Expected value problem

[SGS]

Here's a problem I can't figure out how to get going on. Suppose we put 7 balls randomly into 5 boxes. What is the expected number of empty boxes?

[pzhon]

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Suppose we put 7 balls randomly into 5 boxes. What is the expected number of empty boxes?

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The expected number of empty boxes is the sum of the probabilities that the boxes are empty.

If by "randomly," you mean that each ball is independently placed into each box with probability 1/5, the probability that a particular box is empty is (4/5)^7, so the expected number of empty boxes is 5(4/5)^7.

[BruceZ]

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If by "randomly," you mean that each ball is independently placed into each box with probability 1/5, the probability that a particular box is empty is (4/5)^7, so the expected number of empty boxes is 5(4/5)^7.

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Nice. I was thinking weighted sum of Maxwell-Boltzman.

[probman]

Your analysis is wrong. You assume that the event that the boxes are empty are iid events and therefore the expected number of empty boxes is simpley the probability that a given box is empty times the number of boxes. This is not the case. The simplest way to see that your model fails is to realize that all five boxes cannot be empty at the same time. This clearly violates the independent model.

[jason1990]

His analysis is correct. The expected value of a sum is the sum of the expected values, always. No need for independence. And he does not assume independence. Namely, if N is the number of empty boxes and Xi is 1 if the ith box is empty and 0 otherwise, then

N = X1 + ... + X5,

so

EN = EX1 + ... + EX5.

But EXi is just the probability the ith box is empty. And this is (4/5)^7, which does not depend on i.

[probman]

You're right. I over thought the problem.