This is the so-called matching problem, in which
individuals mix their hats up and then randomly make a selection. The random variable
is the number of individuals that select their own hat. The permutations that lead to
are called derangements. The distribution of
is given by
,
. Both the expectation and the variance of
equal 1, regardless of the value of
. As
goes to infinity, the distribution of
converges to the Poisson distribution with parameter 1.
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