If a process is truly random, how can we begin to understand its behavior? We turn to probability theory to model stochastic processes and to answer important questions about how these systems work. Specifically, we turn to Markov chains. Markov chains have the property that, given the current state, the past has no affect on the future. Both for finite and infinite chains, studying the expected value, G(x, y), is critical to get a sense of what’s going on. Expectation allows us to understand how the random walk behaves in various dimensions. Furthermore, the limiting process of the random walk gives us Brownian motion, which is a useful model in physics, biology, and finance.