I was just reading the impressive paper by Tim Gowers The Two Cultures of Mathematics when I noticed the various connections between combinatorics and randomness. As a non-mathematician, it is not intuitively obvious to me what is the scope of relationship between them, apart from the idea that combinatorics involves counting and counting is needed for preliminary probability (I work in machine learning).
What are the major results that connect combinatorics and randomness, and how do they connect?