Mutually Exclusive Events

Study guide: Exam P-1

Two events $A$ and $B$ are Mutually Exclusive (disjoint) if they cannot both occur simultaneously.

For mutually exclusive events, the Probability Addition Rule simplifies to $P(A \cup B) = P(A) + P(B)$
The addition rule simplification extends to any finite collection: if $A_1, A_2, \ldots, A_n$ are pairwise mutually exclusive, then $P\!\left(\bigcup_{i=1}^n A_i\right) = \sum_{i=1}^n P(A_i)$

$A \cap B = \emptyset \implies P(A \cap B) = 0$

Insurance Claim Type Each claim is classified as exactly one of property damage ( $D$ ), bodily injury ( $B$ ), or other. $P(D) = 0.60$ and $P(B) = 0.35$ . Find the probability that a claim is property damage or bodily injury, and the probability that it is classified as other.

Answer Since $D$ and $B$ are mutually exclusive: $P(D \cup B) = P(D) + P(B) = 0.60 + 0.35 = 0.95$ The three classes are mutually exclusive and cover every claim, so "other" is the complement of $D \cup B$ : $P(\text{other}) = 1 - 0.95 = 0.05$