Packs to finish an album you have nearly completed

Question

An album has 10 distinct stickers. Each pack contains one random sticker, equally likely and independent. You already own 8 different stickers. On average, how many more packs do you need to complete the album (get the last 2 missing stickers)?

Accepted Answer

You only need to walk through the final two stages of the coupon collector, which are the slowest ones. Right now you own 8 of the 10 types. A pack is one of your 2 missing types with probability \frac{2}{10}, so getting the first missing one averages \frac{10}{2} = 5 packs. After that you own 9 types, and a pack is the single remaining missing type with probability \frac{1}{10}, averaging \frac{10}{1} = 10 packs. By linearity of expectation, total additional packs: E = \frac{10}{2} + \frac{10}{1} = 5 + 10 = 15 	ext{ packs.} The tail of the collection dominates: those last two stickers cost …

Packs to finish an album you have nearly completed

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