Can a losing strategy lead to overall success? This review delves into Parrondo's Paradox, a seemingly counterintuitive phenomenon where alternating between two losing games results in a winning expectation. Inspired by the flashing Brownian ratchet, Parrondo's games are often realized as coin-tossing events and challenge conventional wisdom about combining unfavorable odds. The review provides mathematical analyses and explanations to illuminate how such a process is possible, making it highly relevant to the fields of **probability** and **mathematics**. The core of Parrondo's Paradox lies in the interaction between a single biased coin (Game A) and a state-dependent game (Game B) that relies on the player's current capital. The paper investigates the games' dependence on various properties, rather than solely the player's capital. It also offers a concise overview of recent advances in Parrondian ratchet or discrete-time ratchet theory, further enhancing its scientific value. This review enriches our understanding of complex systems and has implications for fields ranging from economics to evolutionary biology. Its exploration of Parrondo's Paradox and its extensions opens new avenues for research and applications in diverse scientific domains.
Published in Fluctuation and Noise Letters, this review fits the journal's focus on stochastic processes and complex systems. By examining Parrondo's Paradox, it contributes to understanding how seemingly random fluctuations can lead to unexpected outcomes, aligning with the journal's interest in phenomena driven by noise and statistical mechanics. The references show its connection to diverse fields, including statistical physics and game theory.