  Choose a random linear operator on a vector space of finite cardinality N:
then the probability that it is nilpotent is 1/N. This is a linear analogue of
the fact that for a random self-map of a set of cardinality N, the probability
that some iterate is constant is 1/N. The first result is due to Fine, Herstein
and Hall, and the second is essentially Cayley's tree formula. We give a new
proof of the result on nilpotents, analogous to Joyal's beautiful proof of
Cayley's formula. It uses only general linear algebra and avoids calculation
entirely.
