An equation is an identity. Even when you seems to find a solution you did not find nothing. You are just another way what really is "in".
As we say in french : "tu ne peux y trouver que ce tu y apportes comme lauberge espagnole"
You use the infinity concept, the indepedency of events and so on as basis of thinking.
But do not forget that infinity, independency are not "real".
The roulette, the coins and other systems are physical machines producing something we call "random", unpredictable.
What make me think that the probability theory is wrong is the fact that we use it in real life.