Packs to collect all four team badges

Question

A sticker series has 4 distinct team badges. Each pack contains one badge, equally likely and independent. You buy packs until you have all 4. On average, how many packs does that take?

Accepted Answer

Split the collection into stages by how many distinct badges you own. With k owned, the chance a new pack is a fresh badge is \frac{4 - k}{4}, so that stage takes \frac{4}{4 - k} packs on average. Adding stages k = 0, \ldots, 3: E = \frac{4}{4} + \frac{4}{3} + \frac{4}{2} + \frac{4}{1} = 4\left(1 + 	frac12 + 	frac13 + 	frac14ight) = 4 \cdot \frac{25}{12} = \frac{25}{3} \approx 8.33. Takeaway. Collecting all n equally likely types averages n Hn; for n = 4 that is 4 \cdot 	frac{25}{12} = 	frac{25}{3} \approx 8.33 packs.

Packs to collect all four team badges

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