Mutually Exclusive Events

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:

$$P\left(\bigcup_{i=1}^{n} A_i\right) = \sum_{i=1}^{n} P(A_i)$$

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

Events $A_1, \ldots, A_n$ form a partition of $S$ if they are:

• Mutually exclusive: $A_i \cap A_j = \emptyset$ for $i \neq j$
• Exhaustive: $A_1 \cup \cdots \cup A_n = S$

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 $P(A_i)/P(\text{union})$. 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

• 1d

Browse all free Exam P lessons or jump into free Exam P practice questions.