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Here is a cooler problem, imo: show that the expected value of the # of people who get their hat back is 1, independent of n.

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Hmm, how about n(1/n)=1!
uA

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Ok, mr genius, show that the outcomes are independent. I know this is "intuitively obvious" but actually providing a proof is an altogether different matter.


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Why do we care about whether or not the outcomes are independent? The question is about expected values, and the expected value of the sum is the sum of the expected values regardless of whether or not the variables are independent.

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n(1/n) = 1/n + 1/n + .... + 1/n (n terms in the sum) where the ith term represents the probability that the ith person receives his hat. If the events were not independent, the 2nd through nth terms could be different, yielding a different sum.