Introduction to Probability

What is a sample space in probability?

Quick answer

A sample space is the complete set of all possible outcomes of a random experiment, where each outcome appears exactly once. An event is any subset of that sample space, containing one or more outcomes you want to track. For example, rolling a standard die has a sample space of six outcomes, while rolling an even number is an event containing three outcomes.

The topic explained

The list behind every probability

Probability begins with counting, and counting begins with a complete list. In probability, a sample space is the collection of every single outcome that could happen in a random experiment, written so that no two outcomes can occur at the same time and none are omitted. We usually write a sample space using braces. For a standard six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.

An event is simply a subset of the sample space containing the specific outcomes you are tracking. If you roll a die and want to know the chance of getting a number greater than 4, your event is E = {5, 6}. An event can contain a single outcome, several outcomes, or even all of them. An event with all outcomes is certain to happen, while an event with zero outcomes is impossible.

Why order matters in multi-stage experiments

Most probability errors happen during listing, not calculating. When an experiment has two stages, like tossing two coins, listing 'two heads, two tails, or one of each' creates an error. The result 'one of each' can happen in two distinct ways: heads on the first coin and tails on the second, or tails first and heads second.

To list every outcome fairly, treat the stages as distinct. The true sample space for two coins is S = {HH, HT, TH, TT}. Because there are four distinct outcomes, each has a 1 in 4 chance. Grouping HT and TH together hides the fact that getting one of each is twice as likely as getting two heads.

Using a grid to avoid missing outcomes

When an experiment grows, writing outcomes in a single row makes it easy to skip a possibility or double-count one. A two-way table organizes two-stage experiments neatly. For rolling two standard dice, create a table with six rows for the first die and six columns for the second.

The resulting grid has 36 cells, and each cell represents one outcome, such as (1, 4) or (4, 1). Finding the event 'the sum is 5' becomes a simple matter of checking the diagonal cells: (1, 4), (2, 3), (3, 2), and (4, 1). Because the grid holds every outcome in its own cell, you never miss a pair or count the same pair twice.

Free printable worksheet

Sample Spaces and Events Practice

List the sample space for each experiment and identify the specific outcomes that make up the given event.

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Sample Spaces and Events PracticeName:

You flip a fair coin once. Write the sample space S.
You roll an eight-sided die numbered 1 through 8. Write the event E of rolling a prime number.
You toss two coins. List the full sample space S.
From a standard deck, you draw one card and record only its suit. Write the sample space S.
You roll two four-sided dice, each numbered 1 to 4. How many total outcomes are in the sample space?
Using two four-sided dice (numbered 1 to 4), list the outcomes that form the event E where the sum of the dice equals 5.
A spinner has three equal sections labeled Red, Blue, and Green. You spin it twice. List all outcomes where the color changes on the second spin.

Common questions

Can an event contain only one outcome?

Yes. An event with exactly one outcome is called an elementary or simple event. For example, rolling a 3 on a standard six-sided die is the elementary event E = {3}.

Why are (1, 4) and (4, 1) treated as two different outcomes when rolling two dice?

Because the two dice are separate objects. Even if both dice are the same color, rolling a 1 on die A and a 4 on die B is physically different from rolling a 4 on die A and a 1 on die B. Counting them separately keeps every listed outcome equally likely.

What is the difference between an outcome and an event?

An outcome is a single final result of an experiment, such as rolling a 4. An event is a set of one or more outcomes grouped by a condition, such as rolling an even number, which contains the outcomes 2, 4, and 6.

Can a sample space have zero outcomes?

No. A random experiment must produce a result. A sample space always contains at least one possible outcome. However, an event can be empty if it describes an impossible condition, such as rolling an 8 on a standard six-sided die.

How do you find the total number of outcomes in a multi-step experiment?

Multiply the number of possibilities at each stage. For instance, tossing a coin (2 outcomes) and rolling a die (6 outcomes) produces 2 times 6, which equals 12 total outcomes in the sample space.

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