Expected Number of Dice Rolls Until an Increasing Run of Three

2023/08/28 by Daniel Chen, Chen, Daniel
Engineering · Mathematics · #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #Diverse Scientific and Engineering Research #FOS: Mathematics #Probability (math.PR) #Statistical Distribution Estimation and Applications

paper · pdf · doi:10.48550/arxiv.2308.14635

openalex publication_date 2023/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A closed form is found for the expected number of rolls of a fair n-sided die until three consecutive increasing values are seen. The answer is rational, and the greatest common divisor of the numerator and denominator is given in terms of n. As n goes to infinity, the probability generating function is found for the limiting case, which is also the exponential generating function for permutations ending in a double rise and without other double rises. Thus exact values are found for the limiting expectation and variance, which are approximately 7.92437 and 27.98133 respectively.

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