Sample Space Examples in Statistics

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 Statistics.

Concept Recap

The sample space is the complete set of all possible outcomes for a probability experiment, listed without repetition. It forms the foundation for every probability calculation because the probability of any event is a fraction of the sample space.

Before calculating probability, list every possible outcome. For a die: \{1, 2, 3, 4, 5, 6\}. For two coins: \{HH, HT, TH, TT\}. That's your sample space - the complete menu of what could happen.

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 the denominator of every probability calculation. Missing even one possible outcome makes all your probability calculations wrong.

Common stuck point: Students often miss outcomes when multiple events occur together — for two dice there are 36 outcomes, not 11 (the possible sums).

Sense of Study hint: When finding a sample space, first identify all possible outcomes for the first event. Then, if there are multiple events, use a tree diagram or organized list to combine outcomes systematically. Finally, count the total to verify using the counting principle: if event A has m outcomes and event B has n outcomes, the combined sample space has m \times n outcomes.

Worked Examples

Example 1

easy
List the sample space for rolling a standard six-sided die and flipping a coin simultaneously.

Solution

  1. 1
    Step 1: The die has outcomes {1, 2, 3, 4, 5, 6} and the coin has outcomes {H, T}.
  2. 2
    Step 2: The sample space is all possible combinations: pair each die outcome with each coin outcome.
  3. 3
    Step 3: S = \{(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)\}. Total outcomes: 6 \times 2 = 12.

Answer

The sample space has 12 outcomes: \{(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)\}.
The sample space is the set of all possible outcomes of an experiment. For combined experiments, the total number of outcomes equals the product of the individual outcome counts (multiplication principle). Listing the sample space systematically ensures no outcomes are missed.

Example 2

medium
A restaurant offers 3 starters (soup, salad, bread), 4 mains (chicken, fish, beef, pasta), and 2 desserts (cake, fruit). How many different 3-course meals are possible? Do you need to list them all?

Practice Problems

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

Example 1

medium
A password consists of one letter (A–E) followed by one digit (1–3). (a) List the entire sample space. (b) How many passwords contain the letter 'C'? (c) What is the probability of randomly generating a password that starts with 'C'?

Example 2

hard
Two dice are rolled. (a) How many outcomes are in the sample space? (b) How many outcomes give a sum of 7? (c) How many outcomes give a sum greater than 10? (d) Which sum is more likely: 7 or greater than 10?
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Related Concepts

Compound Events

Background Knowledge

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

countingsets