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Possible Duplicate:
Number of permutations where n ≠ position n
Probability of matching events

  1. There are n children in a classroom and each child has exactly one toy (so there are n toys in total). They break for recess, leaving their toys in the classroom. When they come back to the classroom, each child picks a toy by random. a) What the expected number of children who pick the toy they had before leaving for recess? b) What's the probability no child picks the toy they had initially?
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marked as duplicate by Did, Mike Spivey, Asaf Karagila, t.b., J. M. Dec 5 '11 at 18:21

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Presumably when they come back each toy is only picked by one child (not that it affects the expectation) – Henry Nov 17 '11 at 10:24

You haven't specified a distribution, so I'll assume that with "by random" you meant "randomly with uniform distribution over all permutations".

Didier has already pointed out another question of which part b) is a duplicate. For a), you can use linearity of expectation. Each child has probability $1/n$ of picking the same toy again, and the expected number is just $n$ times that, i.e., $1$.

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