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What Is a Sample Space in Math?

The probability of an event with $n$ favorable outcomes will be equal to the relative frequency when the number $N$ of trials becomes very large.

You write the probability of an event $A$ as $P (A)$ . The $P$ stands for “probability”, so $P (A)$ is read as “the probability of $A$ ”. Here you will learn some terms that are key when talking about—and understanding—probability.

Outcome and Sample Space

An outcome is a possible result of a trial. When you roll a die, there are six possible outcomes: 1, 2, 3, 4, 5 or 6. A sample space is the set of all the possible outcomes of a trial or an experiment. The sample space when you roll a die is therefore $S = {1, 2, 3, 4, 5, 6}$ , which is the set of all the possible outcomes 1, 2, 3, 4, 5 and 6.

When you flip a coin, there are two different outcomes: heads or tails. The sample space when you toss a coin is therefore $S = {heads, tails}$ , which is the set of the outcomes heads and tails.

Rule

Outcomes and Sample Space

The sample space is the set of all possible outcomes of an experiment.
The sum of the probabilities of all of the outcomes equals 1.
The probability of an outcome can never be less than $0$ or greater than 1.
The probability of an outcome is less than or equal to 1.

Example 1

You roll a die. What is the sample space? What is the probability of each outcome? What is the sum of the probabilities?

The sample space is

S = {1, 2, 3, 4, 5, 6}

When rolling a die, the probabilities of the different outcomes 1, 2, 3, 4, 5 or 6 are written as:

\begin{array}{l} P (1) = \frac{1}{6} P (2) & = \frac{1}{6} \\ P (3) = \frac{1}{6} P (4) & = \frac{1}{6} \\ P (5) = \frac{1}{6} P (6) & = \frac{1}{6} \end{array}

\begin{array}{l} P (1) = \frac{1}{6} P (2) & = \frac{1}{6} P (3) = \frac{1}{6} \\ P (4) = \frac{1}{6} P (5) & = \frac{1}{6} P (6) = \frac{1}{6} \end{array}

The sum of the probabilities of the outcomes in the sample space is

\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{6}{6} = 1

Example 2

You’re at a birthday party, and there are four bowls: The first is filled with Oreos, the second with Doritos, the third with Milk Duds and the fourth with peanuts. You close your eyes and grab a snack at random. What are the possible outcomes, and what is the sample space? What’s the probability of taking something from the bowl with Oreos, Doritos, Milk Duds or peanuts?