Sample Spaces

A precise guide to outcomes, sample spaces, events, equally likely models, and structured enumeration.

Events are subsets of a sample space, and conditional probability changes the reference set. A sample-space rectangle contains two overlapping event regions and a small two-stage probability tree. The overlap marks joint occurrence, while branch labels show how later probabilities can depend on an earlier outcome.

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.

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.