Gambler’s Fallacy Not As Crazy As Mathematician’s Fallacy

A new walks-into-a-bar joke begins (thanks to reader Kip Hansen for the tip):

A mathematician, a philosopher and a gambler walk into a bar. As the barman pulls each of them a beer, he decides to stir up a bit of trouble. He pulls a die from his pocket and rolls it ostentatiously on the bar counter: it comes up with a 1.

The mathematician says: ‘The probability that 1 would come up is 1/6, and at the next throw it will be the same. If we roll the die infinitely many times, the relative frequency of the number 1 will converge to 1/6, that is, to one occurrence every six throws.’

The philosopher strokes her chin, and remarks: ‘Well, this doesn’t mean we won’t get the number at the next throw. Actually, it’s physically possible to have the same number on the next 1,000 throws, although that’s highly improbable.’

The gambler says: ‘I know you’re both right, but I wouldn’t bet on that number for the next throw.’

‘Why not?’ asks the mathematician.

‘Because I trust mathematics, and so I expect that number to come up about once every six throws,’ the gambler answers. ‘Having the same number twice in a row is a rare event. Why would that happen right now?’

The joke, the article goes on to say, is on the gambler whose “argument is a mix of conceptual inadequacy, misinterpretation, irrelevant application of mathematics, and misleading use of language.” And in the spirit of gender parity, the gambler is a she. So is the authoress.

The article continues with other sins of gamblers, such as the eponymous gambler’s fallacy, “where someone believes that a series of bad plays will be followed by a winning outcome, in order for the randomness to be ‘restored'”, plus there are cautions about serotonin and addiction. The authoress also wonders if exposing gamblers to naked mathematics will cure them of their bad habits and thinking. It hasn’t worked for the philosopher or mathematician, as we’ll see, so that’s unlikely.

Now the authoress acknowledges “statistical models are grounded in probability theory, one of the fields in mathematics most open to philosophical debate”, which is true. But that’s because everybody hasn’t yet read—and assimilated—Uncertainty, where the true understanding of probability is given (stating it this way riles people).

I don’t agree with anybody, really, but my sympathies are closest to the gambler’s. The mathematician and the philosopher have committed the Deadly Sin of Reification. The gambler alone sought to understand the cause of the roll, in a vague way, with his idea of a restorative force, a cause. The gambler was the only scientist among the three (where I use that word in its old-fashioned sense).

The die had no probability whatsoever of coming up anything. The die was caused to come up 1. To say it has a probability is to reify a model of the die and say the model is reality itself. This is, as said, a deadly sin.

Here is one possible model of a die, out of (as far as I know) an infinite number of them: “An object has six different sides, labeled 1-6, which when tossed has one side come up.” Given that model, the probability of a 1 is, as both the mathematician and the philosopher say, 1/6.

Does that model apply, in real life, to real throws or real dice by bartenders on wine-soaked bars?

Who knows? Nobody, that’s who. The only guide is to try it and see. The model has loose similarities to real dice, but real dice are rough and real; the model is infinitely smoother. Real dice are thrown on strange surfaces with varying amounts of force and spin. Real dice are never symmetric, except grossly. They wear through use. Throwing conditions are non-uniform. People know how to manipulate throws. On and on. Dice exist. Throws exist. The model does not.

Are there other models that are better than our simple one, as the gambler thought?

Why, yes. Yes, there are.

The best model is the one which delineates all the causes of each particular throw, a model which gives “extreme probability”, i.e. 0 or 1, to each outcome. Since the causes depend on the milieu, which is ever-changing, this Reality model must change with every throw, too. It can be done. It’s just that real dice are sensitive to initial conditions, which makes measuring all the causes difficult. That’s why real dice are useful in gambling. Not knowing causes makes throws unpredictable to some degree.

Casinos try to force both unpredictability of cause and symmetry of forces operating on dice in ways we all know. That enforcement brings the simple model above closer to Reality in some aspects, while never matching it. Experience with actual throws is what gives us a notion the simple model does an adequate job abstracting Reality in controlled conditions.

The mathematician has the bartender throwing the die an infinite number of times, which is an impossibility. Not a light one, either since any finite number of throws is infinitely far from infinity. We should have been able to deduce from talking of infinite anything we’re dealing with a model and not Reality. No number of finite throws will match the model except by coincidence, and unless the real number of throws is divisible by 6, matching is impossible. The philosopher mixes up Reality of the “physically possible” with the simple model’s probabilities.

