Still no six after six rolls

Question

You roll a fair die repeatedly, stopping at the first six. What is the probability that it takes more than six rolls to get your first six?

Accepted Answer

"More than six rolls" means the first six has still not appeared after six rolls, which is exactly the event that the first six rolls are all non-sixes. Each roll avoids a six with probability 	frac56, and the rolls are independent, so P(	ext{no six in first six rolls}) = \left(\frac56ight)^6 = \frac{15625}{46656} \approx 0.335. So there is about a 33\% chance you are still waiting after six rolls, even though the average wait is exactly six rolls. That gap between the average and the "more than average" probability is typical of geometric waiting: a few very long waits pull the average u…

Still no six after six rolls

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