Sample Space Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sample Space.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The sample space S is the set of all possible outcomes of a random experiment — every outcome that could conceivably occur.

Before you can calculate any probability, you need the complete menu of possibilities. The sample space is that menu—like listing every face of a die or every possible hand in a card game. Missing even one outcome throws off every probability you calculate, because all probabilities must add up to exactly 1 over the full sample space.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Probabilities of all outcomes in sample space must sum to 1.

Common stuck point: The sample space depends on how you define the experiment and what counts as an outcome — listing it explicitly before computing probabilities prevents errors.

Sense of Study hint: Draw a tree diagram or grid to list every outcome systematically. Check that nothing is missing before you count.

Worked Examples

Example 1

easy
List the sample space for rolling a fair six-sided die, and verify that all probabilities sum to 1.

Solution

  1. 1
    Identify all outcomes: S = \{1, 2, 3, 4, 5, 6\}
  2. 2
    Each outcome is equally likely with probability P(\text{each}) = \frac{1}{6}
  3. 3
    Sum all probabilities: P(1)+P(2)+P(3)+P(4)+P(5)+P(6) = 6 \times \frac{1}{6} = 1
  4. 4
    Conclusion: The probabilities sum to 1, confirming a valid probability model

Answer

S = \{1,2,3,4,5,6\}; each with P = \frac{1}{6}; total = 1.
A sample space contains all possible outcomes of a random experiment. The fundamental rule is that all probabilities must sum to exactly 1 — this axiom ensures the model is complete and consistent.

Example 2

medium
Two coins are flipped. Write out the sample space, assign probabilities to each outcome, and find P(\text{exactly one head}).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A bag contains one red, one blue, and one green marble. You draw one marble. Write the sample space and assign probabilities so they sum to 1.

Example 2

medium
A spinner has 3 sections: Red (probability 0.5), Blue (probability 0.3), and Green. Find P(\text{Green}) and verify the sample space probabilities sum to 1.
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Related Concepts

Probability

Background Knowledge

These ideas may be useful before you work through the harder examples.

probability