Parrondo's Paradox Calculator

Parrondo's paradox describes a counterintuitive probability result: two games that each lose money when played alone can produce a winning expectation when they are alternated or randomly mixed. The effect does not come from creating probability out of nothing. It comes from changing how often the player's capital visits different states. Game B uses one coin when capital is divisible by the cycle length and another coin otherwise, so its future odds depend on the current remainder. Game A has a constant win probability. Its expected change per play is simply twice its win probability minus one because a win adds one unit and a loss removes one unit. Game B needs more care. Capital moves between remainder states after every play, and the long-run frequency of those states is not generally uniform. This calculator builds the transition process for the selected cycle length, iterates it to a stationary distribution, and weights each state's win or loss expectation by its stationary probability. For the mixed game, the calculator first combines Game A's probability with the appropriate Game B probability in every remainder state. The chance of choosing Game A controls that blend. It then solves the stationary distribution again because mixing changes state occupancy as well as immediate odds. This distinction is essential: averaging the standalone returns of A and B would miss the mechanism that produces the paradox. The classic example uses a slightly unfavorable Game A, a very unfavorable bad-state coin in Game B, and a favorable good-state coin. Both standalone games have negative long-run drift, while an even random mixture can have positive drift. Results are expected values rather than guarantees. A finite sequence remains random and may gain or lose capital even when its theoretical drift has the opposite sign. Use the calculator to explore probability models, teach Markov chains, or verify published Parrondo parameters without running a noisy simulation.