What are some examples of applications of the probabilistic method to areas where it might not be expected? I was just wondering how feasible it would be to present different ideas about the probabilistic method to those who don't necessarily have a lot of math background.
One really interesting example I found was hat games, and the applications of Hamming codes and probabilities to find really slick ways to think about different hat puzzles. I was just wondering if you had any other specific examples that don't take a lot of build up to get to some interesting results.
I plan to introduce the basics of probability and expected value, but I don't think I can get to topics like the Local Lemma and Martingales. Also, I don't think I'd like to go very deep into graphs or analysis either, given that a lot of people aren't very familiar with those concepts.