re: Please answer the following math/philosophy question

You're referring to the Monte Carlo fallacy, which is a little different. The MC fallacy pertains to the probability of an event happening as one step in a causal chain rather than the entire chain.

Going back to the nomenclature I used earlier -- Event X with Y probability and Z attempts -- we can see that as Z increases, the chances of X happening go up. But there are two things that fuel the chance of X happening: Y probability, and Z attempts. It's not fallacious to say that as Z increases, the chance of X happening increases. Where it is fallacious, and where the MC fallacy comes into play, would be if you argued that as Z increases, Y also increases. Y stays the same, the chance of X happening just goes up (and eventually reaches 1) because of Z going up.

Does that make sense?

EDIT: Just thought of an example that fits. You would agree that I have a better chance of flipping a coin and getting heads if I got to flip the coin 10 times instead of 1. But that's because I got 9 more tries. We would have a problem if I said that because I didn't get heads before, my odds of getting heads go up to 60% or 70% or whatever have you. The former (my chances improving because of more attempts) is just basic logic, the latter (the odds themselves changing because I failed previously) is the MC fallacy.