Rounds until at least one of two dice shows a six

Question

Each round you roll 2 fair dice together. You keep rolling rounds until at least one die shows a 6. On average, how many rounds does that take?

Accepted Answer

Compute the chance a round yields at least one six, then apply the wait-for-first-success rule. The chance neither die shows a six is \left(	frac56ight)^2 = 	frac{25}{36}, so the chance of at least one six is p = 1 - 	frac{25}{36} = 	frac{11}{36}. The expected wait is 1/p: E = \frac{1}{11/36} = \frac{36}{11} \approx 3.27 	ext{ rounds.} Takeaway. At least one six on 2 dice has probability 1 - (5/6)^2 = 11/36, so the wait is 36/11 \approx 3.27 rounds.

Rounds until at least one of two dice shows a six

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