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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