Sample Space

College/University

Definition

All the possible outcomes or results for an experiment. Sample space is usually written as a set where the possible results are listed out as elements. As an example, if you're rolling a die, you have 6 possible outcomes and you will get a sample space of {1, 2, 3, 4, 5, 6}.

Worked examples

\(S = \{1, 2, 3, 4, 5, 6\}\)

Rolling a standard die has six equally likely outcomes, so the sample space lists all of them.

\(S = \{HH, HT, TH, TT\}\)

Flipping two coins has four possible outcomes; order matters so HT and TH are distinct.

\(S = \{\)red\(, \)blue\(, \)green\(\}\)

Spinning a spinner with three equal sections gives three possible results in the sample space.

Common mistakes

\(S = \{1, 2, 3, 4, 5, 6, 6\}\) for rolling a die → \(S = \{1, 2, 3, 4, 5, 6\}\) Each outcome is listed once; sets don't repeat elements even if some outcomes feel more likely.
\(S = \{H, T\}\) for flipping two coins → \(S = \{HH, HT, TH, TT\}\) You must list every distinct sequence; two coins produce four outcomes, not two.
Writing sample space as a list without set braces → Always use set notation \(S = \{\ldots\}\) Sample space is formally a set, so curly braces are required notation.

Where you'll use it next

Sample space is the foundation for calculating probabilities, working with compound events, tree diagrams, and understanding independent vs. dependent events in statistics and probability units.

Found in 1 StudyPug lesson

Mastering Probability: Organizing Outcomes with Tree Diagrams

Grade 7 Math

Unlock the power of probability with our comprehensive guide on organizing outcomes. Learn to create and interpret tree diagrams, calculate probabilities, and tackle complex scenarios with confidence.

7th Grade7th