Rolls until a one is immediately followed by a two

Question

You roll a fair six-sided die until the last two rolls are a 1 immediately followed by a 2 (the pattern "1,2"). On average, how many rolls does it take for "1,2" to first appear?

Accepted Answer

Track progress toward "1,2": S0 (nothing useful), S1 (last roll was a 1), done (S2). Let ei be expected remaining rolls, e2 = 0. Each face has probability 1/6. - From S0: roll a 1 (	frac16) to S1; anything else (	frac56) stays S0. - From S1 (last was 1): roll a 2 (	frac16) completes it (done); roll a 1 (	frac16) keeps you at S1 (a fresh 1); any of the other four faces (	frac46) drops you to S0. e0 = 1 + 	frac16 e1 + 	frac56 e0, \qquad e1 = 1 + 	frac16 \cdot 0 + 	frac16 e1 + 	frac46 e0. From the first, 	frac16 e0 = 1 + 	frac16 e1, so e0 = 6 + e1. The second gives 	frac56 e1 = 1 + …

Rolls until a one is immediately followed by a two

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