We all know that randomness has several useful applications, ranging from quantum physics to cryptography to experimental design. What, though, does it offer the poor pure mathematician, who cannot afford a quantum computer, has nothing to hide, and has even less interest in real-world phenomena? In this talk we will give a partial answer to this question by demonstrating the use of the probabilistic method in extremal combinatorics. No prior knowledge of combinatorics is necessary, although some familiarity with colours — in particular, red and blue — could be useful.