Moments for generalizations of a coin flip game

TL;DR

This paper develops recursive and closed-form formulas for the moments of the number of coin flips needed to produce specific binary strings, extending to dice with multiple faces and using advanced combinatorial methods.

Contribution

It introduces new recursive and closed-form formulas for moments of pattern occurrences in biased coin and die roll sequences, utilizing Eulerian numbers and combinatorial techniques.

Findings

Derived recursive formulas for moments of pattern occurrences in biased coin flips.

Simplified formulas using extended Eulerian numbers and combinatorial methods.

Extended results to dice with multiple faces and arbitrary probabilities.

Abstract

We derive a recursive formula for the moments of the number of flips using a possibly biased coin to produce a prescribed finite binary string $S$ when $S$ is either a run of heads or a run of heads followed by a tails. Our recursive formula involve certain sums, which we simplify by using a one-parameter extension of the well-studied Eulerian number, which belongs to the two-parameter family of numbers introduced by Graham, Knuth, and Patashnik. We also use the Goulden--Jackson cluster method and Fa\`a di Bruno's formula to establish a closed formula for the moments in a more general situation where a die having an arbitrary number of faces with possibly different probabilities is rolled repeatedly until a prescribed finite word occurs.