So I’ve been reading some critiques of frequentism and p-value tests, and one thing that stuck out was how the “intuitive” understanding of frequentism will necessarily lead to fallacies like The Gambler’s Fallacy. This is one of those fallacies that is not weak Bayesian evidence since it violates Bayes Theorem. The Gambler’s Fallacy is thus:
The Gambler’s fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a Monte Carlo Casino in 1913), and also referred to as the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are then more likely.
Someone subscribing to the Gambler’s Fallacy would think that, since I flipped three heads in a row, the probability of me flipping a fourth heads is 6.25%… since the probability of flipping heads four times in a row is 6.25%. But no, that is incorrect. The coin doesn’t have a memory; the probability of flipping heads that fourth time is the same as flipping tails.
The problem with frequentism is that this leads people to think that probability is part of the essence of an object instead of a description of your internal state of knowledge. So someone who thinks that 50% is a property of the coin being flipped will more than likely subscribe to the Gambler’s Fallacy. The 50% property of the coin builds up some sort of probability equity if you flip too many heads or tails in a row!
On a message board I frequented about 15 years ago, a huge argument that lasted for months on end went on about airplane disasters. One faction on the board argued that since the probability of being in a plane crash is [X%], the more you fly on planes, the probability of being in a plane crash will eventually move to 100%. This makes no sense from a Bayesian point of view (what new knowledge is being input in the system on each flight that increases the probability?), but makes sense from an intuitive Frequentist point of view where probability is based on n number of trials (though an actual Frequentist who used probability regularly would not make that mistake).
The n number of trials being used to establish a fact about the object in question instead of your own subjective state of knowledge.
Thinking of things in terms of n number of trials leads one to believe that in some unspecified point in the future, the improbable thing must happen. This is the one huge critique of frequentism that Bayesians have, that frequentism is based on an infinite set, and relying on infinity abandons empiricism (we will never empirically verify the infinitieth trial).
This is different than some event that is not independent, like picking cards from a deck. In that instance, 1/52 is a fact about the object in question; in that case both your subjective knowledge and a fact about pulling a particular card from the deck are the same. But even so, something like the Monty Hall problem still makes you realize that Bayes Theorem wins the day.