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.