What is a Sample Space?
The sample space (often denoted by Ω or S) is the set of all possible outcomes of a random experiment. Each outcome in the sample space is called a sample point.
Types of Sample Spaces
Discrete Sample Space
Contains a finite or countably infinite number of outcomes. Examples include coin tosses, dice rolls, or number of students in a class.
Continuous Sample Space
Contains an uncountably infinite number of outcomes. Examples include time measurements, heights, or temperatures.
Examples of Sample Spaces
| Experiment | Sample Space | Number of Outcomes |
|---|---|---|
| Tossing a coin once | S = {H, T} | 2 |
| Rolling a six-sided die | S = {1, 2, 3, 4, 5, 6} | 6 |
| Tossing two coins | S = {(H,H), (H,T), (T,H), (T,T)} | 4 |
| Drawing a card from a standard deck | S = {52 different cards} | 52 |
| Temperature in a room (°C) | S = {x | x ∈ ℝ, typically 15 ≤ x ≤ 30} | Uncountably infinite |
Understanding Events
An event is a subset of the sample space. It represents a collection of outcomes that we're interested in.
Types of Events
- Simple Event: Contains exactly one outcome from the sample space.
- Compound Event: Contains multiple outcomes from the sample space.
- Empty Event (∅): Contains no outcomes.
- Certain Event (S): Contains all outcomes in the sample space.
Example: Rolling a Die
Consider the experiment of rolling a six-sided die once.
Sample Space: S = {1, 2, 3, 4, 5, 6}
Some possible events:
- A = {2, 4, 6} (rolling an even number)
- B = {1, 3, 5} (rolling an odd number)
- C = {5, 6} (rolling a number greater than 4)
- D = {1} (rolling a 1)
Set Operations with Events
Since events are sets, we can perform various set operations on them.
Basic Set Operations
Union (A ∪ B)
The event that either A or B (or both) occurs.
Intersection (A ∩ B)
The event that both A and B occur.
Complement (Ac)
The event that A does not occur.
Difference (A - B)
The event that A occurs but B does not.
Example: Set Operations with Die Events
Using the die rolling events from earlier:
- A = {2, 4, 6} (even numbers)
- C = {5, 6} (numbers greater than 4)
We can calculate:
- A ∪ C = {2, 4, 5, 6} (even OR greater than 4)
- A ∩ C = {6} (even AND greater than 4)
- Ac = {1, 3, 5} (not even)
- C - A = {5} (greater than 4 but not even)
Sample Space & Probability Calculator
Practice Problems
Test your understanding of sample spaces and events with these practice problems.
Problem 1
A fair die is rolled twice. What is the probability of getting a sum of 7?
Problem 2
Two cards are drawn without replacement from a standard deck of 52 cards. What is the probability that both cards are aces?
Further Reading
To deepen your understanding of sample spaces and probability, explore these related topics:
Conditional Probability
Learn how to calculate probabilities when additional information is known.
Learn More