$S = \{\omega_1,\, \omega_2,\, \ldots\}$ $\text{where each } \omega_i = \text{an elementary outcome of the experiment}$ A Sample Space ($S$, also written $\Omega$) is the set of all possible outcomes of a random experiment; every conceivable result of the experiment must appear as exactly one element of $S$. Outcomes in $S$ must be mutually exclusive (no two can occur simultaneously) and collectively exhaustive (together they cover every possibility). The sample space can be finite, countably infinite, or uncountably infinite depending on the experiment. > [!example]- Sample Space for Claim Occurrence and Size {💡 Example} > An experiment records whether a policyholder files a claim and, if so, classifies the loss as small ($\leq\$1{,}000$) or large (gt;\$1{,}000$). Write the sample space. > > > [!answer]- Answer > > There are three mutually exclusive, exhaustive outcomes: > > $S = \{\text{No Claim},\; \text{Small Claim},\; \text{Large Claim}\}$ > > Each outcome is distinct, they cannot co-occur, and every possible result of the experiment is represented.