Applications of Probability Theory in pure mathematics My (maybe wrong) impression is that while probability is widely used in science (for example, in statistical mechanics), it is rarely seen in pure mathematics. Which leads me to the question - 
Are there some interesting application of Probability Theory in pure mathematics, outside Probability Theory itself?

In addition to examples, the overall "gestalt" answer to the question is: probability is used everywhere in mathematics.  It is a basic idea on the level of algorithm, algebraic structure, geometry, calculus, or other very ubiquitous things.  It has become a very popular source of questions and intuitions in research, in all fields.   Knowing that this is true, it is not surprising that many examples of theoretical uses of probability can be posted.
Being a basic language, it is also true that many of the uses of probability are basic, and do not go far beyond the idea of a probability distribution, frequencies of events, expectations, related combinatorics and so on.  But in some fields, advanced results in probability theory are constantly being used.
