## Definition A ==Set Function== is a function whose domain is a collection of sets and whose range is the real numbers. Probability is the most important example: it maps each event (a set of outcomes) to a real number between 0 and 1 that measures its likelihood. > [!example]- Why is probability considered a set function rather than a point function? > Probability assigns values to *events* (sets of outcomes), not to individual points. For example, when rolling a die, we write $P(\{2,4,6\}) = 0.5$ — the input is the set $\{2,4,6\}$, not any single number. This set-based definition allows probability to satisfy the axioms (non-negativity, normalization, and countable additivity) in a consistent way.