Now you will hear some say “the dice have no memory” when discussing the so-called gambler’s fallacy. The gambler seems to think they do; or, if not the dice, than whatever causes are operating on the dice, material or spiritual, hence his idea of a restorative force. We can’t prove to him he’s wrong. Especially when the finite groupings of tosses he witnesses provide confirmatory evidence he’s right. These groupings will have distributions with wide departures from the model’s theoretical limit.

The philosopher and mathematician also believe certain spiritual forces operate on the dice, which they call randomness. That force imbues the dice with a different kind of directing force, which ensures the relative frequency of actual tosses confirms to the model, which you recall they think is real.

The randomness force is real to them, which is why they speak of tossing “fair” dice. What in the world could that be, except a die that matches the imagined simple model exactly, an impossibility in Reality. Yet they say that fairness is (or can be) a property of the dice, like its weight or ink spot color. Fairness is real but, strangely, cannot be measured. It’s in there somewhere, no one knows where. Or how. Or maybe it’s in the dice-throwing milieu somewhere. Again, no one knows where. Or how.

If this doesn’t convince you everybody has a problem with reification, answer this question, “An unfair die is tossed. What is the probability it comes up 1?” I leave the answer to homework.

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Categories: Statistics

27 replies »

  1. It stands to reason, then, that if gambling in dice throws, observing the thrower, the venue, etc and noting the outcomes before one bets makes the most sense. You get a feel for the causes of the outcomes and can make better choices (assuming a gambler can make better choices, of course). The gambler has to be careful not to see what he wants to see, of course.

    An unfair or fair dice is not reality. All dice are neither fair nor unfair, they just exist. Nothing more. “Fair” applies to the desired outcome of the gambler or the casino owner. (So there is no answer to your question.)

  2. The gambler says: ‘I know you’re both right, but I wouldn’t bet on that number for the next throw.’
    ‘Why not?’ asks the mathematician.
    ‘ Because I trust mathematics, and so I expect that number to come up about once every six throws,’ the gambler answers. ‘Having the same number twice in a row is a rare event. Why would that happen right now?’

    This isn’t a fallacy. It is an observable reality. Even assuming a perfectly “fair” die and roll, 5/6 > 1/6. The gambler is correct. The odds really are against the die coming up a “1” on the next roll. (Assuming the die and roll are not crooked/fixed, of course.) Therefore, the authoress has no understanding of the topic on which she is opining.

    How typical. Stereotypes exist for a reason.

  3. The unfair die has to die. Not only is it unequal, but it does not obey the model maker.


    Be the unfair die

  4. I had a gambler friend who only played crap. He won nearly all the time. He only bet against the shooter, i.e., with the house. Before starting to play he set a dollar limit on the amount to be won or lost (no more than 25% of win limit). He knew the all the odds against the shooter winning. In a high odds situation he would increase his bet. If the shooter won more than twice he would stand down and wait for a new shooter. The whole process took no more than an hour at which point he left the casino. He was an amazing guy in many ways.

  5. This article reminds me of that famous Richard Feynman quite: “The first principle is that you must not fool yourself and you are the easiest person to fool.” This applies to the mathematician, the philosopher and the gambler. (And probably the bartender, too. Did the bartender just lose a good tip by stirring up a bit of trouble?)

  6. If the dice were completely un”fair”, 1’s all the way

    If any other number can came up, nothing more can be said

  7. “An unfair die is tossed. What is the probability it comes up 1?”

    Fairness is not a property of a die.

  8. And of course people who defend p-values will defend them on the reasoning that “data with a p-value this low would be a rare occurrence if we assume the given distribution. Why should it happen now?” But that’s okay since it’s an established statistical procedure.

  9. I have always been amused by the “trick” question.

    “You have seen a coin toss come up heads 20 times in a row- what are the odds the 21st toss will be heads?”

  10. “An unfair die is tossed. What is the probability it comes up 1?”

    I think only God knows the answer to this. But I would say 1 in 6.

    Also I may have misunderstood, but I think the one who write the article linked above is a male and therefore an author. Catalin from Romania.

  11. A bit of a tangent, but that is why if materialism is true, then free will is an illusion and there is no point debating anyone about anything, for every thought we think is caused by the initial conditions at the Big Bang; nor does true randomness exist at all. I do believe in free will, however, so I will look for meaning and purpose in life.

  12. The Map is not the Territory. So many focused on the map they don’t look around ay the beauty outside.

  13. Well I think that you would call de Finetti a mathematician and he was well aware of the necessity of assumptions. Dicing with Death describes a “puzzle” of his as follows.

