Probabilistic chip-collecting games with modulo winning conditions

TL;DR

This paper analyzes a probabilistic chip-collecting game with modulo winning conditions, settling two existing conjectures and providing insights into the game's probabilistic structure and outcomes.

Contribution

It resolves two open conjectures in the literature about the game's behavior and winning probabilities under specific modulo conditions.

Findings

Confirmed conjectures regarding winning probabilities

Derived formulas for game outcomes based on parameters

Provided new theoretical insights into probabilistic chip-collecting games

Abstract

Let $a$, $b$, and $n$ be integers with $0<a<b<n$. In a certain two-player probabilistic chip-collecting game, Alice tosses a coin to determine whether she collects $a$ chips or $b$ chips. If Alice collects $a$ chips, then Bob collects $b$ chips, and vice versa. A player is announced the winner when they have accumulated a number of chips that is a multiple of $n$. In this paper, we settle two conjectures from the literature related to this game.