Probabilistic combinatorics is a fast developing area started with works of Paul Erdos and coauthors in the 1950s and 1960s. It has deep impact on theoretical computer science, especially in the theory of randomized algorithms and also on number theory, through the notion of pseudo-randomness (which, for instance, plays the key role in the proof of Green-Tao theorem on long arithmetic progressions in primes; see Additive Combinatorics for more details).
Here are the two main streams in this areas: