What Is Probability in Math?

Probability is the branch of math that measures how likely something is to happen. Instead of saying an event is "likely" or "unlikely" in vague words, probability gives that likelihood an exact number. Every probability is a value between 0 and 1, where 0 means the event can never happen and 1 means the event is guaranteed to happen. Everything in between describes how likely or unlikely the event is.

You can write a probability as a fraction, a decimal, or a percent. For example, a probability of \( \dfrac{1}{2} \) is the same as 0.5, which is the same as 50 percent. All three describe the exact same chance, just in different formats.

It helps to picture probability as a position on a number line stretching from 0 to 1. Events near 0 are unlikely, events near 1 are likely, and an event sitting right in the middle at \( \dfrac{1}{2} \) has an even chance of happening or not happening.

For an event \(A\) made up of equally likely outcomes, probability is calculated with a simple ratio:

\( P(A) = \dfrac{f}{n} \)

Here \(f\) stands for the number of favorable outcomes (the outcomes that count as a "success" for event \(A\)), and \(n\) stands for the total number of possible outcomes. This ratio is always between 0 and \(n\) divided by \(n\), which simplifies to a value between 0 and 1.

A standard die has 6 faces, numbered 1 through 6, so there are 6 total outcomes. Suppose event \(A\) is rolling a number greater than 4. The favorable outcomes are 5 and 6, so there are 2 favorable outcomes.

\( P(A) = \dfrac{2}{6} = \dfrac{1}{3} \)

So the probability of rolling a number greater than 4 is \( \dfrac{1}{3} \), or about 33 percent. This kind of single-trial calculation is exactly what you practice in simple probability experiments, where you count outcomes from a spinner, coin, or die.

The complement of event \(A\), written \(A'\), is everything that counts as \(A\) not happening. Since an event either happens or it does not, the two probabilities always add to 1:

\( P(A) + P(A') = 1 \)

In the die example above, \( P(A) = \dfrac{1}{3} \), so \( P(A') = 1 - \dfrac{1}{3} = \dfrac{2}{3} \). This shortcut is useful whenever it is easier to count the outcomes that do not satisfy an event than the ones that do.

Once you understand what a probability number represents, you are ready to apply it. That skill shows up when you predict outcomes of repeated trials, and when you work through probability word problems that describe real situations in words instead of numbers. Both build directly on the same core idea covered here: probability is just a number, between 0 and 1, that tells you how likely an event is.

Related lessons

• Math 3