Rolls of a die until the first six

Question

You roll a fair six-sided die repeatedly until a six comes up. Let N be the number of rolls up to and including the first six. What are the mean and variance of N?

Accepted Answer

Each roll is a success (a six) with probability p = 1/6, and N counts trials until the first success, so N is geometric with mean 1/p and variance (1-p)/p^2: \mathbb{E}[N] = \frac{1}{1/6} = 6, \qquad \operatorname{Var}(N) = \frac{5/6}{(1/6)^2} = \frac{5}{6} 	imes 36 = 30. Standard deviation \sqrt{30} \approx 5.48. On average it takes 6 rolls, but the spread is large: streaks with no six are common. Takeaway. For a fair die, the wait for the first six has mean 1/p = 6 and variance (1-p)/p^2 = 30.

Rolls of a die until the first six

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