## Probability 101, the intuition behind martingales and solving problems with them

This post was originally written on Codeforces; relevant discussion can be found here. Recently someone asked me to explain how to solve a couple of problems which went like this: “Find the expected time before XYZ happens”. Note that here the time to completion is a random variable, and in probability theory, such random variables are called “stopping times” for obvious reasons. It turned out that these problems were solvable using something called martingales which are random processes with nice invariants and a ton of properties....