Sample spaces, events, unions, intersections, complements
Sample Spaces and Events
Sample Spaces and Events is the opening topic of Probability & Statistics in the California Common Core State Standards. It is aligned to Standard S-CP.1, which requires students to describe events as subsets of a sample space using unions, intersections, and complements.
The sample space lists all outcomes; events combine with unions (or), intersections (and), and complements (not).
Theory
The sample space \(S\) is the set of all possible outcomes; an event is a subset of \(S\). Events combine with set operations:
- Union \(A\cup B\): outcomes in \(A\) or \(B\).
- Intersection \(A\cap B\): outcomes in both.
- Complement \(A'\): outcomes not in \(A\).
Basic probability facts:
How to work with events
- Write the sample space \(S\).
- List each event as a subset.
- Combine with union, intersection, or complement.
- Divide favorable by total for the probability.
The sample space is \(S=\{1,2,3,4,5,6\}\); the even outcomes are \(\{2,4,6\}\).
| \(P(\text{even})\) | \(=\) | \(\dfrac{3}{6}=\dfrac12\) |
The intersection is the outcomes in both sets.
| \(A\cap B\) | \(=\) | \(\{4,6\}\) |
| \(P(A\cap B)\) | \(=\) | \(\dfrac{2}{6}=\dfrac13\) |
The union is the outcomes in either set (list each once).
| \(A\cup B\) | \(=\) | \(\{2,4,5,6\}\) |
| \(P(A\cup B)\) | \(=\) | \(\dfrac{4}{6}=\dfrac23\) |
The complement's probability is \(1\) minus the event's.
| \(P(A')\) | \(=\) | \(1-\dfrac12=\dfrac12\) |
Common pitfalls
Frequently asked questions
What is a sample space?
The set of all possible outcomes of an experiment.
What is the difference between union and intersection?
Union \(A\cup B\) is outcomes in either event; intersection \(A\cap B\) is outcomes in both.
What is the complement of an event?
All outcomes not in the event; its probability is \(1-P(A)\).
How do you find a probability with equally likely outcomes?
Divide the number of favorable outcomes by the total number of outcomes.