How many Dice Rolls Would It Take to Reach Your Favorite Kind of Number?

TL;DR

This paper explores a generalized dice-rolling game where the goal is to reach a prime number, using symbolic computation to extend previous work and discuss the practical relevance of error estimates.

Contribution

It introduces several generalizations of the game, demonstrating the utility of symbolic computation and challenging the emphasis on rigorous error bounds.

Findings

Extended the game to multiple variants using symbolic computation

Provided practical, non-rigorous estimates for game outcomes

Critiqued the focus on theoretical error bounds in prior work

Abstract

Noga Alon and Yaakov Malinovsky recently studied the following game: you start at 0, and keep rolling a fair standard die, and add the outcomes until the sum happens to be prime. We generalize this in several ways, illustrating the power of symbolic, rather than merely numeric, computation. We conclude with polemics why the beautiful rigorous error estimate of Alon and Malinovsky is only of theoretical interest, explaining why we were content, in our numerous extensions, with non-rigorous, but practically-certain, estimates.