Expected Number of Dice Rolls Until an Increasing Run of Three

TL;DR

This paper derives a closed-form expression for the expected number of fair die rolls until three consecutive increasing values occur, including limiting behavior as die sides grow large, with exact expectations and variances calculated.

Contribution

It provides a novel closed-form solution for the expected rolls until a specific increasing pattern, linking it to permutation statistics and limiting generating functions.

Findings

Exact expected number of rolls for finite n

Limiting expectation approximately 7.92437

Limiting variance approximately 27.98133

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.