Distinct stickers from six packs

Question

A sticker line has 8 designs, each equally likely and independent per pack. You open 6 packs. On average, how many distinct designs do your 6 stickers cover?

Accepted Answer

Linearity of expectation: the expected number of distinct designs is the sum over the 8 designs of the probability each appears at least once. A specific design is absent from all 6 packs with probability \left(\frac{7}{8}ight)^6 \approx 0.4488, so it appears at least once with probability 1 - 0.4488 = 0.5512. With 8 designs, E[	ext{distinct}] = 8\left(1 - \left(	frac78ight)^6ight) \approx 8 	imes 0.5512 = 4.410. Takeaway. Expected distinct types after m draws is n\left(1 - \left(\frac{n-1}{n}ight)^might); six packs from eight designs average about 4.4 distinct designs.

Distinct stickers from six packs

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