Expected rolls until the first six

Question

You roll a fair six-sided die over and over, stopping as soon as a 6 appears. What is the expected number of rolls, including the roll that shows the six?

Accepted Answer

Let E be the expected number of rolls. Take the first roll: it always happens, so it contributes 1. With probability 	frac16 it is a six and you stop. With probability 	frac56 it is not, and you are right back where you started, needing E more rolls on average. E = 1 + 	frac56 E. Solving, 	frac16 E = 1, so E = 6. Takeaway. For repeated independent trials with success probability p, first-step analysis gives E = 1 + (1-p)E, hence E = 1/p. A one-in-six event takes six tries on average.

Expected rolls until the first six

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