Sample Space¶

Definition¶

Let a random experiment be specified. The sample space $S$ is the set of all possible outcomes of that experiment.

Interpretation¶

The sample space lists everything that can happen in the model: each run of the experiment produces exactly one of its elements. Events are subsets of $S$ , so $S$ has to be fixed before any probability can be discussed.

Two sample spaces: the six faces of a die roll, and the four outcomes HH, HT, TH and TT of two coin tosses.

The left panel is one die roll, $S=\{1,2,3,4,5,6\}$ ${1, 2, 3, 4, 5, 6}$ . The right panel is two coin tosses, $S=\{HH,HT,TH,TT\}$ ${H H, H T, T H, T T}$ , where the order of the tosses is recorded, so $HT$ and $TH$ are different outcomes; by the multiplication rule there are $2\cdot2=4$ of them.

Choosing $S$ is part of modelling. For two coin tosses, $\{0,1,2\}$ (the number of heads) is also a valid sample space, but its outcomes are not equally likely, and "the first toss is heads" cannot be expressed in it. $S$ need not be finite: tossing a coin until the first head gives $\{1,2,3,\ldots\}$ . Some texts write $\Omega$ instead of $S$ .

Key Results¶

The sample space must include every possible outcome in the model.
Different modelling choices can lead to different sample spaces for the same real situation.
Events are subsets of the sample space.

Examples¶

For one coin toss, the sample space can be heads and tails.
For two dice, the sample space can be ordered pairs of die faces.
For a waiting-time experiment, the sample space can be all nonnegative times.

Prerequisites¶

Set
Outcome

Sources¶

Probability Theory for EOR · Week 1

Aliases: outcome space Keywords: possibility space Symbols: omega

Last updated: 20 October 2026 03:02