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Suppose that there are $n$ bins, each have capacity $C$. What is the expected number of balls can be thrown until one is full?

This question is asked and answered here, but I am interested for an approximation given that $C$ is very large (approx $n=10$, and $C=128,000$ ).

user3563894
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  • cf. Theorem 1 of this paper where the maximum number of balls in any bin after $m$ throws into $n$ bins is considered. You probably want the fourth regime since $n$ is fixed and $m \to \infty$. It may also be possible to perform a Poisson approximation, but I don't have time for the calculations. – Sarvesh Ravichandran Iyer Jun 05 '23 at 11:10
  • It does not gives a direct approximation, only an order of the result – user3563894 Jun 05 '23 at 11:13
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    To be clear... are you asking for some sort of probabilistic measure here... like the expected wait time until one of the bins is full? Or are you asking for the worst case scenario until one is full? If you are looking for the worst case scenario... this is very simply when all bins are just one ball shy from full capacity, the next ball will make a bin full. That is to say, if you have $n\times (C-1)$ balls you might not have a full bin but if you have $n\times (C-1)+1$ then you must have a full bin. This is one of the versions of the pigeonhole principle. – JMoravitz Jun 05 '23 at 12:02
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    If you do want the expected number of balls that can be thrown until... then you really must say so in your post. Make sure you understand why what you have written currently is not that question. – JMoravitz Jun 05 '23 at 12:04
  • @JMoravitz thanks for your comment! – user3563894 Jun 05 '23 at 12:45

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