# Chapter 3: An Introduction to Probability

**Advanced**Created by: CK-12

Introduction

The concept of probability plays an important role in our daily lives. Assume you have an opportunity to invest some money in a software company. Suppose you know that the company’s past records indicate that in the past five years, the company’s profit has been consistently decreasing. Would you still invest your money in it? Do you think the chances are good for the company in the future?

Here is another illustration: suppose that you are playing a game that involves tossing a single die. Assume that you have already tossed it \begin{align*}10\;\mathrm{times}\end{align*} and every time the outcome was the same, a \begin{align*}2\end{align*}. What is your prediction of the eleventh toss? Would you be willing to bet \begin{align*}\$ 100\end{align*} that you will not get a \begin{align*}2\end{align*} on the next toss? Do you think the die is “loaded”?

Notice that decisions concerning a successful investment in the software company and the decision of not betting \begin{align*}\$ 100\end{align*} for the next outcome of a die are both based on probabilities of certain sample results. Namely, the software company’s profit has been declining for the past five years and the outcome of rolling a \begin{align*}2\end{align*} ten times in a row is quite strange. From these sample results, we might conclude that we are not going to invest our money in the software company or continue betting on this die. In this chapter you will learn mathematical ideas and tools that can help you understand such situations.

- 3.1.
## Events, Sample Spaces, and Probability

- 3.2.
## Compound Events

- 3.3.
## The Complement of an Event

- 3.4.
## Conditional Probability

- 3.5.
## Additive and Multiplicative Rules

- 3.6.
## Basic Counting Rules