TITLE: Applications of Probability Theory in pure mathematics
QUESTION [12 upvotes]: 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?

REPLY [6 votes]: The Erdős–Kac theorem shows that the (log of the log of the) prime factors of a number are Poisson/normally distributed.