There is a good connection between probability and Venn diagrams: Both are about area. Probability is about area where the area of everything under considerat

There is a good connection between probability and Venn diagrams: Both are about area. Probability is about area where the area of everything under consideration is 1. So, there is a set of trials. It has area 1. Each subset of the set of trials is an event and has an area, its probability. Then we can move on to random variables, distributions of random variables, independence of events and random variables, the event that a random variable has value <= some real number x, etc. In pure math, since H. Lebesgue in about 1900, the usual good theory of area is Lebesgue's measure theory. The ordinary ideas of area we learned in grade school, plane geometry, and calculus are all special cases. But Lebesgue's theory of area handles some bizarre, pathological, extreme cases. And we can show that there can be no really perfect theory of area -- e.g., there have to be some bizarre subsets of the real line to which no nice theory of area can assign a length. But, once we have the Lebesgue theory, the usual way to show that there is a subset of the real line without an area uses the axiom of choice. Well, in 1933, A. Kolmogorov wrote a paper showing how Lebesgue's theory of area would make a solid foundation for probability, and that approach is the standard one for advanced work in probability, statistics, and stochastic processes.