2026-04-02

Problem — 2026-04-02

Hard

Hardprobability

What is the expected number of rolls of a fair six-sided die needed to see all $6$ faces at least once?

Express as a fraction or decimal

Show solution

After seeing

k

distinct faces, the probability of a new face on the next roll is

\frac{6-k}{6}

, so the expected rolls until a new face is

\frac{6}{6-k}

. Total:

\frac{6}{6} + \frac{6}{5} + \frac{6}{4} + \frac{6}{3} + \frac{6}{2} + \frac{6}{1} = 1 + 1.2 + 1.5 + 2 + 3 + 6 = 14.7 = \frac{147}{10}