Flips until three of the same face in a row

Question

You flip a fair coin until you see 3 of the same face in a row, that is, either HHH or TTT. A flip that differs from the one before it starts a brand-new streak of length 1 (it is not a full reset to zero). On average, how many flips does that take?

Accepted Answer

The twist here: because either face can form the run, the very first flip always establishes a streak of length 1 for free. After that, each new flip either matches the previous one (streak grows) or differs (streak restarts at 1, since that differing flip itself begins a new run). Let aj be the expected additional flips to reach a run of 3 given a current same-face streak of length j (j = 1, 2), with a3 = 0. Each flip matches with probability 1/2 (advance) or differs with probability 1/2 (back to streak 1): a1 = 1 + 	frac12 a2 + 	frac12 a1, \qquad a2 = 1 + 	frac12 a3 + 	frac12 a1 = 1 + \…

Flips until three of the same face in a row

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