In the reality where we exist, there is no way that "each number is equally likely from one spin to the next" because ALL numbers are FORCED to strive towards a statistical balance, so in practice this means that every number that is a continuation of the distribution of numbers, must obey the rules prevailing in this reality, so this means that each such number is only part of a larger whole which in the final effect always must show a specific "picture", which in this case is the STATISTICAL BALANCE.

So you're saying that the wheel somehow knows what numbers came up previously and adjusts future outcomes so the statistical balance is formed? It's an appealing thought, but what actually 'forces' the proportion in the long run to conform to balance isn't any mysterious force, but just the characteristics and properties of the physical system as a whole, and primarily the wheel. Because this is true, there is no

**necessity** for balance at all, because if the wheel became biased for some reason, there would be no balance, which proves that it's the physical variables which cause the long-run pattern, not 'statistical pressure'.

The assertion that there no any connection between one spin and the next is illogical from a mathematical point of view, because all numbers must constantly strive to maintain a balance with each other, which may be disturbed ONLY temporarily, but in the end ALWAYS MUST BE A STATISTICAL BALANCE, which could not exist without this "dependence", because there is no effect without a cause. (Of course, this does not mean that the next spin will be depend on the previous spin, because in this case the connection between these two events will be only partial/indirect)

It's not mathematics which is at issue here, but physics, as I've already explained. Yes, there is no effect without a cause, and the cause of statistical BALANCE is primarily the symmetry of the wheel (the fact that it's unbiased). If the symmetry is removed, there will be no balance. Logically, then, what is the cause? The statistical 'pressure' must obey the physics, not the other way round.

You misunderstood me, the point was that IN THE CONTEXT we are talking about, this part of the text "the occurrence of one does not affect the probability of occurrence of the other" - is only an ASSUMPTION, because it has never been proven, so if it does influence in any degree, it means that it CANNOT BE independent.

No, I don't think I misunderstood your previous post. The statement was a definition of independence, so it's beside the point to talk about proof. But since you mention it, are you saying that it has never been proven that spins are independent? It's actually been proven many many times. How many systems have been created which use triggers based on events A and B which are supposedly related, only to find that there is no connection at all, and that you might as well have bet a random set of numbers? Gambling forums are full of such systems.

Actually, what has never been proven is that there is any dependence, not independence.

It doesn't matter, because in this case it is enough to use the method of deduction to reach the appropriate conclusions: "An accumuIation of smaII groups wiII form a Iarge group; therefore, anything that appIies to a Iarge group wiII aIso appIy to a smaII group, in a smaIIer way. So, the statisticaI pressure for numbers to conform to their probabiIities wiII be feIt in aII numbers that form any smaII group, just as they do for a Iarge group." - "It comes down to this: in a controIIed environment that invokes a statisticaI certainty, there has to be a cause, and an effect. The effect is that the numbers conform to their statisticaI expectation."

Ok, so as I said in my previous post, Ellison is making the argument that spins aren't independent because outcomes conform to probabilities. But if there really was a 'law of small numbers', as you are suggesting, then he would be correct.

If outcomes conformed to their long-run probabilities in the short run (say 50 spins if playing the ECs), then you could easily make a fortune very quickly, because knowing that in 50 spins the probability of at least 20 reds is over 90%, you would only have to wait for say 35 spins with only 10 reds, then you would be overwhelmingly likely to make a flat bet profit in the next 15 spins. Therefore, this means that spins would be effectively dependent, because knowing that event A occurred, the probability of event B has changed.

The problem is, there are countless systems based on this kind of logic, and none of them work. There is a reason why 'The Law of Small Numbers' is called the gambler's fallacy, it's because there is no such thing. You can argue that what happens in the long-run must happen in the short run, to a 'lesser extent', but then you haven't really understood what 'the long run' means. Mathematically, it means infinity. Try simulating some events (like the one I suggested above) and see how many spins you need to get close to the theoretical probabilities. And realize that probabilities are proportions, they don't tell you anything about exactly how many reds or whatever will be in the next X spins, and they certainly can't tell you which ORDER the reds will come in.