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If we have $n$ bucket and $\beta$ balls and each ball is thrown to a bucket randomly, then what is the probability of "Every bucket at least has a ball in it"?

I tried 2 ways:

  1. I try to calculate from the inverse, and then I can give the probability of bucket $i$ has at least one ball $$p_i=1-\left(\frac{n-1}{n}\right)^\beta,$$ but they seem not independent that I can't just multiply them up;
  2. I try to calculate the number of samples that $\beta$ balls cover $m$ buckets, but I can't see the pattern in it.

Where should I go?

Thank you.

Hance Wu
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  • @DavidK thanks I was making the same mistake described here:https://math.stackexchange.com/questions/1712367/probability-of-all-boxes-containing-a-ball-when-distributing-n-balls-into-k-boxe – 3rdMoment Nov 13 '22 at 06:08
  • @3rdMoment You're welcome. I figured you'd know what I meant. – David K Nov 13 '22 at 06:29
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    The answer of $$1 - \frac{\sum_{k=0}^n (-1)^k \binom{n}{k} \left(n - k\right)^\beta}{n^\beta}$$ can be manually derived by a study of Inclusion Exclusion. See this article for an introduction to Inclusion-Exclusion. Then, see this answer for an explanation of and justification for the Inclusion-Exclusion formula. – user2661923 Nov 13 '22 at 06:58
  • Exactly! Thanks a lot everyone! – Hance Wu Nov 13 '22 at 08:21

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