Mutually Exclusive Events

Events are mutually exclusive when they share no outcomes, so $P(A \text{ and } B) = 0$ and one occurring rules the other out. The general addition rule $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ removes the double count of the shared outcomes, and it collapses to a plain sum only because that intersection is zero for disjoint events. An event and its complement are the canonical disjoint pair, summing to 1, and any categorical variable's categories partition the sample space the same way, which is why a two-way table's joint proportions total 1.

Two errors dominate. The overlap is dropped: $P(\text{heart or face card})$ computed as $\frac{13}{52} + \frac{12}{52} = \frac{25}{52}$ instead of $\frac{22}{52}$, with the three face cards of hearts counted twice, and or-probabilities in surveys that quietly climb above 1. And mutually exclusive is read as independent, when disjointness with positive probability forces $P(A \mid B) = 0 \ne P(A)$, making such events maximally dependent. Independent events with positive probability must actually overlap, since $P(A \text{ and } B) = P(A)P(B) > 0$, so the two conditions can never hold together.