Sample Space and Events

Learn how to identify all possible outcomes and describe events in probability experiments.

Intermediate25 minLesson
Printable worksheet

Definition

In probability, a sample space is the set of all possible outcomes of an experiment. We write it as SS or use set notation with curly braces.
An event is any subset of the sample space: the outcomes we are interested in (one, several, all of them or even none).
Example - Rolling a die:
S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}
The event "rolling an even number" would be:
E={2,4,6}E = \{2, 4, 6\}

Try it now

What is the sample space when flipping a single coin?

Worked Examples

What is the sample space when flipping a coin? List the event of getting heads.
1

Identify the experiment

We are flipping a single coin once → Coin flip experiment
2

List all possible outcomes

A coin can land on heads (H) or tails (T) → S={H,T}S = \{H, T\}
3

Define the event

The event 'getting heads' contains only one outcome → E={H}E = \{H\}

Common Mistakes

Forgetting outcomes when listing the sample space

Why it's wrong: When listing outcomes manually, it's easy to miss some. For example, forgetting TH when flipping two coins.

Correct: Use a systematic approach: tree diagrams, tables, or organized lists to ensure you capture all outcomes.

Confusing the sample space with an event

Why it's wrong: The sample space contains ALL possible outcomes, while an event is a SUBSET of outcomes we care about.

Correct: Remember: SS = everything that CAN happen; EE = outcomes that match a specific condition.

Treating different orderings as the same outcome

Why it's wrong: When flipping two coins, HT and TH are different outcomes (first coin vs second coin).

Correct: Unless the problem says otherwise, treat the order as important. HT means first coin heads, second coin tails.

Interactive Visual

Flip a coin, roll a die: 2 × 6 = 12 outcomes

CoinDieH123456T123456(H, 1)(H, 2)(H, 3)(H, 4)(H, 5)(H, 6)(T, 1)(T, 2)(T, 3)(T, 4)(T, 5)(T, 6)

Multiply the choices at each step: 2 × 6 = 12 (12 outcomes)

Tap an outcome on the right to follow its path.

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Practice Problems

15 problems
Problem 1 of 15
Easy
What is the sample space when flipping a single coin?

Why It Matters

Understanding sample spaces and events is the foundation of all probability:
  • Games: When playing board games, knowing all possible dice rolls helps you strategize
  • Weather: Meteorologists consider all possible weather outcomes to make forecasts
  • Medicine: Doctors analyze possible outcomes of treatments to help patients make decisions
  • Sports: Coaches plan plays by considering all possible scenarios
Without clearly defining the sample space, we cannot calculate probabilities accurately!

Real World Applications

Quality Control

Factories inspect products and track outcomes like 'defective' or 'non-defective' to ensure quality.

Example:

A light bulb factory tests 3 bulbs. Sample space: {DDD,DDN,DND,NDD,DNN,NDN,NND,NNN}\{DDD, DDN, DND, NDD, DNN, NDN, NND, NNN\} where D=defective, N=non-defective.

1Try It Yourself

A factory tests 2 items. Each item is either defective (D) or non-defective (N).

Which outcomes have exactly one defective item?

Step 1: Choose the right answer

Choose the outcomes with exactly one D.

Weather Forecasting

Meteorologists consider all possible weather conditions to make accurate predictions.

Example:

For tomorrow's weather, outcomes might be: sunny, cloudy, rainy, or stormy. This is the sample space for weather prediction.

2Try It Yourself

A weather model predicts conditions for 2 days. Each day can be sunny (S) or rainy (R).

What is the event of having at least one rainy day?

Step 1: Choose the right answer

Which outcomes have at least one R?

Key Takeaways

  • 1A sample space (SS) is the set of all possible outcomes of an experiment
  • 2An event (EE) is a subset of the sample space - outcomes we're interested in
  • 3Use set notation: S={outcome1,outcome2,…}S = \{\text{outcome}_1, \text{outcome}_2, \ldots\}
  • 4For compound experiments, multiply the number of outcomes: n(S)=n1×n2n(S) = n_1 \times n_2
  • 5Systematic listing (tree diagrams, tables) helps avoid missing outcomes

Frequently Asked Questions

An outcome is a single possible result (like rolling a 3). An event is a collection of one or more outcomes (like rolling an odd number, which includes 1, 3, and 5).
An outcome is a single possible result (like rolling a 3). An event is a collection of one or more outcomes (like rolling an odd number, which includes 1, 3, and 5).
Yes! This is called a 'certain event' because it will definitely happen. For a die roll, the event 'rolling a number from 1 to 6' equals the entire sample space.
An impossible event has no outcomes - it cannot happen. For example, rolling a 7 on a standard die. We write this as E={}E = \{\} or E=∅E = \emptyset (empty set).

Glossary

Sample space
The set of all possible outcomes of a probability experiment, denoted by SS
Outcome
A single possible result of an experiment (e.g., rolling a 4)
Event
A subset of the sample space: the outcomes we are interested in
Equally likely outcomes
Outcomes that have the same probability of occurring
Compound experiment
An experiment with two or more stages (e.g., flip a coin AND roll a die)
Certain event
An event that includes all outcomes in the sample space (probability = 1)
Impossible event
An event with no outcomes; it cannot happen (probability = 0)
Today's challengeOne problem, three tries

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