Sample space

In probability theory, the sample space or universal sample space, often denoted S, &Omega; or U (for "universe"), of an experiment or random trial is the set of all possible outcomes. For example, if the experiment is tossing a coin, the sample space is the set {head, tail}. For tossing a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Any subset of the sample space is usually called an event, while subsets of the sample space containing just a single element are called elementary events.

For some kinds of experiments, there may be two or more plausible sample spaces available. For example, when drawing a card from a standard deck of 52 playing cards, one possibility for the sample space could be the rank (Ace through King), while another could be the suit (clubs, diamonds, hearts, or spades). A complete description of outcomes, however, would specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above.

Sample spaces appear naturally in an elementary approach to probability, but are also important in probability spaces. A probability space (&Omega;, F, P) incorporates a sample space of outcomes, &Omega;, but defines a set of events of interest, the &sigma;-algebra F, for which the probability measure P is defined.