When I was in graduate school, in my stochastic signals course we were assigned homework to give an example of a ‘random variable’. I went by the book, and gave some kind of mapping employing the ‘modern definitions’ described at http://en.wikipedia.org/wiki/Probability_theory and that build up probability theory as a kind of integral calculus. It turned out neither the TA nor the professor could understand what I wrote, and had expected an answer like ‘rolled dice’. I got credit, eventually, mainly because I cried for them and wouldn’t go away, and because they liked me. I was very proud to have figured out what the book was talking about.
Since then I have become attached to another abstract view of probabilities, which is that, if you have certain excellent requirements of what a generalization of Aristotelian logic to numbers between 0 and 1 would look like, then the only possible solution turns out to be probability theory. Aristotelian logic is completely agnostic about its subject matter, so this form of probability theory is exactly the same. It has the advantage over the ‘integral calculus’ approach of being more obviously applicable to logic-like problems, such as pattern-detection; however, anything calculus-like tends to be easier to visualize, if you ask me, and so perhaps may be more productive for theoreticians.