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: The sample space is every single thing that could possibly happen in one trial — listed without leaving any out.

Common stuck point: The procedure for sample space is the easy part; the trap is merging distinct outcomes. Asking "Have I listed every distinct outcome that could occur, with none missing or merged?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Have I listed every distinct outcome that could occur, with none missing or merged?

Worked Examples

Example 1

easy

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

Answer

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

; each with

P = \frac{1}{6}

; total

= 1

First step

Identify all outcomes:

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

Full solution

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

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})

Example 3

medium

List the sample space for flipping three coins.

Example 4

medium

Three students

A, B, C

are seated in a row. List the sample space of seatings.

Example 5

hard

A bag has marbles numbered

1, 2, 3

. Two are drawn at once. List the sample space.

Example 6

hard

A child has a red, a blue, and a green ball. She picks two balls in a row, replacing after each. List the sample space.

Example 7

challenge

Three coins are tossed. Find the probability of getting exactly

2

heads using the sample space directly.

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})

0.1, 0.2, 0.3, x

. Find

x

Example 23

easy

List the sample space for flipping a single coin.

Example 24

easy

A drawer has

4

shirts of different colors. You pick one. How many outcomes are in the sample space?

Example 25

easy

List the sample space for rolling a four-sided die.

Example 26

medium

A coin is flipped, then a die is rolled. How many outcomes are in the sample space?

Example 27

medium

Two dice are rolled. How many outcomes have a sum of

7

Example 28

medium

You draw two cards (without replacement) from a

52

-card deck. How many ordered outcomes are in the sample space?

Example 29

medium

A coin is flipped and a 4-sided die is rolled. List one outcome and state the size of the sample space.

Example 30

medium

A jar has

5

red and

3

blue marbles. You draw one. What is the size of the sample space if outcomes are 'red' and 'blue' (not individual marbles)?

Example 31

hard

You flip a coin until you get heads or reach

3

flips, whichever first. List the sample space.

Example 32

hard

How many outcomes are in the sample space for rolling three distinct dice?

Example 33

hard

A die is rolled. What is the size of the event 'rolling a prime number'?

Example 34

hard

Two dice are rolled. What is the size of the event 'both dice show the same number'?

Example 35

hard

A code consists of

2

digits from

0

–

9

. How many outcomes are in the sample space if digits may repeat?

Example 36

hard

A code consists of

2

distinct digits from

0

–

9

. How many outcomes are in the sample space?

Example 37

challenge

A bag has

1

red,

1

blue, and

1

green marble. You draw all three one by one. How many ordered outcomes are in the sample space?

Example 38

challenge

You flip a fair coin until you get a tail. What is the sample space?

← Back to Sample Space Formula Explained Practice Mode

Related Concepts

Probability

Background Knowledge

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

probability