A General Formula for the Probability of Winning in Sequential Turn-Based Games

Abstract: This paper presents a general formula for calculating a player's probability of winning in a sequential, turn-based game with a constant success probability per trial. The problem extends the classical two-player probability models of dice tossing or coin flipping to an arbitrary number of n players. A compact proof based on the summation of a geometric series is provided, and examples demonstrate the correctness and applicability of the result. This formulation can serve as an educational tool for understanding probabilistic reasoning, sequences, and infinite series.