A Note on the Closed-Form Solution for the Longest Head Run Problem of Abraham de Moivre

Abstract

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.