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What terms do you need to know to understand probability?

You may use and hear the word probability in casual conversation. The meteorologist on the news may forecast a high probability of rain, or your neighbors might discuss the low probability of winning the lottery. People typically use probability to describe how likely something is to happen. While probability does mean the likelihood of an event, it actually has a mathematical representation. Before you can calculate probability, you need to understand a few other terms.

Outcome

First, each probability is a ratio of outcomes. An outcome is a possible result of an action or experiment.

Consider the probability that you will have a pop quiz in your pre-algebra class. The outcomes are yes (you have a quiz) and no (you do not have a quiz). Having a quiz and not having a quiz are the possible outcomes from the act of you going to class.

Simple Event

Next, each probability determines the mathematical likelihood of an event. A simple event is the result from an action or experiment. Each simple event will have one outcome.

Consider the previous example—whether or not you have a pop quiz in pre-algebra. You have two possible outcomes: yes and no. Therefore, one simple event is yes (you have a quiz). The other simple event is no (you do not have a quiz).

Probability

Now that you understand outcomes and simple events, you can calculate probability. The probability of a simple event is the ratio of the number of desired outcomes to the number of all possible outcomes.

Consider the probability of NOT having a pop quiz. The number of desired outcomes is 1 (no pop quiz), and the number of possible outcomes is 2 (yes and no). Therefore, the probability of not having a pop quiz is \(\mathsf{ \frac{1}{2} }\), or 50%.

Random

When calculating probability, you should also consider whether the simple event occurs at random. A random event occurs solely by chance.

Because random events occur by chance, they are independent of any other factors. A coin toss is random because each flip of the coin is independent of any other toss. Rolling dice is another example of a random event. Each roll is independent of the others, and you never know what number you'll get.

Sample Space

When determining the probability of a simple event, you often have to consider the sample space. The sample space is the set of all possible outcomes of the action or experiment. Similarly, one of those possible outcomes is referred to as a sample point.

In the pop quiz example, the sample space was [yes, no]. When flipping a coin, the sample space is [heads, tails]. When rolling a die, the sample space is [1, 2, 3, 4, 5, 6].

Complementary Events

Sometimes your sample space will contain complementary events. Two events are complementary if one event must occur when the other event does not occur.

Throughout this slide show, you've seen sample spaces with complementary events. The sample space for pop quizzes contains complementary events. Either the teacher gives a quiz, or she does not give a quiz. One of these events has to happen. Also, consider flipping a coin—if the coin does not land on heads, then it must land on tails. These two events are complementary. Here's another example: Consider the probability of you going to class. Either you go or you do not go—you have no other options; therefore, the sample space containing [go to class, do not go to class] has complementary events.

To calculate the probability of a simple event, you should consider the sample space of all possible outcomes, whether the events are random, and if the sample space contains complementary events.

Question

What is the mathematical definition for the probability of a simple event?

P(event) = number of desired outcomes/number of possible outcomes