  The problem of the longest head run was introduced and solved by Abraham de
Moivre in the second edition of his book Doctrine of Chances (de Moivre, 1738).
The closed-form solution as a finite sum involving binomial coefficients was
provided in Uspensky (1937). Since then, the problem and its variations and
extensions have found broad interest and diverse applications. Surprisingly, a
very simple closed form can be obtained, which we present in this note.
