USA - Geometry
Probability and statistics
Sample spaces, events, unions, intersections, complements
20 practice questions
2 video lessons
Theory + worked examples
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 \(A\).
For equally likely outcomes, \(P(A)=\dfrac{\text{favorable}}{\text{total}}\), and \(P(A')=1-P(A)\).
The overlap is \(A\cap B\); the whole shaded pair is \(A\cup B\).
Set language for events.
Basic probability facts:
\[P(A)=\dfrac{|A|}{|S|},\qquad P(A')=1-P(A)\]
List the union's outcomes only once — don't double count the overlap.
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.
Example 1 — List a sample space
A fair die is rolled. Give the sample space and \(P(\text{even})\).
Solution
The sample space is \(S=\{1,2,3,4,5,6\}\); the even outcomes are \(\{2,4,6\}\).
| \(P(\text{even})\) | \(=\) | \(\dfrac{3}{6}=\dfrac12\) |
Example 2 — Intersection
With \(A=\{2,4,6\}\) and \(B=\{4,5,6\}\), find \(A\cap B\) and its probability.
Solution
The intersection is the outcomes in both sets.
| \(A\cap B\) | \(=\) | \(\{4,6\}\) |
| \(P(A\cap B)\) | \(=\) | \(\dfrac{2}{6}=\dfrac13\) |
Example 3 — Union
For the same \(A\) and \(B\), find \(A\cup B\) and its probability.
Solution
The union is the outcomes in either set (list each once).
| \(A\cup B\) | \(=\) | \(\{2,4,5,6\}\) |
| \(P(A\cup B)\) | \(=\) | \(\dfrac{4}{6}=\dfrac23\) |
Example 4 — Complement
If \(P(A)=\dfrac12\), find \(P(A')\).
Solution
The complement's probability is \(1\) minus the event's.
| \(P(A')\) | \(=\) | \(1-\dfrac12=\dfrac12\) |
Common pitfalls
Union is \(or\), intersection is \(and\) — keep them straight.
Count each outcome once in a union.
The complement is \(1-P\), not \(-P\).
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.
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