Sample Space in Probability

The sample space is the set of all possible outcomes of an experiment. It’s usually denoted as S or Ω (omega).

In Venn diagrams, the sample space is depicted as a rectangle.

The outcomes are represented by circles.

Example

When rolling a die, the sample space includes 6 possible outcomes:

$$ S = \{ 1, 2, 3, 4, 5, 6 \} $$

Subsets of the sample space are called events.

An event E is a subset of the sample space (E⊂S), consisting of certain possible outcomes.

Example

In a die roll, one possible event E is getting an even number:

$$ E = \{ 2, 4, 6 \} $$

Another event F could be rolling an odd number:

$$ F = \{ 1, 3, 5 \} $$

This can be illustrated using a Venn diagram:

Both events are subsets of the sample space S = {1, 2, 3, 4, 5, 6}.

$$ E ⊂ S $$

$$ F ⊂ S $$

And so on.