Mutually Exclusive Events
Statistics & ProbabilityMutually 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.
Example
Rolling a die: "rolling a $3$" and "rolling a $5$" are mutually exclusive. You can only roll one number at a time. Flipping a coin: heads and tails are mutually exclusive.
Key Insight
When events are mutually exclusive, you can just add their probabilities together. $P(3 \text{ or } 5) = P(3) + P(5) = 1/6+1/6 = 2/6 = 1/3$.
Definition
Two events $A$ and $B$ are mutually exclusive (or disjoint) if they share no outcomes: $A \cap B = \emptyset$. For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$. This rule extends to any number of mutually exclusive events.
Example
Drawing one card: "drawing a heart" and "drawing a spade" are mutually exclusive (a card cannot be both). $P(\text{heart or spade}) = P(\text{heart}) + P(\text{spade}) = 13/52+13/52 = 26/52 = 1/2$.
Key Insight
Mutually exclusive events and independent events are different concepts. Mutually exclusive events cannot both happen; independent events do not influence each other. In fact, if $P(A) > 0$ and $P(B) > 0$, mutually exclusive events cannot be independent.
Definition
Events $A$ and $B$ are mutually exclusive if $A \cap B = \emptyset$, implying $P(A \cap B) = 0$. For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$ by the additivity axiom. A partition of $\Omega$ is a collection of pairwise mutually exclusive and exhaustive events: their probabilities sum to $1$.
Example
The law of total probability: for a partition $\{B_1, \ldots, B_k\}$ of $\Omega$, $P(A) = \sum P(A|B_i)P(B_i)$. This decomposes $P(A)$ into contributions from each partition element, enabling computation when direct calculation is difficult.
Key Insight
Mutual exclusivity is a stronger condition than independence: mutually exclusive events with nonzero probabilities are always negatively dependent (knowing one occurred makes the other impossible). This is why "mutually exclusive" and "independent" are often confused but are fundamentally different concepts.