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.  
 A: 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.
A: In my opinion, pure mathematics may well predict natural phenomenons, given that the phenomenon is proven to have a certain mathematical property. For instance, gravity field has the mathematical property of being a vector field - one cannot invoke pure mathematics to prove that, but, having proven that, may learn a great deal about gravity. This doesn't mean that multivariable analysis must be derived from esperimenting with gravity field.
I suggest that randomness is one such mathematical property that an object may or may not have - one cannot use pure maths to justify that the object is random, but if it is, its behavior is predictable by probability theory.
A: Probability theory is a mathematical abstraction that makes no claims to prove properties of the physical world. It turns out to be an abstraction that works very well in applications of mathematics. However, the mathematical abstraction is not a theory in the sense of physics as it deals with concepts like an infinite sequence of experiments that cannot be realised or tested in the real world. Conversely, a precise analysis in terms of physics of an experiment like tossing a coin would involve issues that are outside the scope of the mathematical abstraction given by probablity theory.
