Which toy will be the last one you complete?

Question

A set has 7 distinct toys, each equally likely and independent per box. You keep buying boxes until you own all seven. One of the toys is a special gold toy.

What is the probability that the gold toy is the last new toy you obtain (that is, the toy that completes your set)?

Accepted Answer

It is tempting to think the special toy is somehow more likely to lag behind, but the labels do not affect the process at all. Every toy is drawn with the same probability on every box, so the seven toys are completely interchangeable. By symmetry, each of the 7 toys is equally likely to be the one that shows up last for the first time. Since exactly one toy finishes the set, and all 7 have the same chance, P(	ext{gold toy is last}) = \frac{1}{7} \approx 0.1429. A cleaner way to see it: imagine listing, for each toy, the box number on which it first appears. These seven "first-appearance" ti…

Which toy will be the last one you complete?

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