You roll a six-sided fair die repeatedly until you obtain each face at least once. On average, how many times do you need to roll the die?

Free worked solution · Quant Blueprint

147/10 = 14.7 rolls on average.

1. After k different faces have appeared, the probability the next roll is new is (6 − k)/6.
2. The waiting time to the next new face is geometric with mean 6/(6 − k), counting its successful roll.
3. Add the six stages: 1 + 6/5 + 6/4 + 6/3 + 6/2 + 6 = 14.7. Linearity of expectation justifies adding their means.

Common mistake

Six is the minimum, not the mean. Repeats become especially costly when only one face remains unseen.

Follow-up

For n equally likely faces, the expected collection time is nHₙ, where Hₙ = 1 + 1/2 + ⋯ + 1/n.

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.