Three connected token bowls and two contrasting generic coins. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Game B depends on where the capital currently sits
In the classic capital-dependent model, game A uses win probability 0.5−ε. Game B uses 0.1−ε when capital is divisible by three and 0.75−ε in the other two residue states. Each win adds one unit and each loss subtracts one. The relevant state is capital modulo three, not merely which game was chosen.
Track probability moving between states
From residue j, a win moves to (j+1) modulo three and a loss to (j−1) modulo three. A three-component distribution records the chance of occupying each residue before the next play. Expected one-round change is the weighted average of 2p−1 across those states. Because game choice changes the occupancy distribution, averaging isolated long-run game profits is not the same as analyzing the mixed state-dependent process.
Define the random mixture precisely
If A is selected independently each round with probability q, the win probability within each residue is q times A's win probability plus (1−q) times B's state-specific win probability. Propagating this mixture transition rule can produce a different long-run drift from either standalone game. This state redistribution is the mechanism behind the paradox, rather than an arithmetic rule that two negative numbers add to a positive number.
Check the model rather than a slogan
The Parrondo's Paradox Calculator compares A, B and the random mixture under one entered epsilon. It calculates finite-state expectations, not guaranteed winnings or investment performance. The finite-horizon guide explains why early behavior depends on the initial residue. For ordinary outcome-weighted means without state dependence, use the existing Expected Value Calculator; do not apply a standalone average to a changing state process.
Check the model behind your result
Use the Parrondo's Paradox Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26