Cereal boxes to collect all five toys

Question

A cereal brand puts one of 5 distinct collectible toys in each box, each type equally likely and independent from box to box. You buy boxes until you own all 5 types. On average, how many boxes must you buy to complete the set?

Accepted Answer

This is the coupon collector problem. Break the quest into stages by how many distinct toys you already own. When you own k distinct toys, a new box gives you a fresh type with probability \frac{5 - k}{5}. Getting that next new type is a wait-for-first-success with that probability, so it takes on average \frac{5}{5 - k} boxes. Summing the stages from k = 0 up to k = 4: E = \frac{5}{5} + \frac{5}{4} + \frac{5}{3} + \frac{5}{2} + \frac{5}{1} = 5\left(1 + 	frac12 + 	frac13 + 	frac14 + 	frac15ight). The bracket is the harmonic sum H5 = \frac{137}{60}, so E = 5 \cdot \frac{137}{60} = \frac{…

Cereal boxes to collect all five toys

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