Expected packs to collect all three stickers

Question

A promotion hides one of three different stickers in each pack, each equally likely and independent. You buy packs until you own all three distinct stickers. What is the expected number of packs you must buy?

Accepted Answer

Break the hunt into stages by how many distinct stickers you already hold. When you hold j of the three, a new pack is a new sticker with probability \frac{3 - j}{3}. Let ej be the expected packs to get the next new sticker from stage j. First-step analysis: you buy one pack; it is new with probability \frac{3-j}{3} (advance) or a repeat otherwise (stay): ej = 1 + \frac{j}{3} ej \;\Longrightarrow\; ej = \frac{3}{3 - j}. You pass through stages j = 0, 1, 2: E = e0 + e1 + e2 = \frac33 + \frac32 + \frac31 = 1 + 1.5 + 3 = 5.5. Takeaway. Collecting n equally likely types costs n\big(1 + 	frac12 +…

Expected packs to collect all three stickers

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