# 7.1 Introduction

"The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along."

-Sir Francis Galton

Mathematics is often thought of as an exact discipline. In fact, many people who practice math are drawn to it because it tackles situations in which there are clear and predictable answers. There is a certain comfort in the idea that we can use mathematics to make exact predictions about what happens in the future. For example, we could use the mathematical formulation of physical laws to predict the outcome of a coin flip if we knew enough about its size, weight, shape, initial velocity, initial angle, and its other initial conditions. In practice, however, we have a very hard time knowing all of the conditions that contribute to the outcome of a coin flip. In the face of such complexity, we call the flip a "random" event, one in which the outcome is based solely on chance and not on any immediate knowable cause. Nonetheless, mathematics has a broad set of tools to explain and describe events that appear, like the coin toss, to be random. This set of tools makes up the mathematics of probability.

Does the past determine the future? If an event is truly random, the answer must be "no." There would be no way to predict the outcome of a specific event given knowledge about its previous outcomes. Although it might seem that situations like this are beyond the reach of mathematics, the truth is that random events behave quite predictably, as long as one has no interest in the outcome of any single event. Taken on average, random events are highly predictable.

Probability theory manifests itself in many ways in our daily lives. Most of us have insurance of some form or another-house, car, life, etc. These are products that we purchase to help mitigate risk in our lives. We often associate risk with unpredictable outcomes. This could be in the context of a small business opening in an up-and-coming neighborhood, a commodities trader making decisions based on how global political situations affect prices, or a teenager getting behind the wheel for the first time. All of these situations involve a certain amount of complexity that is functionally unpredictable on a case-by-case basis. Probability theory, however, shows that there is paradoxically a large amount of structure and predictability when these individual situations are examined on a larger scale.

Probability theory shows that we can indeed make useful analyses and predictions of events that are unpredictable on a case-by-case basis, provided we look at the bigger picture of what happens when these events are repeated many times. Concepts such as the Law of Large Numbers and the Central Limit Theorem provide the machinery to make predictions about these types of situations with confidence. One of the most ubiquitous, and familiar, uses of probability is in gambling. Casinos are the ultimate "players" in using mathematics to foresee the results of a series of events that, taken individually, are functionally random. Indeed, the mathematics of probability ensures that while an individual gambler may have a good night or a lucky streak, in the long run, "the house always wins." Have you ever wondered how Las Vegas seems to have vast amounts of money to spend on glitzy hotels and golf courses in the middle of the desert? Gambling is a large, lucrative business, and its success is due, in part, to the laws of probability. In this unit we will see how probability, the mathematical study of the seemingly unpredictable, has developed over a period of time to become an extremely valuable tool in our modern world. We will see its relatively late origins in European games of chance and its most recent applications in modeling and understanding our increasingly complex and unpredictable world. We will ponder how it is that news networks are able to predict the winners of elections before all the votes have been counted. By the end of this unit, we will have a sense of how mathematics can be used to make accurate predictions about unpredictable events.