Expected rolls to get two sixes in a row

Question

You roll a fair die repeatedly and watch for two sixes in a row. You stop the instant a six is immediately followed by another six. What is the expected number of rolls until you stop?

Accepted Answer

Track your progress with two states: state A means the previous roll was not a six (no progress), and state B means the previous roll was a six (one six banked). Let eA and eB be the expected additional rolls to finish from each. A six has probability p = 	frac16, a non-six q = 	frac56. From A, one roll happens; with probability 	frac16 you move to B, otherwise stay in A: eA = 1 + 	frac16 eB + 	frac56 eA. From B, one roll happens; with probability 	frac16 you get the second six and finish, otherwise you fall back to A: eB = 1 + 	frac16 (0) + 	frac56 eA. The first equation gives 	frac…

Expected rolls to get two sixes in a row

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