Does Probability theory require empirical research to justify or is it pure math? We can prove the fundamental theorem of calculus without any reference to empirical data. We can't "prove" coulomb's law, we make experiments and describe the phenomena using mathematical tools. Where does Probability fall on that spectrum? if Probability suppose to predict natural phenomenon wouldn't you need to verify it works via experiments? If so, why is it considered a part of Math and not Physics? If empirical observations aren't needed, that means we can prove that the limit of(a(n)\n), a- number of times a coin lands on heads, n- number of throws, converges to 0.5 as n goes to infinity. Same as we can prove any other Theorem using only a pen and paper, But to me, that doesn't make any sense. I don't see how pure Math can be used to PROVE physical phenomenons.  

These days it is easy to simulate many probability problems with computers and come up with a good idea about the correct answer. 
If the pure math is not helping due to complexity of problem, then the simulation is very helpful, specially if there is a debate on which solution is the correct one.
