The phrase “sample space definition” refers to the mathematical set that lists (or describes) every possible outcome of a random experiment. Correct probabilities start with a correct sample space.
Sample space definition
A sample space (often written \(S\) or \(\Omega\)) is the set of all possible outcomes of a random experiment. Each individual outcome is a sample point (or simply an outcome), typically written \(\omega\in S\).
An event is any subset of the sample space: \(A \subseteq S\). If the outcome \(\omega\) occurs and \(\omega \in A\), then the event \(A\) occurs.
Key properties of a well-defined sample space
- Collectively exhaustive: every outcome that could happen is included in \(S\).
- Mutually exclusive outcomes: in a single trial, exactly one sample point \(\omega\) occurs (no overlap between distinct outcomes).
- Right level of detail: outcomes must be defined with the resolution needed for the question (too coarse or too fine can cause errors).
How the sample space is used to compute probabilities
A probability model assigns probabilities to events \(A \subseteq S\). Two common cases:
Finite, equally likely outcomes:
If \(S\) is finite and each outcome has the same probability, then for any event \(A \subseteq S\),
Finite, not equally likely outcomes:
If probabilities differ by outcome, assign \(P(\{\omega\})\) to each \(\omega \in S\) with \(\sum_{\omega \in S} P(\{\omega\}) = 1\). Then
Standard event operations are defined using set operations inside the same sample space: \(A^c = S \setminus A\) (complement), \(A \cup B\) (union), and \(A \cap B\) (intersection).
Examples of sample spaces and events
| Random experiment | Sample space \(S\) | Example event \(A\) | Probability idea |
|---|---|---|---|
| Roll a fair six-sided die | \(\{1,2,3,4,5,6\}\) | \(A=\{2,4,6\}\) (even) | \(P(A)=3/6\) (equally likely) |
| Toss two coins (order matters) | \(\{\text{HH},\text{HT},\text{TH},\text{TT}\}\) | \(A=\{\text{HT},\text{TH}\}\) (exactly one head) | \(P(A)=2/4\) if fair coins |
| Draw 1 card from a standard deck | All 52 distinct cards | \(A=\{\text{hearts}\}\) (any heart) | \(P(A)=13/52\) |
| Measure a lifespan (continuous) | \([0,\infty)\) | \(A=[60,80]\) (between 60 and 80) | Probabilities use intervals and a density model |
Step-by-step method to define a sample space correctly
- State the experiment precisely (what is observed and when the trial ends).
- List outcomes or describe them as a set (finite list, countable set, or interval/region).
- Check exhaustiveness: no possible outcome should be missing.
- Check exclusivity: outcomes should not overlap (one trial corresponds to one outcome).
- Define events as subsets that answer the question (e.g., “even,” “at least two,” “between”).
Visualization: events inside a sample space
Common pitfalls
- Changing the sample space mid-solution: probabilities must be computed with a single fixed \(S\).
- Forgetting order when it matters: \(\{\text{HT},\text{TH}\}\) are different outcomes if order is recorded.
- Mixing outcomes and events: an outcome is \(\omega\in S\); an event is a set \(A\subseteq S\).
- Assuming equally likely without justification: if outcomes have different probabilities, use outcome weights rather than \(|A|/|S|\).