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Assume n distinguishable balls are independently and uniformly distributed among m distinguishable bins. What is the probability that at least one bin gets exactly k balls?

This problem generalizes a previous question which considered indistinguishable balls.

user1185597
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    Can you show some of your efforts at solving? – Piita May 28 '23 at 07:59
  • Distinguishable and indistinguishable balls don't affect this question. You are still assigning the balls randomly. Indistinguishable just means the resulting counts entirely determine the result. – Thomas Andrews May 28 '23 at 10:53
  • This is wrong. For example, consider the number of possible outcomes. Assume m=3 n=2 . Indistinguishable balls results in 6 outcomes while distinguishable balls is 9. So for r=1, the former gets you a probability of 1/2 while for the latter it is 2/3. – user1185597 May 28 '23 at 11:39
  • I think you're assuming that for indistinguishable balls all outcomes are of equal probability, which isn't the case - what Thomas is saying is that since the events depend on each ball and bin symmetrically, it doesn't matter whether they can be distinguished. But for example with 2 indistinguishable 6 sided dice, 7 is still the most common sum. – George May 28 '23 at 12:01
  • OK, let's try to clarify: We have a uniform probability of alphabet size m. We draw n i.i.d. samples. What is the probability that at least on symbol appears k times? This corresponds to distinguishable balls and for m=3, n=2 and k=1 the answer should be 2/3 (you may easily verify it with full enumeration or simulation). – user1185597 May 28 '23 at 12:43
  • The solution is given here:

    https://math.stackexchange.com/questions/116426/multinomial-distribution-probability-that-at-least-one-variable-takes-a-certain

     – user1185597 May 30 '23 at 07:52

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