Parrondo Paradox Calculator
Simulate Parrondo's paradox coin games and compare strategies to see how alternating losing games can produce a winning outcome.
What Is Parrondo's Paradox?
Parrondo's paradox shows how two individually losing games can produce a winning strategy when played in a specific sequence. Juan Parrondo discovered this while studying random processes in physics. This simulator uses the classic coin-flip setup where Game A always uses fixed odds, and Game B switches between two sub-games based on your current capital. Related tools: Monty Hall Problem Calculator and Expected Value Calculator.
How the Games Work
Game A gives a 49.5% chance to win $1 each round. Game B checks whether your capital is a multiple of 3: if yes, you play B1 with 9.5% win odds; if not, you play B2 with 74.5% win odds. Each game alone has negative expected value, yet alternating patterns like A-B or AABB can drift upward over many rounds.
Why Alternating Can Win
The key is state dependence. Game B's odds depend on capital modulo 3, so the order of play changes how often you land in favorable B2 states. Markov chain analysis shows that mixed strategies can increase the long-run probability of winning above 50%, even though each isolated game stays below 50%.
Frequently Asked Questions
Does Parrondo's paradox work in real casinos?
No. Casino games are designed so combined strategies cannot turn negative expected value into positive expected value. Parrondo setups require carefully linked games with specific state-dependent rules.
Which strategy usually performs best?
In the classic parameters, alternating A and B or playing AABB often produces positive average capital, while playing only A or only B trends downward.
Why does capital start at zero?
Zero is the standard starting point in Parrondo demonstrations because Game B's sub-game depends on whether capital is divisible by 3.
Will every simulation run look identical?
No. Individual runs vary because coin flips are random. Averages over hundreds or thousands of runs reveal the paradox more clearly.
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