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I have a problem from the "discrete mathematics and its applications" textbook.

What is the expected number of balls that fall into the first bin when m balls are distributed into n bins uniformly at random?

This is my current attempt: Let X be a random variable that represents the number of balls in the first bin. Let E(X) be the expected number of balls in the first bin. Then, E(X) = p(X=1)1 + p(X=2)2 + ... + p(X=m)m

How do I figure out p(X=r), and then find their sum?

Peter Yan
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    The expected value will be the same for all $m$ bins. if you add them up they will sum to $n$ – WW1 Mar 27 '23 at 20:24
  • See Linearity of Expectation, which includes a proof that the formula applies even when the events are not independent of each other. Then, what is the probability that the very first ball goes into bin-1? Alternatively, the analysis inherent in the comment of @WW1 also solves the problem. – user2661923 Mar 27 '23 at 21:13
  • Could you please give a detailed answer? – Peter Yan Mar 27 '23 at 22:28

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