Probabilistic chip-collecting games with modulo winning conditions

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.