How many boxes to collect all six toys?

Question

Each cereal box contains one toy, chosen uniformly at random from 6 different toys, independently of every other box. You keep buying boxes until you own all 6 toys.

On average, how many boxes do you buy in total?

Accepted Answer

The total wait is hard as one lump, but it splits into stages, and linearity of expectation adds the stages even though the stage lengths are random. Break the collection into 6 stages by how many distinct toys you already own. Stage k starts when you have k - 1 toys and ends when you first get a new one. During stage k, a new toy appears with probability pk = \frac{6 - (k-1)}{6}, since 6 - (k-1) of the 6 toys are still new. The number of boxes in stage k is geometric with mean 	frac{1}{pk} = 	frac{6}{6 - (k-1)}. By linearity, add the stage means: E[	ext{boxes}] = \sum{k=1}^{6} \frac{6}{6-…

How many boxes to collect all six toys?

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