Definition
The sample space of a random experiment, usually written as Ω (omega), is the set of every possible outcome that could result from the experiment. For a coin flip, Ω is {Heads, Tails}; for a single roll of a die, Ω is {1, 2, 3, 4, 5, 6}; for measuring the lifetime of a light bulb, Ω is the set of non-negative real numbers.
Every event whose probability you want to compute is a subset of Ω. Choosing the sample space is the first — and often the most consequential — modelling decision in any probability problem.
Why it matters
How it works
To analyse any random experiment, you first enumerate Ω. Then you identify each event of interest as a subset of Ω. The probability function assigns a number to each (admissible) subset, subject to the axioms that the whole space has probability 1 and that disjoint subsets add. From that point on, every calculation — conditional probability, expected value, variance — is built out of operations on these subsets.
For a fair die, the event 'even number' is the subset {2, 4, 6}, with probability 3/6 = 1/2. For a continuous measurement, events become intervals or regions, and probabilities are computed as integrals of a density function.