A Note on the Closed-Form Solution for the Longest Head Run Problem of Abraham de Moivre
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Malinovsky, Yaakov. “A Note on the Closed-Form Solution for the Longest Head Run Problem of Abraham de Moivre.” The Mathematical Intelligencer 44, no. 3 (September 1, 2022): 267–68. https://doi.org/10.1007/s00283-021-10125-2.
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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.