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.