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I was in the store earlier, and I saw wrapped figurine collectibles that have 10 unique kinds. What distribution represents the probability that you have collected all 10 figurines after k figurines purchased? (Under the obvious assumption that the figurines are "replaced" because you can buy already purchased ones.)

More generally, given $n$ classes and $k \geqslant n$ draws, what is the probability you collected 1 (or more) of each class from those draws? I played around with models myself, but I am sure there is a canonical distribution to model this situation.

Calvin Elder
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  • This is known as the Coupon collector's problem, and the distribution is given in the accepted answer to this question, I believe. – ConMan Aug 22 '21 at 23:45
  • @ConMan does this assume that each coupon have the same probability to appear? it doesn't seem the case here – Tortar Aug 23 '21 at 00:02
  • Hmm, yes you would need to adjust the distribution if the figurines aren't distributed evenly. It also assumes independence of probabilities between boxes which is often not the case in reality (at minimum, there's usually a fixed number of each figurine across the entire production, but also it's common for each pallet of boxes to have fixed numbers). – ConMan Aug 23 '21 at 00:05
  • yes that is true, didn't think about the pallet, maybe we can skip the problem and assume random sampling :D – Tortar Aug 23 '21 at 00:09
  • Thanks! I’m willing to make the assumption they are uniformly distributed because there doesn’t appear to be a figurine desirability hierarchy. – Calvin Elder Aug 23 '21 at 00:23

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