The question is if in 30 spins where 25 numbers are red and 5 are black, what will happen in the next few spins. You show me a window from a live roulette set of numbers that has this condition, and then 5 more red spin out. You are not going to find it any time soon. Maybe once in the next 1 million spins.

palestis,

What you find is that the distribution of blacks and reds is the same whether or not you wait for 30 spins with only 5 blacks. Have you actually tried doing this for purposes of comparison?

The trouble is, we tend to assume that, just because we don't often see such deviations, the next few spins will see a return to normality. This is a remarkably stubborn cognitive bias. The next 30 spins WILL probably generate more than 5 blacks, not BECAUSE the previous 30 spins didn't but simply because this is a more likely event ANYWAY, regardless of past spins. You're attributing some kind of cause-and-effect to the previous 30 spins which doesn't exist.

From a philosophical point of view, GF is interesting because it's not easy to pinpoint exactly where the error lies. In my view, it's not really about independence of events, but consistency. GF is a logical error, but independence is about the world out there, and the outcomes may not be independent! (although in 99.9% of cases were are justified in believing that they are). One who commits GF is contradicting himself because he implicitly assumes that the wheel is fair, but then says that his triggers are valid. Being "fair" doesn't mean only that the wheel has a tendency to equilibrium, it also means that there is no regular pattern to the outcomes. These two kinds of unfairness (bias and dependence) can both exist and are independent of each other.

Bias: There is a tendency for outcomes to favor some events more than others.

Dependence: Outcomes are affected by previous events and therefore exhibit regular patterns.

So theoretically, a wheel could be

1. biased, dependent

2. unbiased, dependent,

3. unbiased, independent

4. biased independent

We can assume, for the most part, that no. 3 is true. One who commits GF correctly assumes that the wheel is unbiased, but he forgets about independence. So equal distribution can generally be counted on, EVENTUALLY, (but certainly not in the short term, and even this equal distribution is in regard to PROPORTIONS, not absolute numbers of even chances, dozens and so on. In fact, absolute numbers tend to DIVERGE over time, not converge to a limit).

So GF is a double whammy: the player assumes that he is dealing with a fair wheel, but doesn't take into account that this means there should be no regular pattern in the outcomes (which, if true, means that his triggers are invalid), and also he tends to assume that equilibrium will manifest in the short term, and additionally, in regard to absolute numbers, neither of which is true.