Sample Space

probability-theory

Definition

Sample Space

The sample space of a random experiment is the set of all its possible outcomes. It is denoted by

$$\Omega =\{\omega \mid \omega \text{ is a possible outcome of the experiment}\}.$$

Each $\omega \in \Omega$ is an elementary event, and an event is a subset $A\subseteq \Omega$ containing the possible outcomes for which that event occurs.

Choosing the Sample Space

A sample space must include every possible result and distinguish possible outcomes at the level of detail required by the question. Different descriptions of the same experiment may therefore use different sample spaces.

Two dice

If only the sum of two dice matters, one may use

$${\Omega }_{\mathrm{sum}}=\{2,3,\dots ,12\}.$$

If the value of each die matters, the more detailed space is

$${\Omega }_{\mathrm{pair}}=\{1,\dots ,6\}\times \{1,\dots ,6\}.$$

The first representation merges all ordered pairs with the same sum into one outcome.

Warning

A sample space states what can happen, not how likely each event is. Probabilities arise only after a $\sigma$-algebra of events and a probability measure are added, producing a probability space.