Expected number of fixed points of permutation equals one

Statement
Suppose $$n$$ is a natural number. Consider the uniform distribution on the symmetric group of degree $$n$$, i.e., the symmetric group on a set of size $$n$$. The expected number of fixed points for a permutation picked according to the uniform probability distribution equals $$1$$.

See also probability distribution of number of fixed points of permutations.

Proof
We note that the number of permutations that fix a particular element $$i$$ in the set is $$(n - 1)!$$, hence the probability that $$i$$ is fixed is $$1/n$$. By linearity of expectation, the expected number of fixed points is the sum, for each point, of the probability that it is fixed. This sum is $$n(1/n) = 1$$.

Note that expectation is linear even when the random variables in question are not independent.