$A \cap B = \emptyset \implies P(A \cup B) = P(A) + P(B)$ Two events $A$ and $B$ are Mutually Exclusive (also called disjoint) if they cannot both occur simultaneously; their intersection is the empty set $\emptyset$. Mutual exclusivity and independence are distinct and often confused: mutually exclusive events with positive probability are **never** independent, because if $A$ occurs it guarantees $B$ did not ($P(B \mid A) = 0 \neq P(B)$). The addition rule simplifies to a plain sum only when events are mutually exclusive. > [!example]- Mutually Exclusive Claim Types {💡 Example} > A policy classifies each claim as either Minor ($M$) or Major ($J$), never both. $P(M) = 0.7$ and $P(J) = 0.3$. Are $M$ and $J$ independent? > > > [!answer]- Answer > > Since every claim is exactly one type, $M \cap J = \emptyset$ — the events are mutually exclusive. For independence we would need: > > $P(M \cap J) = P(M) \cdot P(J) = 0.7 \times 0.3 = 0.21$ > > But $P(M \cap J) = 0 \neq 0.21$, so they are **not** independent. Mutual exclusivity (with positive probabilities) always precludes independence.