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

2020/09/16 by Yaakov Malinovsky, Malinovsky, Yaakov · 1 citation
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #History and Overview (math.HO) #Mathematical and Theoretical Analysis #Other Statistics (stat.OT) #math.HO #stat.OT

paper · pdf · doi:10.48550/arxiv.2009.07765

openalex publication_date 2020/09/16 · arxiv created 2021/02/21 · arxiv updated 2021/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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