Chapter 1

Basic Probability Theory

# 1.1 Introduction

Probability theory is the mathematics of randomness. This statement immediately invites the question “What is randomness?” This is a deep question that we cannot attempt to answer without invoking the disciplines of philosophy, psychology, mathematical complexity theory, and quantum physics, and still there would most likely be no completely satisfactory answer. For our purposes, an informal definition of randomness as “what happens in a situation where we cannot predict the outcome with certainty” is sufficient. In many cases, this might simply mean lack of information. For example, if we flip a coin, we might think of the outcome as random. It will be either heads or tails, but we cannot say which, and if the coin is fair, we believe that both outcomes are equally likely. However, if we knew the force from the fingers at the flip, weight and shape of the coin, material and shape of the table surface, and several other parameters, we would be able to predict the outcome with certainty, according to the laws of physics. In this case we use randomness as a way to describe uncertainty due to lack of information.^{1}

Next question: “What is probability?” There are two main interpretations of probability, one that could be termed “objective” and the other “subjective.” The first is the interpretation of a probability as a limit of relative frequencies; the second, as a degree of belief. Let us briefly describe each of these.

For the ...