Juan Parrondo, a Spanish physicist, described the effect in 1996. Game A is a slightly unfair coin: you win one unit with probability 0.495 and lose one with probability 0.505, so you slowly lose. Game B uses two coins. If your capital is a multiple of three you must flip a bad coin that wins only 9.5% of the time; otherwise you flip a good coin that wins 74.5% of the time. Because the bad coin is used a little more than a third of the time, game B also loses on average.
Now mix the games. Playing A, A, B, B, A, A, … or choosing A or B at random each time produces a steady profit. Game A acts as a randomiser that knocks your capital out of the pattern in which game B keeps hitting its bad coin, so B gets to use the good coin more often. Neither game is good alone, but the combination is – an idea later applied to biology, economics and Brownian ratchets in physics.
The calculator computes the exact expected capital for each strategy using a Markov chain on the capital modulo M, simulates many runs of the strategy you choose and plots both curves. You can change the bias ε and the modulus to see when the paradox appears.
How to use the Parrondo’s Paradox Calculator
- Strategy
- Games per run
- Simulated runs
- Bias ε — The original example uses ε = 0.005.
- Modulus M for game B
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The exact result tracks the probability distribution of the capital modulo M game by game; the expected gain on each game is Σ π(s)·(2p(s) − 1), where p(s) depends on the game played and the state. The simulation plays the chosen strategy from capital 0 for the given number of games and runs, and averages the capital after each game.
Worked example
With ε = 0.005 and 100 games starting from 0, the exact expected capital is −1.00 for game A alone and −1.39 for game B alone. Playing A, A, B, B, … gives +1.39 and choosing A or B at random gives +1.29. Over 500 games the gaps widen: A −5.00, B −4.87, AABB +7.25, random +7.57, while strict alternation (A, B, A, B) ends at −2.58.
The two games
| Game | Rule | Win chance (ε = 0.005) |
|---|---|---|
| A | One slightly unfair coin | 0.495 |
| B, capital a multiple of 3 | Bad coin | 0.095 |
| B, otherwise | Good coin | 0.745 |
Formula
Game A: win with p = 1/2 − ε · Game B: win with 1/10 − ε if capital is a multiple of M (= 3), else 3/4 − ε · each game wins or loses 1 unit
Worked example
Inputs — Strategy: A, A, B, B, … (two of each); Games per run: 100; Simulated runs: 1,000; Bias ε: 0.005; Modulus M for game B: 3.
Results — Expected capital after the last game = 1.3917; Verdict = Winning in the long run; Expected gain per game = 0.01392; Simulated average capital = 1.214; Simulated runs that finished ahead = 49.7 %; All strategies compared = Strategy Expected capital after 100 Per game
─────────────────────────────────────────────────────
Only game A −1 −0.01
Only game B −1.392 −0.01392
A, B, A, B, … 0.119 0.00119
A, A, B, B, … 1.392 0.01392
Random A or B 1.287 0.01287.
Key terms
- Markov chain
- A random process whose next step depends only on the current state – here the capital modulo 3.
- Capital-dependent game
- A game whose odds depend on the player's current capital, like game B.
- Ratchet
- A mechanism that turns random motion into directed motion; Parrondo's games are a discrete version.
Frequently asked questions
What is Parrondo's paradox?
Two games that each lose in the long run can produce a winning combination when played alternately or at random.
Why does game B lose on its own?
Its bad coin (9.5% win) is used whenever the capital is a multiple of 3, which happens about 38% of the time, enough to make B a losing game.
How can two losing games win together?
Game A shuffles the capital so game B lands on a multiple of 3 less often and uses its good coin (74.5%) more, turning B's average result positive.
Which combination works best?
With ε = 0.005, in the long run playing A, A, B, B gains about 1.5 units per 100 games and random switching about 1.6; alternating A, B wins slightly at first but loses in the long run.
Does this mean I can beat a casino?
No. Game B depends on your capital, which casino games do not, and real casino games are independent of each other.
Where is the paradox used?
In models of molecular motors (Brownian ratchets), population dynamics, investment rebalancing and evolution.