A note on the closed-form solution for the longest head run problem of Abraham de Moivre

TL;DR

This paper revisits de Moivre's classic problem of the longest head run, providing a surprisingly simple closed-form solution that enhances understanding and potential applications of the problem.

Contribution

It presents a new, simplified closed-form expression for the longest head run problem, improving upon previous complex formulas.

Findings

A new simple closed-form solution is derived.

The solution involves a finite sum with binomial coefficients.

The result broadens the applicability of the problem in probability theory.

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.