Applications of Probability Theory in pure mathematics

The topic of Probabilistic Combinatorics. See http://en.wikipedia.org/wiki/Probabilistic_method

This is a powerful way of giving non-constuctive existence proofs for lots of different (finite) mathematical structures and determining their properties.

I found this the highlight of all my undergraduate probability courses. If you are interested in learning more I would recommend: "The Probabilistic Method" by Alon and Spencer.

The Erdős–Kac theorem shows that the (log of the log of the) prime factors of a number are Poisson/normally distributed.

Probability that a random permutation has no fixed point among the first $k$ elements

Is the complex exponential function injective, surjective and/or bijective - and why?

What could be an intuitive explanation for $ \sum\limits_{k=1}^{\infty}\frac{1}{k2^k} = \log 2 $?

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

Why were filters and nets in topology named filters and nets?

For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.

Can you explain this card trick?

What is an additive group?

Finding :$x_1^8+x_2^8+\cdots+x_8^8$

Are there algebraic structures with more than one neutral element and/or more than one inverse element?

independent, identically distributed (IID) random variables [closed]