Sample Space
The sample space is the complete set of all possible outcomes of a probability experiment.
Definition
The sample space is the complete list of all possible outcomes of an experiment; it includes every result that could possibly happen. The sample space S (or $\Omega$) is the set of all possible outcomes of a probability experiment; events are subsets of S, and for two experiments combined, the sample space is the Cartesian product of their individual sample spaces. Formally, $\Omega$ is the fundamental set in a probability space $(\Omega, \mathcal{F}, P)$; its structure determines the appropriate sigma-algebra, for finite or countable $\Omega$ the power set works, while for uncountable $\Omega$ (e.g., $\mathbb{R}$) a Borel sigma-algebra is required to avoid unmeasurable sets, and the choice of $\Omega$ implicitly encodes all modeling assumptions about what outcomes are possible.
Example
Flipping a coin gives sample space $= \{\text{Heads}, \text{Tails}\}$; rolling a die gives $\{1,2,3,4,5,6\}$; flipping two coins gives $\{HH, HT, TH, TT\}$. Flipping a coin and rolling a die together gives sample space $\{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6\}$, $12$ equally likely outcomes, so $P(H \text{ and } 3) = 1/12$. For modeling a single continuous measurement, $\Omega = \mathbb{R}$ with Borel sigma-algebra $\mathcal{B}(\mathbb{R})$ and Lebesgue-based measure; for sequences of coin flips, $\Omega = \{0,1\}^\infty$ (the infinite product space), with the product sigma-algebra enabling probabilities for all cylinder sets and their limits.
Key Insight
You must know the full sample space before you can calculate any probability, since the probabilities of all outcomes in it must add up to 1; drawing a tree diagram is a useful technique for listing sample spaces when a multi-stage experiment has many branches. Kolmogorov's extension theorem guarantees that a consistent family of finite-dimensional distributions uniquely determines a probability measure on the infinite product space, enabling rigorous treatment of stochastic processes like Brownian motion.