Rolls of a die to see four sixes

Question

You roll a fair six-sided die repeatedly. You stop once you have rolled a 6 four times in total (they need not be consecutive). On average, how many rolls does that take?

Accepted Answer

Each roll shows a 6 with probability p = 1/6. Waiting for 4 sixes is four independent single-six waits placed end to end, each with expected length 1/p. By linearity of expectation, E = \frac{r}{p} = \frac{4}{1/6} = 4 \cdot 6 = 24 	ext{ rolls.} Takeaway. Expected rolls to the r-th six is r/p = 4 \cdot 6 = 24.

Rolls of a die to see four sixes

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