The goal of this course is to develop some of the major ideas of probability theory. Emphasis will be placed on specific examples and on ways to compute expectation values. We begin with the most basic of all processes, the simple random walk. This is equivalent to studying the tosses of a fair coin. We prove the strong law of large numbers, the central limit theorem, recurrence property and the law of the iterated logarithm for this system. The proofs of these theorems depend on an ability to compute expectation values of various random variables. In the next part of the course we develop systematic methods for computing expectation values. This leads to the study of finite difference equations. We construct the continuous process Brownian motion by taking a limit in which the finite difference equations become partial differential equations.

The second part of the course is concerned with some ideas which have wide application. These represent an abstraction of ideas involved in studying the simple random walk. The first of them is the idea of a measure preserving transformation and the notion of ergodicity. We shall prove the vonNeumann and Birkhoff ergodic theorems. We also shall prove the Poincare recurrence theorem and show how recurrence times can be estimated from the invariant measure. The second is the idea of a Markov process. We shall discuss Markov chains on a finite state space, obtain an invariant measure for the chain and prove ergodicity.

The final part of the course is an introduction to Ito's stochastic integration theory. We shall rigorously define a stochastic integral and prove Ito's lemma. Stochastic differential equations and their solutions will be discussed in a heuristic manner. The ideas involved will be illustrated by simple examples, in particular linear equations.