Expected rolls to see two sixes in a row

Question

You roll a fair six-sided die repeatedly. What is the expected number of rolls until you first see two sixes in a row?

Accepted Answer

Let success be rolling a six, probability p = 	frac16. Use states S0 (last roll not a six, or start) and S1 (last roll a six), with expected remaining rolls E0, E1. - From S0: a six moves to S1, a non-six stays: E0 = 1 + 	frac16 E1 + 	frac56 E0. - From S1: a six finishes, a non-six returns to S0: E1 = 1 + 	frac16 \cdot 0 + 	frac56 E0. Using the general result derived for two consecutive successes, E0 = \dfrac{1+p}{p^2}: E0 = \frac{1 + 	frac16}{(	frac16)^2} = \frac{	frac76}{	frac{1}{36}} = \frac76 \cdot 36 = 42. So 42 rolls on average. Contrast with the wait for a single six, which is…

Expected rolls to see two sixes in a row

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