2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/84975We present some new analytical expressions for the so-called Parrondo effect, where simple coin-flipping games with negative expected value are combined into a winning game. Parrondo games are state-dependent. By identifying the game state variable, we can compute the stationary game state probabilities. Mixing losing games increases the probability of entering into a game state with positive value, as can be seen clearly from our analytical expressions for the Parrondo game value.6 pages Changed title, abstract, refs + corrected typos and one algebraic mistake (optimal value approximation)Classical PhysicsGeneral Physicstext