## Definition ==Mutually Exclusive Events== (disjoint events) are events that cannot occur simultaneously — their intersection is empty: $A \cap B = \emptyset$. If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$. > [!example]- Are the events "rolling a 2" and "rolling a 5" on a single die mutually exclusive? Are they independent? > These events are **mutually exclusive** because a single die roll cannot show both 2 and 5: $P(A \cap B) = 0$. > > They are **not independent** because $P(A) \cdot P(B) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36} \neq 0 = P(A \cap B)$. In fact, non-trivial mutually exclusive events can never be independent (if one occurs, the other certainly does not).