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$m$ balls are placed into $n$ boxes at random with uniform probability. What is the expected number of empty boxes?

The link below presents the solution for the case when $m = n$. Unfortunately, I do not understand the solution sufficiently to extend it to the non-equal case.

Find: The expected number of urns that are empty

Aleksejs Fomins
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1 Answers1

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By linearity of expectation, the answer is $n$ (the number of boxes) times the probability that a given box remains empty (that is $(1-1/n)^m$).

I'm assuming the positions are independent.

Angina Seng
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  • Thanks. Your answer makes sense, but it does not reduce to the linked answer when $n=m$. Does that mean that the linked answer is wrong? – Aleksejs Fomins Jun 27 '18 at 06:39
  • The linked problem is different @AleksejsFomins – Angina Seng Jun 27 '18 at 06:48
  • I fail to see how. The objects are labelled, but it is explicitly stated in the solution that it is irrelevant. If we were to take the special case $m=n$, how is my problem different from the linked problem? – Aleksejs Fomins Jun 27 '18 at 06:51
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    In the linked problem, the first ball can go into the first box. The second ball can go into the first or the second box. The third can go into the first, second, or third box, and so on. – forget this Jun 27 '18 at 07:16
  • Ufff, I did not notice difference in range between random variables originally. Thanks :) – Aleksejs Fomins Jun 27 '18 at 07:20