In probability, the mutually exclusive meaning is that two events cannot occur in the same trial: there is no shared outcome that belongs to both events. Equivalently, the intersection of the events is empty.
Definition (set and probability form)
Let \(A\) and \(B\) be events (subsets of a sample space \(S\)). Events \(A\) and \(B\) are mutually exclusive (also called disjoint events) if:
\[ A \cap B = \varnothing \quad\Longleftrightarrow\quad P(A \cap B) = 0. \]
How it changes the addition rule
For any two events, the union-of-events addition rule is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]
Under the mutually exclusive meaning, \(P(A \cap B)=0\), so the formula simplifies to:
\[ P(A \cup B) = P(A) + P(B). \]
Worked example (die roll)
Consider one roll of a fair six-sided die with sample space \(S=\{1,2,3,4,5,6\}\) (all outcomes equally likely).
| Event | Definition | Probability | Mutually exclusive with the other? |
|---|---|---|---|
| \(A\) | Even number \(\{2,4,6\}\) | \(P(A)=\frac{3}{6}=\frac{1}{2}\) | No, because \(A \cap B=\{2\}\neq\varnothing\) |
| \(B\) | Prime number \(\{2,3,5\}\) | \(P(B)=\frac{3}{6}=\frac{1}{2}\) | |
| \(C\) | Roll a 1 \(\{1\}\) | \(P(C)=\frac{1}{6}\) | Yes, because \(C \cap D=\varnothing\) |
| \(D\) | Roll a 6 \(\{6\}\) | \(P(D)=\frac{1}{6}\) |
For \(A\) and \(B\), the addition rule must subtract the overlap:
\[ P(A \cup B)=P(A)+P(B)-P(A \cap B) =\frac{1}{2}+\frac{1}{2}-\frac{1}{6} =\frac{5}{6}. \]
For \(C\) and \(D\), the mutually exclusive meaning applies, so the overlap term is zero:
\[ P(C \cup D)=P(C)+P(D)=\frac{1}{6}+\frac{1}{6}=\frac{1}{3}. \]
Common pitfalls and a key contrast
Mutually exclusive vs independent
- Mutually exclusive means \(P(A \cap B)=0\).
- Independent means \(P(A \cap B)=P(A)\cdot P(B)\).
- If \(P(A)>0\) and \(P(B)>0\), then mutually exclusive events cannot be independent because \(0 \neq P(A)\cdot P(B)\).