Does the concept of probability/possibility apply to mathematics?

That's a very interesting question. Philosophers have been arguing about the concept of possibility (and its dual, necessity) for some time.

There's a sense of "necessary randomness", that Chaitin has written very extensively about.

http://www.umcs.maine.edu/~chaitin/

Corresponding to this notion, there are stochastic models of (generally-agreed) necessary truths. The best known is probably the "probability of n being prime":

http://primes.utm.edu/glossary/xpage/PrimeNumberThm.html

But there are plenty others - e.g. the 3n+1 problem

http://en.wikipedia.org/wiki/Collatz_conjecture

or the question of who wins (first or second player) an integer-parametrized family of combinatorial games.

More exotically, Neal Stephenson's "Anathem" and Greg Egan's short stories "Luminous" and "Dark Integers" explore the possibility that what we think of as "necessary truths" are in fact contingent truths, frozen at some point in the distant past, and exerting a pervasive influence. (Note: I think this might sound ridiculous to a logician, but moderately reasonable to a cosmologist.) It is quite difficult to tell the difference between a necessary truth and a contingent truth which has always been true.

More prosaically, we do make errors and (given things like cosmic rays and other low-level stochastic processes) it seems unlikely that any physical process could be absolutely free of errors. We might believe something to be impossible, but erroneously. Your answer to the question "Are there any necessary truths?" probably depends on your degree of Platonism.