin repeatedly. What is the expected number of flips until you see three heads in a row?

Accepted Answer

Track progress toward the run. Let Ej be the expected flips remaining when you currently have j heads in a row (j = 0, 1, 2), and finishing means reaching j = 3. Each flip is heads (advance) or tails (reset to 0) with probability 	frac12 each: E0 = 1 + 	frac12 E1 + 	frac12 E0, \quad E1 = 1 + 	frac12 E2 + 	frac12 E0, \quad E2 = 1 + 	frac12\cdot 0 + 	frac12 E0. From the first equation, 	frac12 E0 = 1 + 	frac12 E1, so E0 = 2 + E1. Work from the bottom up: E2 = 1 + 	frac12 E0, \quad E1 = 1 + 	frac12 E2 + 	frac12 E0 = 1 + 	frac12(1 + 	frac12 E0) + 	frac12 E0 = 	frac32 + 	frac34 E0…

Expected flips until three heads in a row

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