An abacus beside three state markers and a small hourglass. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Initial capital matters through its residue
Capital values zero and three start in the same modulo-three state, so they have the same expected gain in this model, though different final expected capital. Initial capital one starts in a different state. For game B, the first-step expected change can therefore differ before the chain approaches its long-run occupancy distribution.
Compare a one-round check
At ε = 0.005, game A wins with probability 0.495 and has expected one-round gain −0.01. Starting at capital zero, game B uses probability 0.095 and has first-step expected change −0.81. Starting at capital one, it uses probability 0.745 and has first-step expected change +0.49. Those transient values are not the stationary average drift of B.
Read the two outputs separately
The Parrondo calculator propagates the exact three-state distribution for the chosen finite number of rounds. It also estimates the converged stationary drift for A, B and the random mixture. A long-run drift times the number of rounds need not equal the finite-horizon gain because the process may not start in stationarity. The displayed expected capital adds the initial capital to the finite expected gain.
Keep omitted constraints visible
This mathematical chain allows capital below zero. It includes no bankruptcy boundary, transaction fee, finite funding limit or real market mechanism. Changing those conditions can change the result. A periodic schedule such as AABB is also different from the independently randomized mixture currently supported. Read the residue-occupancy explanation to understand the state mechanism before interpreting a positive mixed drift. This example demonstrates model-dependent stochastic behavior, not a general recommendation to combine losing choices.
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