Expected rolls of a die until the first six

Question

You roll a fair six-sided die repeatedly until it shows a six for the first time. What is the expected number of rolls?

Accepted Answer

Let E be the expected number of rolls until the first six. Condition on the first roll: - With probability 	frac{1}{6} it is a six, and we are done after 1 roll. - With probability 	frac{5}{6} it is not: we have used 1 roll and, because rolls are independent, we face the same problem again, expecting E more rolls. This gives E = 	frac{1}{6}(1) + 	frac{5}{6}(1 + E). Solving: E = 1 + 	frac{5}{6}E, so 	frac{1}{6}E = 1, which means E = 6. This is just the geometric-distribution mean 1/p with p = 	frac{1}{6}: an event of probability p takes 1/p trials on average. Takeaway. Waiting for a pro…

Expected rolls of a die until the first six

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