Mutually Exclusive Events
Mutually exclusive events cannot both occur at the same time; if one happens, the other cannot.
Formula
P(A \text{ or } B) = P(A) + P(B) \quad [\text{when } A \text{ and } B \text{ are mutually exclusive}]
Definition
Mutually exclusive events cannot happen at the same time: if one occurs, the other definitely does not. Two events $A$ and $B$ are mutually exclusive (or disjoint) if they share no outcomes, $A \cap B = \emptyset$, and for mutually exclusive events $P(A \text{ or } B) = P(A) + P(B)$, a rule that extends to any number of them. Formally, mutual exclusivity implies $P(A \cap B) = 0$, and by the additivity axiom $P(A \cup B) = P(A) + P(B)$; a partition of $\Omega$ is a collection of pairwise mutually exclusive and exhaustive events whose probabilities sum to $1$.
Example
Rolling a die, "rolling a $3$" and "rolling a $5$" are mutually exclusive since you can only roll one number at a time, so $P(3 \text{ or } 5) = P(3) + P(5) = 1/6+1/6 = 1/3$. Drawing one card, "drawing a heart" and "drawing a spade" are mutually exclusive too: $P(\text{heart or spade}) = 13/52+13/52 = 1/2$. The law of total probability states that for a partition $\{B_1, \ldots, B_k\}$ of $\Omega$, $P(A) = \sum P(A|B_i)P(B_i)$, decomposing $P(A)$ into contributions from each partition element when direct calculation is difficult.
Key Insight
When events are mutually exclusive, you can just add their probabilities together; mutually exclusive events and independent events are different concepts, since mutually exclusive events cannot both happen while independent events do not influence each other, and in fact, if $P(A) > 0$ and $P(B) > 0$, mutually exclusive events cannot be independent. Mutual exclusivity is actually a stronger condition than independence: mutually exclusive events with nonzero probabilities are always negatively dependent, since knowing one occurred makes the other impossible.