In my years of reading on forums, this is the one error I see time and time again. It concerns confusing the probability of a series with that of a single.
For example, there is a famous example in probability called the "Birthday Problem" which states that in a room of 23 people, there is a 50% chance that at least two will share a birthday. Does this suggest a system for roulette? what is the probability that in a sequence of spins, you will get at least one repeat?
If you do the math, it turns out that in any 8 spin sequence, there is a roughly 56% chance that there will be at least one repeat in the sequence. No problem with that, but the error occurs when you make a statement like this:
"So it appears that after 7 consecutive numbers have appeared and none are repeats there is a 56% chance that one of the 7 will repeat on the next spin."
This is saying that there is a 56% chance that you will get a win when betting on 7 numbers! but any casino which offered the equivalent of such odds would soon be out of business. The mistake lies in assuming that you can apply the probability of the series (7 spins with at least one repeat) to that of a single outcome. But once the 7 spins have gone, probability applies to the next spin only, so the original probability is now meaningless. All you can do is bet the last 7 numbers and hope that one of the last 7 repeats. What is the chance of that?
The answer is 7/37, no more and no less. If you doubt this, assume that the last 7 numbers were 17,1,32,25,8,12,28. The chance that 17 will hit on the next spin (and so result in a repeat) is 1/37, the chance that 1 will repeat is again 1/37. Similarly for each of the others. Since these are mutually exclusive outcomes, we can add the results, which gives 7/37.
You can indeed make a system out of the knowledge that there is at least one repeat with probability 56% in the last 8 spins, but in order for the probability to remain valid, you have to place your bets from spin 1, not after spin 7. So on spin 1 you put one chip on the last outcome, on spin 2 add another chip to whatever just hit, and so on, until you get a repeat (a win). But in that case, your profit will vary according to when you get the repeat, assuming you do get it. 56 times out of 100 you will indeed get at least one, but what you cannot say is that you will win 29 chips 56 times out of 100, betting 7 numbers!
Making this mistake is no different, in principle, to "calculating" that because there is a 99.9% chance of getting at least 1 black in 10 spins, then after 9 spins with no blacks the chance of a black on the next spin is 99.9%. This is of course, none other than the gambler's fallacy, but it may not be so easy to recognize it in more unusual or complex scenarios such as the probability of repeats.