Introduction to Probability

Probability is a way of measuring how likely something is to happen. It is always a number between 0 and 1, where 0 means an event is impossible and 1 means it is certain. The closer a probability is to 1, the more likely the event is to occur.

The basic formula for the probability of an event \(A\) is:

\(P(A) = \dfrac{\)number of favorable outcomes\(}{\)number of possible outcomes\(}\)

Probabilities are often written as fractions, but they can also be expressed as decimals or percents. If you need a refresher on switching between these forms, see the lesson on converting among decimals, fractions, and percents.

The sample space is the list of every possible outcome of an experiment. For example, if you flip a coin twice, the sample space contains four equally likely outcomes:

Knowing the full sample space is the first step in almost every probability question, because it tells you the total number of possible outcomes to put on the bottom of the probability fraction.

Once you know the sample space, you can pick out the outcomes that match the event you care about. For a single coin toss, the sample space is {Heads, Tails}, so:

\(P(\)Heads\() = \dfrac{1}{2}\)

For rolling a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}, so the probability of rolling a 3 is:

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

These simple examples show the pattern used throughout probability: count the outcomes you want, count all the outcomes that are possible, and write the ratio.

An independent event is one whose outcome does not change the probability of another event. Tossing a coin twice is independent, because the result of the first toss has no effect on the second. When two events \(A\) and \(B\) are independent, you find the probability that both happen by multiplying:

\(P(A \) and \( B) = P(A) \times P(B)\)

For example, the probability of getting heads on two separate coin tosses is \(\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}\).

A dependent event is one where the outcome of the first event does affect the probability of the second. Drawing two cards from a deck without putting the first one back is dependent, because removing a card changes the number of cards left to choose from.

Two events are mutually exclusive (also called disjoint) if they cannot both happen at the same time. Rolling a 2 and rolling a 5 on the same die roll are mutually exclusive, since a single roll can only show one number. For mutually exclusive events, the probability that either one occurs is found by adding:

\(P(A \) or \( B) = P(A) + P(B)\)

A compound event combines two or more simple events, such as "rolling an even number or a number greater than 4." When the events overlap instead of being mutually exclusive, you have to subtract the overlap so it is not counted twice:

\(P(A \) or \( B) = P(A) + P(B) - P(A \) and \( B)\)

Drawing this kind of overlap is much easier to see with a picture, which is exactly what the lesson on probability with Venn diagrams covers in detail.

It helps to picture every probability sitting somewhere on a scale from 0 to 1.

Every probability you calculate, whether for a simple event or a compound one, should land somewhere on this scale. If your answer is less than 0 or more than 1, it is a sign to check your work.

This introduction gives you the tools to describe an experiment, list its sample space, and calculate the probability of basic, independent, dependent, mutually exclusive, and compound events. From here, you can compare how well theory matches real trials in the lesson on theoretical and experimental probability, which builds directly on the ideas covered here.

Related lessons

• Statistics Help