Rolling until two sixes in a row

Question

Roll a fair six-sided die repeatedly and stop the first time you see two sixes in a row. What is the expected number of rolls?

Accepted Answer

Let p = 	frac16 be the chance of a six on any roll, q = 	frac56 otherwise. Track your progress toward the goal with three states: state 0 (no six pending), state 1 (just rolled one six), state 2 (done). Let e0, e1 be the expected additional rolls. From state 1 (one six banked). Roll a six (prob p) 	o done. Roll a non-six (prob q) 	o progress lost, back to state 0: e1 = 1 + p\cdot 0 + q\, e0 = 1 + q\,e0. From state 0. Roll a six (prob p) 	o state 1. Roll a non-six (prob q) 	o stay in state 0: e0 = 1 + p\,e1 + q\,e0. Substitute e1: e0 = 1 + p(1 + q\,e0) + q\,e0 = 1 + p + (pq + q)e0. So e0…

Rolling until two sixes in a row

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