Expected rolls until a five is followed by a six

Question

You roll a fair six-sided die repeatedly. What is the expected number of rolls until you see a 5 immediately followed by a 6? How does it compare to waiting for two sixes in a row?

Accepted Answer

Set up progress states. Let E0 be the expected rolls with no active 5, and E1 right after rolling a 5 (now hoping for a 6). E0 = 1 + 	frac16 E1 + 	frac56 E0. From state 1, the next roll is a 6 (finish, probability 	frac16), another 5 (stay in state 1, since you still hold a fresh 5, probability 	frac16), or anything else (back to state 0, probability 	frac46): E1 = 1 + 	frac16\cdot 0 + 	frac16 E1 + 	frac46 E0. The first equation gives 	frac16 E0 = 1 + 	frac16 E1, i.e. E0 = 6 + E1. From the second, 	frac56 E1 = 1 + 	frac46 E0, i.e. E1 = 	frac65 + 	frac45 E0. Substitute E0 = 6 + E…

Expected rolls until a five is followed by a six

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