Parrondo’s Paradox Calculator
Enter the win probabilities for Game A and the capital-dependent Game B, plus how often each is played, to see how combining two losing games can produce a winning outcome — Parrondo’s paradox.
What this calculator does
Parrondo’s paradox is a surprising result in probability where two games that are each individually losing (or fair) can be combined into a winning strategy. The classic version uses Game A, a simple coin-flip game, and Game B, a game whose odds depend on whether your current capital is a multiple of 3. This calculator estimates the long-run outcome of alternating between the two games.
How the calculation works
The calculator models capital modulo 3 as a three-state Markov chain, since only whether your capital is a multiple of 3 (and which remainder it has) affects Game B’s odds. It computes the long-run steady-state probability of being in each of the three capital states using iterative refinement of the state distribution, then combines that with your chosen mix of Game A and Game B to find the overall probability of winning each round. Since each round moves your capital up by 1 (win) or down by 1 (loss), the expected change per round is 2 x (win probability) – 1, which is then multiplied by the number of rounds.
With the classic default values, Game A and Game B are each losing on their own, but randomly alternating between them produces a small positive expected gain per round — the paradox. This tool is a simplified mathematical model for educational purposes and does not predict any real gambling outcome.