## Definition The ==Axioms of Probability== (Kolmogorov's axioms) are three foundational rules that any valid probability measure $P$ must satisfy: 1. **Non-negativity**: $P(A) \geq 0$ for every event $A$. 2. **Normalization**: $P(S) = 1$ where $S$ is the entire sample space. 3. **Countable Additivity**: For any countable sequence of mutually exclusive events $A_1, A_2, \ldots$, $ P\!\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i) $ > [!example]- A probability model assigns $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cup B) = 0.9$ where $A$ and $B$ are mutually exclusive. Does this violate the axioms? > If $A$ and $B$ are mutually exclusive, countable additivity requires $P(A \cup B) = P(A) + P(B) = 0.3 + 0.5 = 0.8$. Since the model states $P(A \cup B) = 0.9 \neq 0.8$, **yes, it violates the third axiom**.