The not mutually exclusive meaning in probability is that two events can occur at the same time, so their overlap (intersection) is not empty. In symbols, “not mutually exclusive” means \(P(A \cap B) > 0\) (at least in typical applications).
Key definitions
Mutually exclusive (disjoint) events: \(A\) and \(B\) cannot occur together, so \(A \cap B = \varnothing\) and \(P(A \cap B)=0\).
Not mutually exclusive (overlapping) events: \(A\) and \(B\) can occur together, so \(A \cap B \neq \varnothing\) and typically \(P(A \cap B)>0\).
Why the addition rule changes
When computing \(P(A \cup B)\) (the probability that at least one of the events happens), adding \(P(A)\) and \(P(B)\) counts outcomes in \(A \cap B\) twice. Subtracting the intersection corrects this double-counting.
If events are mutually exclusive, then \(P(A \cap B)=0\), so the formula reduces to \(P(A \cup B)=P(A)+P(B)\).
Visualization of overlap
Worked example with numbers
Suppose two events overlap and the following probabilities are known: \(P(A)=0.55\), \(P(B)=0.40\), and \(P(A \cap B)=0.20\). Compute the probability of “\(A\) or \(B\)” (the union).
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Use the addition rule for not mutually exclusive events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]
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Substitute values:
\[ P(A \cup B) = 0.55 + 0.40 - 0.20. \]
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Compute:
\[ P(A \cup B) = 0.75. \]
The result \(0.75\) is less than \(0.55+0.40=0.95\) because the overlap probability \(0.20\) was counted twice when simply adding \(P(A)\) and \(P(B)\).
Mutually exclusive vs not mutually exclusive
| Relationship | Intersection | Union rule | Common wording |
|---|---|---|---|
| Mutually exclusive | \(P(A \cap B)=0\) | \(P(A \cup B)=P(A)+P(B)\) | “A or B, but not both” |
| Not mutually exclusive | \(P(A \cap B)>0\) (typically) | \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\) | “A or B (possibly both)” |
Common confusions
Not mutually exclusive vs independent
Independence is about whether the occurrence of one event changes the probability of the other. If \(A\) and \(B\) are independent, then \[ P(A \cap B)=P(A)\cdot P(B). \] Mutually exclusive events have \(P(A \cap B)=0\), so they are not independent unless \(P(A)=0\) or \(P(B)=0\).
Mutually exclusive vs complementary
Complementary events \(A\) and \(A^c\) are always mutually exclusive and exhaustive: \(A \cap A^c=\varnothing\) and \(A \cup A^c\) equals the entire sample space. Not mutually exclusive events overlap and therefore cannot be complements.