Mutually Exclusive Events | Free Exam P Lesson

A single claim arises from exactly one peril, fire and theft cannot cause the same claim event. Allow overlap between mutually exclusive perils in your model and the aggregate loss double-counts scenarios that cannot happen.

Two events $A$ and $B$ are mutually exclusive (disjoint) if:

$P (A \cap B) = 0$

On a Venn diagram, the circles do not overlap.

HIGH-FREQUENCY: The simple addition rule and its misapplication are among the most tested concepts in the general probability section.

For mutually exclusive events:

$P (A \cup B) = P (A) + P (B)$

This extends to any finite collection of pairwise mutually exclusive events:

$P (i = 1 ⋃ n A_{i}) = i = 1 \sum n P (A_{i})$

This is a direct consequence of Axiom 3. It is why the axiom exists, to let you add probabilities of non-overlapping events.

Events $A_{1}, \dots, A_{n}$ form a partition of $S$ if they are:

Mutually exclusive: $A_{i} \cap A_{j} = \emptyset$ for $i \neq = j$
Exhaustive: $A_{1} \cup \dots \cup A_{n} = S$

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Common mistakes

Applying the simple addition rule to non-exclusive events. If $P (A) = 0.40$ , $P (B) = 0.35$ , and $P (A \cap B) = 0.10$ , then $P (A \cup B) = 0.65$ , not $0.75$ . Trap: 0.75.
Assuming mutually exclusive means independent. "No overlap" does not mean "no relationship." ME events are maximally dependent. Trap: computing $P (A \cap B) = P (A) P (B) = 0.0375$ when the correct answer is 0.
Forgetting a partition must sum to 1. If $P (A_{1}) = 0.3$ , $P (A_{2}) = 0.4$ , $P (A_{3}) = 0.2$ for a partition, the implicit fourth event has $P = 0.1$ . Missing this makes your total probability calculation incomplete.

Bottom line

Mutually exclusive: $P (A \cap B) = 0$ , the events cannot both occur in one experiment
Simple addition rule: $P (A \cup B) = P (A) + P (B)$ when the events are mutually exclusive
Partition of S: mutually exclusive plus exhaustive, with probabilities that sum to 1
Mutually exclusive $\neq =$ independent: ME events with positive probabilities are maximally dependent

Exam shortcut

When mutually exclusive categories are given, conditional probabilities reduce to $P (A_{i}) / P (union)$ . Always verify the partition sums to the correct total. "Mutually exclusive = mutually destructive." If one happens, it destroys the possibility of the other. The opposite of independence. Quick impossibility check: If $P (A) + P (B) > 1$ , the events overlap, mutual exclusivity is impossible.

The full lesson (about 2,035 words, 14 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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