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A question posed on John D. Cook's blog asks:

Suppose you have a large number of buckets and an equal number of balls. You randomly pick a bucket to put each ball in one at a time. When you’re done, about what proportion of buckets will be empty?

Can you explain the answer to this question in a way understandable to high school students? How about to ten year olds?

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    Do you mean that you randomly select a bucket, then put a ball in it, then randomly select the next bucket (which might again be the same bucket), and so on?2012-10-12
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    Yes, I believe that's what the question intends.2012-10-12
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    @Martin: Yes, that’s fairly clear if you read the discussion in the blog.2012-10-12
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    High school, no problem. Can one explain *anything* to a $10$ year-old?2012-10-12
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    @André: Depends on the individual in both cases. I don’t think that one could explain it to the majority of high school students, and there are a few $10$-year-olds to whom you could explain it reasonably well; I was one of them, once upon a time. I shouldn’t be surprised if Qiaochu was one, and quite possibly several others here.2012-10-12
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    The usual "mean" argument is explainable in terms of gambling: the $i$-th gambler hoping for no hit in her bin. The behaviour of $\left(\frac{n-1}{n}\right)^n$ can be made plausible by a dozen calculator computations.2012-10-12

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