Sample Space in Statistics: Definition & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: X-Bar in Statistics: Theory & Formula

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:02 Sample Space
  • 1:22 Counting Principle &…
  • 2:58 Examples
  • 5:31 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Ria Yambao

Ria has taught College Algebra and Biostatistics. She has a master's degree in Applied Mathematics.

In this lesson, you will learn the definition of sample space - an important concept in the study of probability. Examples and quiz questions will illustrate how this concept exists in the real world.

Sample Space

You decided to perform a random experiment of rolling a single fair die with six sides. A random experiment is one where you cannot be absolutely sure what the outcome would be prior to performing the experiment. An outcome is a possible result of the experiment. Each side of the die has 1, 2, 3, 4, 5 or 6 dots. What are all the possible outcomes for this experiment? In a single roll of the die, it can show 1, 2, 3, 4, 5 or 6 dots. The exhaustive list of possible outcomes for this experiment is: 1, 2, 3, 4, 5, 6. The set whose members are all possible outcome of a random experiment is called the sample space for that experiment. For our die-rolling experiment, we can write the sample space as:

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

An event is a particular set whose members are from a sample space. For example, the event that the die shows an odd number of dots has as its members 1, 3 and 5. We can write

E = {1, 3, 5}.

In this lesson, we will assume that every experiment is a random experiment.

Counting Principle & Tree Diagram

The fundamental counting principle is an important concept that can be used to determine the number of members in a sample space. The fundamental counting principle states that if the first activity can be done in m ways and a second activity can be done in n ways then the first activity followed by the second activity can be done in m times n ways.

Let's discuss the principle using a simple example. Suppose you perform a simple random experiment that involves two steps. First, you toss a coin. Then, you roll a die. How many different outcomes are there for this two-step experiment? In tossing a coin, there are two possible outcomes - Head (H) and Tail (T). In rolling a die, there are six possible outcomes based on the number of dots on each face - 1, 2, 3, 4, 5, 6. The possible outcomes for the two-step experiment are shown in the tree diagram on screen:

Tree Diagram Showing Outcomes of Two-Step Experiment
Diagram Showing Outcomes of Two-Step Expt

A tree diagram is an organized way of writing out all the possible outcomes of an experiment. It is a useful tool in determining the members of a sample space. It can also be used in determining outcomes for an event. Note that since there are two possible outcomes for the coin-tossing activity (step 1) and six possible outcomes for the die-rolling activity (step 2), then there are twelve possible outcomes (2 x 6 = 12) for the two-step experiment.

The fundamental counting principle can be extended to any finite number of activities. When there are three activities:

First Activity Second Activity Third Activity First, Second and Third Activities, in that order
m n k m times n times k


Let's now go over some examples to put what we've learned into practice.

Example 1. A teacher gave a three-question true or false quiz to her class. If students are not allowed to leave a question unanswered and the only allowed answers for each question are T (for true) or F (for false), how many different possible outcomes are there? What are all the different possible ways of answering the quiz?


To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account