    “de Finetti’s objection to any theory of necessary convergence of Bayesians in the light of evidence can be simply illustrated with help of a brilliant concrete illustration of his which we will modify slightly. Suppose that we consider inspecting a batch of 1000 specimens for quality. We examine 100 items at random and ?nd that 15 of them do not work but the speci?cation says that for a batch of 1000 to be acceptable no more than 50 should be defective. What may we expect about the number of defectives in the remaining 900? In particular, for example, what can we say about the probability that the remaining 900 will contain 36 or more defectives, a number that renders the batch unacceptable?”

    There is, of course a standard Bayesian solution involving conjugate priors etc.

    “However, as de Finetti points out there is absolutely no obligation to use rules of this form. Consider three different cases. Case 1: we believe that batches are packed from very large production runs that have been mixed together. Case 2: batches are packed from given runs of particular machines and these tend to drift badly from their settings from time to time. Case 3: we think that the product is inspected prior to packing to ensure that no more than 50 per batch are defective. This might be the case where production costs are high, the average proportion defective is close to that which customers will tolerate and the manufacturer has a ruthless attitude to its customers.”

    See page 81

  14. Stevie,

    All probability requires assumptions, as we agree, because all probability is conditional.

    De Finetti and I further agree that there is no such thing as probability; pace, there is no such thing as parameters, either.

    There is no need of “priors” and so forth unless there are parameters. Statisticians need to get themselves out of the bad habit of turning first to ad hoc parameterized models. Which aren’t needed in this simple dice example, but which, naturally, can be used.

    Since all probability is conditional, the assumptions made about the parameters, if one is going to use them, count just as much as the other assumptions.

    Alas, I’m unfortunately not mathematician enough to deduce the origin of parameters in limiting models in the most commonly used models. I’ve done it for simple success/failure models (in Uncertainty). Jaynes did it when assuming all possible location potentialities can be actualities, which gave the normal. But that never happens in Reality.

    This is a ripe area for research.

  15. Surely the best bet would be on another “1”.

    It wouldn’t be much better than any other bets, but what evidence does the gambler have that any other result is even possible?

  16. Where does this get us? I like the observation that “randomness” is not a cause — or even not a thing. But isn’t “random” a nice, shorthand way of saying, “My predictions do not approach 0 or 1”? Is anybody really *confused* by the term?

    Are these different in a practical way?

    1. Frequentist: If you rolled the die many times, the number of ones would converge on 1/6 of the total throws.

    2. Bayesian: My degree of belief that the next throw will be a one is 1/6.

    3. Reifier: The outcome is random. There’s a one in six chance it’ll be a one.

    4. Briggs (?): The outcome either will be a one or will not be. The result is determined by the conditions at the time of the throw, but those conditions are far too complex to assess in real time. That said, it’s about 1/6 for the obvious reasons.

  17. If anyone wants to read a great novel about chance and gambling with the preternatural thrown in, I recommend Last Call by Tim Powers.

  18. The gambler is dealing with reality of what he has noticed. Just because he thinks he can predict accurately enough, or relies on that observation in order to win, does not mean he is wrong about everything else.

    It means simply that he has taken his observation to the next stage and has yet to admit that he does not know enough, to bet what he can’t afford to loose.

    “Gamblers play to loose” (says my Dad), who’s Dad was a gambler but a wiz at Bridge….wrote a book! threw it on the fire in temper. I think the worst thing that can happen is beginners luck or finding that you’re good at a game of cards to the point where pride gets the better of you.

    The next door neighbour won an Aston Martin Vanquish in Vagus.
    His fiancee now wife had to drag him from the table so that he didn’t. carry on and put all the money back. He was very drunk at the time, apparently. Black Jack was the game.

  19. Coins and die and playing cards:
    Don’t have properties that are outside of physical reality

    The object is always held by a person or a machine. Even if simulated on a modern computer

    So the trick or the catch is somewhere between concepts that are finite, (i.e. numbers of possibilities), overlaid on a situation in real time with an infinite number of variables.
    Theory versus practice!

  20. “Don’t have properties that are outside of physical reality”

    Don’t have “rules” or “behaviour” in isolation, as objects, outside of physical reality
    where the pattern is observed again, and that pattern is theoretical, too

  21. “An unfair die is tossed. What is the probability it comes up 1?”

    Some possible answers.

    1. For an n sided die, sides numbered 1 to n, probably not 1/n. 1/n if and only if the total of the probabilities of tossing all the other numbers 2 to n is 1-1/n.

    2. Depends how many of the sides – if any – were marked 1.

    3.Who cares, it was tossed away. Ambiguity of language allows for this, depending on context.

    For example, if the context were a regulatory inspection of a casino’s dice, and the biased “unfair” ones were tossed into a crusher, melter, shredder etc.

  22. 4. If tossed in outer space, the die may never “land” or “come up” with any result, or may not do so within an acceptable observation period.

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