Probabilistic chip-collecting games with modulo winning conditions

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