Parrondo’s Paradox Explains How Two Shedding Methods Mixed Can Win

A Mathematical Paradox Reveals How Combining Shedding Methods Can Create a Win

By Manon Bischoff edited by Daisy Yuhas

In 1996 Spanish physicist Juan Parrondo made an unimaginable discovery: typically two video games that every finish in loss individually will be mixed right into a profitable technique. This paradox is not any mere mathematical curiosity—it’s scientifically helpful. It helps clarify the numerous life histories of slime molds and will contribute to new most cancers remedy methods.

To grasp this paradox, we have to think about a scenario wherein you play two video games with some very particular parameters. As an example, let’s think about that the primary sport, “A,” includes a coin toss. The coin on this case has a weight distribution has been barely altered in order that it lands preferentially on one facet with a chance of fifty.5 p.c. Now let’s assume that Sport A is considerably rigged in order that I win if it lands on the popular facet and also you win if it lands on the opposite facet. You’ll be able to subsequently solely win with a chance of 49.5 p.c and, on this case, I offers you $1; in any other case you’ll pay me the identical quantity.

If you happen to play Sport A towards me many occasions, you’ll inevitably maintain lots of losses as a result of you need to pay me a median of 1 cent per sport. (We are able to calculate that rapidly by taking the probability of your win and subtracting the probability of my win: 0.495 – 0.505 = –0.01.)

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Then there’s a second, extra difficult sport, “B,” that includes spinning two wheels of fortune. For this sport, you’ll get to spin one or the opposite based mostly on how a lot cash you at present have. In case your accessible capital for the sport (based mostly on the way you’ve been enjoying A) is evenly divisible by 3, then you definitely spin a wheel of fortune that offers you an opportunity of profitable of solely 9.5 p.c. If, alternatively, your capital is just not divisible by 3, then you definitely get higher odds: you spin one other wheel that offers you a 74.5 p.c probability of profitable.

Now issues get fascinating.

Two Video games You Can Solely Lose

In Sport B, the stake is as soon as once more $1. On common, you’ll lose 87 cents per spin.

Let’s dig into that. You would possibly assume that one third of the time, you’ll spin the wheel that’s unfavorable for you and two thirds of the time, you’ll spin the opposite wheel. However that’s incorrect as a result of your cash doesn’t fluctuate evenly. For instance, you probably have $9, you’ll spin the unfavorable wheel and can probably lose, leaving you with solely $8. If you happen to play the sport once more with that quantity, nevertheless, you’ll spin the wheel that’s extra favorable for you and could have a better probability of profitable. So that you’ll find yourself with $9 once more.

The chance that you’ve sum of cash divisible by 3 is subsequently considerably a couple of third. Utilizing a sophisticated process often called a Markov chain, you’ll be able to calculate that your total chance of profitable Sport B is simply 49.565 p.c—and your anticipated revenue per spherical is destructive: 0.49565 – 0.50435 = –0.0087.

A Paradox Seems

If you happen to’re sensible, you wouldn’t play towards me in both Sport A or Sport B. In each instances, you’re certain to lose in the long term. However Parrondo realized {that a} combined technique can repay: by alternating between Video games A and B, you’ll be able to truly win total.

For instance, in case you at all times play two rounds of Sport A adopted by two rounds of Sport B, you’ll win a median of 1.48 cents per spherical. Or in case you comply with every A spherical with two B rounds, you’ll earn a median of 5.8 cents per spherical. So in the long term, you’d see a revenue in each instances.

As Parrondo found, there are extra mixtures of A and B which have a optimistic anticipated worth for you than vice versa. Due to this fact, you emerge as a winner even in case you randomly select whether or not to play A or B every spherical (for instance, by letting a good coin resolve). On this case, your common win is 1.47 cents per spherical.

How is that this attainable? The important thing to the Parrondo’s paradox is that the 2 video games A and B can affect one another as a result of Sport B is determined by the cash you at present have, and that quantity fluctuates as you play Sport A. Due to this fact, A and B can now not be considered as impartial video games. That is the core of Parrondo’s paradox. If Sport B have been modified in order that, for instance, the worth of a die decided which wheel of fortune you’d spin, the paradox would disappear as a result of each video games can be fully impartial of one another.

Functions of Parrondo’s Paradox

Since Parrondo’s stunning publication in 1996, quite a few papers have appeared on the subject. In 2017 two laptop scientists demonstrated that this paradox can clarify the numerous life methods of slime molds, which may alternate between a solitary, nomadic life and a stationary colony.

In some conditions, it’s extra advantageous for these beings to collect collectively to type colonies as a substitute of current as solitary wanderers. However these communal dwelling preparations can’t survive in the long term both: the organisms exploit their setting, and finally the sources begin to deplete. Sticking to 1 technique would inevitably result in demise, however a combined technique gives an answer: the organisms quickly change into cell once more whereas the setting in a selected space regenerates.

Computational physicist Jian-Yue Guan of Lanzhou College in China and her colleagues offered one other software of Parrondo’s paradox in a paper revealed in Bodily Evaluate E in August 2025. For a lot of varieties of most cancers, two completely different approaches to chemotherapy are used. Sufferers both obtain the utmost tolerated dose at particular intervals or they’re handled constantly with a low dose. The primary technique has the drawback that some tumor cells develop resistance and thus don’t reply to the remedy. Within the second technique, the drug focus is just not at all times excessive sufficient to fully eradicate all most cancers cells.

By way of laptop simulations, the researchers demonstrated that switching between the 2 remedy approaches at set occasions may result in higher outcomes even with out detailed monitoring—very similar to a random order of A and B is advantageous within the lottery instance. Whether or not this theoretical method can actually be utilized to most cancers medication wants additional investigation. Guan and her workforce plan to check their concepts with in vitro research.

This text initially appeared in Spektrum der Wissenschaft and was reproduced with permission.