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You mentioned that it can be proved that spins are independent. I say we can only assume it, not prove it. Where is the proof?
In the random game of roulette:
1. The number of pockets on the wheel determines the odds of winning.
2. At each spin of the wheel there are 37 pockets. The ball is equally likely to fall into any one of the pockets.
3. Once a number has hit the dealer does not block a number for hitting again.
4. Since the number of pockets remains the same from one spin to the next, there is no dependence.
Where people are fooled:
The free mode offered by many of the online casinos/cellphones/games sometimes contain program code that's designed to fool people into believing that they can beat the game.