The Gambler’s Fallacy Proves Classical Statistics (Frequentist & Bayes) Fails


Everybody who’s anybody—which makes, as we’ll see, a lot of nobodies—knows the gambler’s fallacy. Gambler watches the roulette wheel come up red six times running and says to himself, “Black is due.”

It happens, too, these musings. Real people make real bets on black convinced that the Law of Averages will restore the black-red balance of the Roulette Wheel of the Universe. Somehow, fallaciers (yes, fallaciers) believe Fortuna herself, or anyway some occult power, reaches in and causes the wheel to adjust itself to maintain Balance.

We call these people frequentists. Bayesians, too.

And not just those people, but anybody who believes in physical probability embraces the Gambler’s Fallacy; frequentists are just their most visible representatives. Physical probability must be causative to make observed frequencies work out in balance. But since probability is only a state of mind, unless one is staunch Idealist, probability-as-cause makes no sense.

Now no frequentist, or at least none I’ve ever met, actually believes in the theory they espouse, which is limiting relative frequency. Probabilities are only defined, in that theory, at the limit: no probability can be known until infinite time has elapsed, and since we, sitting here in 2016, are well short of the mark of Infinity, yet we still see finite relative frequencies, and these observations are everywhere thought to be “well behaved”, it must be that, according to the theory, the Gambler’s Fallacy is operative. What is happening now is either being influenced by what has not yet occurred, or probability is physically real in the same way that mass or charge is. Yet there is, of course, zero evidence, and anyway its absurd, to think probability is material. Saying so is the Deadly Sin of Reification, and would be the same mistake a mathematician makes who thinks his equations are real.

(There is much, much more to be said about the absurdities that obtain in LRF: this wonderful exciting must-read #1 new best seller book has all the sober details.)

But like I said, no frequentist believes in limiting relative frequency in real life. In actual situations, frequentists behave like we pure probabilists do and believe probabilities are defined on evidence in the form of an argument: these premises imply the probability of this proposition. We know this is so because, again, of the Gambler’s Fallacy. How?

Merely state the fallacy. Every student is taught to chuckle at the foolishness of the gambler who believes numbers are “due”—yet numbers being “due” is exactly what LRF teaches. Still, the frequentist is right! We should pity the poor deluded gambler who believes probability is causative. And in that pity is the proof that probability is argument, that it is cause and essence which are of essence, and that probability is not subjective (as in Bayes) or physical (as in LRF).

Incidentally, as is clear, if probability is subjective, as Bayesians teach, then there is no Gambler’s Fallacy: it cannot exist. If probability is subjective, we should instead laugh at the theoreticians who don’t understand that everybody is free to believe whatever probability they like for any situation. Subjectivism makes the gambler right by default. The probability is his subjective belief! That we recognize the gambler is wrong is proof that subjectivism is false. So once more we have a theory that is touted but which nobody actually believes (consistently, anyway).

Why is it the Gambler’s Fallacy a fallacy? There are two points, both of which are of the utmost importance.

Point (1): we have the proposition of interest, “This ball lands black”, which has a probability m/n, a fixed, deduced number based on the evidence, “This ball must land in one of n slots, m of which are black”. This is probability-as-argument, which produces a number that all, frequentist and Bayesian alike, agree upon (look in any textbook for proof). In the fallacy, the gambler states some number (perhaps not strictly quantified) greater than m/n. But this departure alone is not the full fallacy.

Point (2): The evidence from which the probability is deduced is recognized as (observationally) true, because why? Because we know it is due to the essence, or nature, of the wheel to be this way. We know the causes which are operative have not fundamentally changed between spins of the wheel. This tells us the essence of the wheel is unchanged. It is in particular this second point which is held more strongly and which causes (!) us to recognize the fallacy; indeed, knowledge of the essential properties of the wheel and of cause is deeper and more fundamental than knowledge of probability; probability is a routine deduction, a deduction conditional on the knowledge of the essence and cause.

And there it is. Knowledge of cause and essence is at the base of every probability.

What’s that you say? “What if the wheel has gone bonkers or is worn? What of your fancy theory then?”

A Smiley Face to the reader who identifies the flaw behind this question.

There is more on the Gambler’s Fallacy in this book, which I know you’ve already pre-ordered.


  1. Is this the reason global warming predictions keep saying “will happen”, “is coming”, “hasn’t happened YET”, etc? Because probability says they will be right if they say something long enough? It’s “bound to happen”?

