How many packs to complete a five-sticker set?

Question

A sticker album needs 5 different stickers to complete. Each pack contains 1 sticker, equally likely to be any of the 5, independently every time. You buy packs until you have all 5 stickers.

On average, how many packs do you buy?

Accepted Answer

Split the total into 5 stages by how many distinct stickers you own, and add the stage lengths using linearity of expectation. With k - 1 stickers already owned, the next pack is a new sticker with probability pk = \frac{5 - (k-1)}{5}, so that stage lasts on average 	frac{5}{5-(k-1)} packs. Summing over the 5 stages, E[	ext{packs}] = 5\left(\frac11 + \frac12 + \frac13 + \frac14 + \frac15ight) = 5 	imes \frac{137}{60} = \frac{685}{60} = \frac{137}{12} \approx 11.417. Takeaway. Completing a set of n equally likely items takes n\left(1 + 	frac12 + \cdots + 	frac1night) draws on average.…

How many packs to complete a five-sticker set?

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