**Probability theory** provides a numerical framework for measuring *uncertainty*. This area is important for researchers since all inferential statistical results are ultimately based on probability theory. Understanding probability theory provides fundamental insights into all statistical methods beyond univariate descriptive statistics. Probability is heavily based on the mathematical notion of **sets**. A set is just a collection of things. These objects may be people, places, things, or numbers. Several mathematical operations may be applied to sets, such as unions, intersections, and complements. We will not delve too deeply into the mathematics of sets and probability. We must, however, understand the basic ideas of probability theory so we can understand hypothesis tests in later sections.

The **union** of two sets is a new set that contains all the elements in the original two sets. The **intersection** of two sets is a set that contains only the elements contained in both of the two original sets (if there are any.) The **complement** of a set is a set containing elements that are *not* in the original set. For example, the complement of the set of black cards in a normal deck of playing cards (cards and dice figure prominently in discussions of probability since both relate to games of *chance*) is the set containing all red cards.

Probability theory is based on a model of how random outcomes are generated, known as a **random experiment**. Note that behavioral scientists and probability mathematicians use the term “experiment” differently; in math, it just means a trial, such as one toss of the coin or one role of the dice. In social research, an experiment is a research method designed to maximize the likelihood of causal statements being correct. In probability mathematics, the outcomes are generated in such a way that all *possible* outcomes are known in advance, but the *actual* outcome is not known in advance.

We will limit our discussion of probability math to three simple rules that will help you understand the material that is to come in later sections. We will consider:

-The Addition Rule

-The Multiplication Rule

-The Compliment Rule

You use the addition rule to determine the probability of a union of two sets. The multiplication rule is used to determine the probability of an intersection of two sets. The complement rule is used to identify the probability that the outcome of a random experiment will not be an element in a specified set.

A **random variable** assigns numerical values to the outcomes of a random experiment. When you flip a coin twice, for example, you are performing a random experiment: all possible outcomes are known in advance, and the actual outcome is not known in advance. Since the coin is flipped twice, the experiment consists of two **trials**. Of course, we know that for each trial the outcome must either be a “head” or a “tail.”

Last Modified: 02/03/2021