    (Care to bet on whether I have preordered the book or not? You may be committing the sin of overconfidence. 🙂 )

  2. Las Vegas has high confidence that the ball will occasionally land in the zero bins. I don’t think it gives any thought to the philosophy of probability. So is it guilty of the GF or not?

    Worn wheels are as causative as brandy-new ones, just different.

  3. Because I want internet smiley faces, the flaw behind the question is that it changes the evidence. All probability is conditional, and it’s up to the snarky questioner to state clearly the new evidence that they are adding to the problem.

  4. I thought it was the law of large numbers which was the piece of clothing the Statistical Emperor used to hide their embarrassment.

    The law of large numbers is, as far as I understand, quite distinct from the Gambler’s fallacy.

    Is that correct, Prof Briggs?

    You post seems to put all the blame on the Gambler’s fallacy – doesn’t the Law of Large Numbers have a role too?

  5. This is just a wild guess, since we’re talking about gambling anyway. The flaw behind the question is that the wheel having gone bonkers or being worn are accidental properties, not essential ones. These properties are known through the senses. The essence of the wheel is, as Briggs says, deeper, as in its wheeliness, or is it wheeledness?

  6. And this is why I don’t gamble. It’s just a bad habit, like cocaine – you get an excited rush, you blow a lot of money, you wake up the next afternoon with nothing but a headache and regret.


  7. @JMJ
    But not if you know when to hold ’em
    Know when to fold ’em
    Know when to walk away
    And know when to run.

  8. In reply to “Ye Olde Statistician”, unlike roulette, poker is a game of skill, and it’s sometimes possible to play well enough to beat the “house take”.

  9. Back in the day, the Democratic Party of Jefferson County, Colorado, began raising funds by holding regular bingo games. It was pretty successful too, and soon the other counties were doing the same. Then the State Attorney General ruled that since it was a game of chance it was illegal under anti-gambling laws. That included the churches, which had long been running bingo events. So one of the reporters asked what made it a game of chance and the AG replied that the players were passive victims of probabilities. But that wouldn’t apply to games in which the player must use his skills, like the state-sanctioned dog races? No, because there is skill involved in vetting the dogs. What about poker? That too is a game of skill.

    You guessed it. Before the sun went down the political parties were running poker nights.

    Don’t know what the status us nowadays.

  10. The widespread gambling we have today should be very troubling to anyone concerned with their country. Casinos, the lotteries, now online gambling, these are the institutionalization of a very bad habit. We may as well just legalize cocaine and cave in to that pack of @$$#@!$.


  11. Casinos, the lotteries, now online gambling, these are the institutionalization of a very bad habit. We may as well just legalize cocaine and cave in to that pack of @$$#@!$.

    Wanna bet?

  12. JMJ: Whoa—I actually agree with you completely. This is very, very scary! 🙂

    Usually, I go for legalizing prostitution and other such vices as other things we might as well use for revenue, but cocaine works. All gambling does is teach people they can get money for doing nothing. That being said, if we removed the welfare system and let people live with the consequences of betting the paycheck on a horserace, it might decrease. Nah, even outlawing it won’t stop it—people just love money for nothing as long they believe they can get some of the pot.

  13. Regarding the gambler’s fallacy, check out a prior blog post on the Arcsine rule.
    One can get closer and closer to 50% while still getting farther and farther behind.

    For example, after 10 tosses, you[ve won 4, 40% ad you;’re two bets behind,
    After 100 tosses you’ve won 45. You’re now up to 45% but you’re 10 bets behind.

    After 1000 tosses you’ve won 490. You’re up to 49% but you;re 20 bets behind.

  14. My understanding of the gambler fallacy is that the gambler acts as if the roulette wheel can learn and remember how it has behaved previously and then adjust its own behavior accordingly. The wheel can think and is real! Yes, it’s real. (Just to go along with Briggs’ way of explaining things.)

    IOW, the gambler is ignorant of the concept of independence (not causal dependence), which means that the occurrence of one event/proposition is uninformative about the occurrence of the other. I have no idea if the uninformative-ness is caused by “a nature or essential property of the roulette wheel.” That is, a roulette wheel is made of wood and some kind of metal, but what is the essence, or nature or essential property of the roulette wheel?

  15. Suppose that in flipping a coin heads shows up 10 times in a row. The Gambler’s Fallacy is based on an idea that a bias in favor of tails is now needed to make the probability of 11 heads so low as 0.5^11. That’s obvious nonsense. Why it needs discussion among people who are supposedly interested in probability and statistics is beyond me.

  16. Ah, I should have added, of course it’s higher than 0.5^11 *now that 10 heads have appeared*.

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