Sample Spaces

Precise definition

A sample space $\Omega$ is the set of all possible elementary outcomes of a random experiment under a specified level of detail. An event is a subset of $\Omega$. A probability model assigns probabilities to events with $P(\Omega)=1$ and countable additivity for disjoint events.

Notation and mathematical language

For finite equally likely outcomes, $P(A)=|A|/|\Omega|$. Equal likelihood is an additional assumption, not a consequence of listing outcomes. Ordered pairs distinguish first and second trials; unordered sets may be appropriate when order truly has no meaning.

Conceptual picture

A tree, table, or product set organizes multi-stage outcomes. The right granularity makes events expressible and outcomes mutually exclusive. Combining outcomes with unequal probabilities and then counting them equally changes the model.

Conditions and key results

The sample space must be exhaustive and outcomes mutually exclusive. Physical mechanisms determine probabilities. Infinite spaces require measures rather than simple favourable-over-total counting, and continuous individual points can each have probability zero without the entire space having zero probability.

A reliable strategy

1. Describe the random experiment and what counts as one complete elementary outcome.
2. Construct an exhaustive space using a list, tree, Cartesian product, or rule.
3. Define events as subsets and attach probabilities from symmetry or the mechanism.
4. Check total probability and compute events by unions, intersections, or complements without double counting.

Fully worked example

Interpretation and application

Sample spaces model games, reliability, genetics, queues, and experiments. A mathematically uniform model must be checked against actual mechanisms such as weighted dice, unequal selection, or dependence.

Common mistakes

Verification and reasonableness

• Count leaves in an independently built tree.
• Confirm every experimental result maps to exactly one outcome.
• Sum elementary probabilities to 1 and compare complements.

Practice

1. How many outcomes for two ordered die rolls?
2. Is an event an outcome or a set of outcomes?
3. When does favourable over total apply?

Further deduction

Refining a sample space preserves probabilities when coarse outcomes are split according to their correct masses. For two dice, the sum space $\{2,\ldots,12\}$ is convenient but its 11 sums are not equally likely. The ordered-pair space reveals that sum 7 has six representations while sum 2 has one.

Conditional probability changes the effective sample space by restricting attention to event $B$ and renormalizing: $P(A\mid B)=P(A\cap B)/P(B)$. In equally likely finite spaces this becomes $|A\cap B|/|B|$. The denominator is the conditioned set, not the original total, which explains many conditional-counting errors.

For continuous outcomes, individual points can have probability zero without being impossible; probability is assigned to measurable sets such as intervals. A uniform variable on $[0,1]$ satisfies $P(X=1/2)=0$ but $P(0.4<X<0.6)=0.2$. Thus counting equally likely points is inappropriate for a continuum. The event collection and probability measure replace finite enumeration while retaining complement, union, and conditional-probability laws.

Related topics

• Probability
• Counting Principle
• Independent Events