A mathematician, a statistician and a fool observe a roulette table where black has come 20 times in a row, and think about betting.

The mathematician assumes that the next outcome is independent of the previous ones, and that either black or red has a chance of approx 48% to turn up. Unsatisfied by the odds, the mathematican chooses not to bet.

The statistician, seeing that there is a 20-streak of black, believes that the roulette has some "construction bias" that favors blacks. The statistician bets on black.

The fool bets on red.

:rofl:

--- something I had read a long time ago ---

@tim wood @TonesInDeepFreeze @SophistiCat @TheMadFool

Here again is a clarification to my original post.

I was first concerned with an ideal scenario for the dice game. This scenario has to be imagined in heaven, so to speak, where angels gamble. Maybe a Platonic-Pythagorean heaven with pure intellects gambling. Said heaven would be a single actual world in which pure harmony would prevail in the arrangement of everything and pure justice in the distribution of everything. The law of probability would be an infallible law here. The 1/6 probability for each individual die roll becomes a 1/6 surety. For there are no interfering factors and no manipulation in this heaven and no side of the dice would have any advantage in itself independent of the law of probability, which would be a law of certainty in heaven. If an angel now rolls the dice and the 6 appears, then everyone knows that in the next 5 rolls no more 6 will come. The only surprise for the angels, which there must be if it is to be a game of chance, would be that on the second roll they absolutely cannot tell whether the second roll will be 1, 2, 3, 4, 5. And if the second roll is a 5, then you know you don't have to bet on the 5 and 6 on the third roll. The 6 sixth would be clear. In heaven it would not be so exciting. After the 6th roll, it starts all over again and the greatest possible angelic tension sets in, that is, you can't tell if the 1,2,3,4,5,6 would come on the roll. The probability or surety would be 1/6 again. It would be maintained even with constantly changing gamblers. The gambler's fallacy would not be a fallacy here. And maybe there is a fair God behind all this, who sets it all up so deliberately. The angels could not perceive their God and His intentions, they might even be agnostic about His existence.

Now we come to hell and its dice-playing demons and devils. The law of probability here would not only not be a law of surety, but rather a law of improbability or unsurety in every respect. It would be just a world of chaos and disharmony and injustice of all distribution of anything. The 1/6 says nothing here when rolling the dice and is simply based on self-deception. It could be here that for all eternity only the 6 is always rolled or that it never appears for all times. In the former case the probability for 6 would be 100%, in the latter 0%. A superordinate probability calculation, i.e. a meta-probability, would not help here either. Perhaps one could assume that all devils want to influence the dice fall to the disadvantage of all other devils by means of psychokinesis. And now and then Lucifer as ruler will intervene absolutely arbitrarily likewise to the advantage or disadvantage of one or the other.

Now to my original OP question.If our world were more like the heaven I described, the gambler's fallacy would not be an absolute fallacy. In a merely practical or pragmatic sense it would be a fallacy, but not in a theoretical one. This would only be the case in hell.

Ultimately, to better understand probability in our world, you have to better understand the world.
If there is only one world, which is completely determined, then the probability is only a relative value. If the many worlds interpretation corresponds to the truth, then the probability seems to me to be absolute. With every roll of the dice, 5 new real worlds would then begin to exist.

It can only mean that out of 6 times rolling the dice, the 6 will occur one time, right? — spirit-salamander

Wrong.

The only surprise for the angels, which there must be if it is to be a game of chance, would be that on the second roll they absolutely cannot tell whether the second roll will be 1, 2, 3, 4, 5. — spirit-salamander

If it is impossible to get a 6 on the second throw, then the probability of getting 6 on the second throw is 0, not 1/6. And if the probability of getting a 6 on the second throw is 0, you're not really tossing a fair die. The probability of getting a 6 on any throw of a fair die toss is 1/6... so your angels aren't tossing a die.

The results you get are realistic, but they fit a completely different game. This is something more like drawing number tiles from a bag like one does in scrabble. In this scenario, we have 6 tiles in the bag. You don't replace tiles between draws until the bag is empty, at which point you put exactly 6 tiles back. That is the game that gives the results you describe, not die tossing. And it's not the only game in town... the angels could draw from a bag of 12 tiles, with two tiles each numbered from 1 to 6.

In a merely practical or pragmatic sense it would be a fallacy, but not in a theoretical one. — spirit-salamander

Your theory is flawed though. If this is a fair die, the probability of it landing on each number is 1/6 on every throw.

It could be here that for all eternity only the 6 is always rolled or that it never appears for all times. In the former case the probability for 6 would be 100%, in the latter 0%. — spirit-salamander

That would almost never happen with fair die. But it could happen. But you're confusing the theoretical probability with frequentist probability here.

The 1/6 probability for each individual die roll becomes a 1/6 surety. — spirit-salamander

As @InPitzotl makes clear, but in some fewer words, if a surety, then not a probability.

If an angel now rolls the dice and the 6 appears, then everyone knows that in the next 5 rolls no more 6 will come. — spirit-salamander

Wrong.

How would you philosophically explain and describe the probability 1/6 in the dice rolls. What is the 1 here, what is the 6 and what / and how do they relate to the real world?
I have come to the conclusion that it is all very baffling and perplexing because you get to questions of chance and determination. — spirit-salamander

As has been pointed out, your ideas of what probability should mean are very far from what is usually understood by that term. If you are interested in this subject, you may want to read something introductory first.

Now as for philosophical interpretations of probability, there are several, and all of them are contentious. The standard mathematics of probability (known today as Measure Theory) is well understood, but how it should be interpreted in real life is not that clear. Statements like

6 is the cardinality of the set of possible outcomes; that set is the event space. 1 is the particular outcome, which is one of the members of the event space. Division expresses the ratio of the particular outcome to the possible outcomes. — TonesInDeepFreeze

don't actually explain why TonesInDeepFreeze would bet a particular way in a game of dice.

My own preference is for the epistemic interpretation (often loosely referred to as Bayesian, although Bayesianism does not exhaust all epistemic interpretations). I already tipped my hand earlier when I declared that probability is a measure of your uncertainty: that is, in a few words, the essence of the epistemic interpretation. There are others: frequentism, propensity. In my opinion, frequentism is philosophically untenable: it is basically a naive interpretation of mathematics, which invariably ends in hand-waving when the rubber hits the road. However, frequentist heuristics underlie much of the practical statistics that was developed in the last century. Propensity better suits certain metaphysical ideas, and it can be complementary to the epistemic interpretation.

SEP, as always, provides a nice overview of the interpretations of probability.

If the gambler's fallacy were not fallacious, casinos would all close.

Just to be clear, the reason casinos profit is the percentage payouts. People believing the gambler's fallacy helps the casino only to the extent that a person places a bet with more unfavorable payout instead of placing a bet with a payout not so unfavorable or no bet at all. For example, if a person bets on 6 because they think 6 is due then that helps the casino better than if the person bet on Pass.

Just to be even clearer, it doesn't matter what is believed but what is true. If the fallacy were true, the odds would change and become predictable, and the casino would lose.

Yes, I was not qualifying your remark regarding the implication of the counterfactual.

Of course, the gambler's fallacy cannot be mathematically true. But, of course, if the gambler's fallacy turned out to be empirically true, and the casinos did not adjust their payout ratios, then the casinos would lose (and even if people didn't even play with the gambler's fallacy in mind).

I used to watch the roulette wheel and after 5 or 6 reds in a row, I would bet on black, thinking it was due. However, the past does not exist and so there could be 12, 20 or even 50 reds in a row. Doubling ones bet to cover the previous loss may only go so far, before one runs out of money.