Rolls to a Six

Question

On average, how many rolls of a fair die does it take to see the first 6?

Accepted Answer

Each roll succeeds with p = frac16 independently, so the count is geometric: mathbbE = frac1p = 6. Or by first-step recursion: E = 1 + frac56 E Rightarrow E = 6. Worth holding in mind: the median is 4 (since (5/6)^3 approx 0.58 > frac12 > (5/6)^4 approx 0.48), well below the mean, geometric distributions are right-skewed, so "expected 6 rolls" coexists with "more often than not you're done in 4." Interviewers sometimes probe whether you conflate mean and median.

Rolls to a Six

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