Venn Diagram

Study guide: Exam P-1

A Venn Diagram is a visual tool used to represent the relationships between different sets, used for calculating the intersections and unions of multiple events. For two events, the probability of the union is:

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

$S$ : Universal Set — the rectangle representing every possible outcome (Total Probability = 1)
$A, B$ : Sets A and B — the circles representing specific events
$A \cap B$ : Intersection — the overlapping region where both events occur
$A \cup B$ : Union — the total area covered by both circles
Actuarial exams often use specific phrasing that maps directly to regions of a Venn Diagram

Phrasing in Problem	Mathematical Notation	Venn Region
"Both $A$ and $B$ "	$A \cap B$	The central overlap.
"Either $A$ or $B$ "	$A \cup B$	Everything inside both circles.
"Neither $A$ nor $B$ "	$(A \cup B)^c$	The space outside both circles.
" $A$ but not $B$ "	$A \setminus B$	The "crescent moon" of $A$ only.
"Exactly one of $A$ or $B$ "	$(A \setminus B) \cup (B \setminus A)$	Both crescent moons (no overlap).

The Insurance Policyholder Let $A$ be the event that a policyholder has an auto policy and $H$ the event that they have a homeowners policy, with $P(A) = 0.70$ , $P(H) = 0.40$ , $P(A \cap H) = 0.20$ . Find the probability of a policyholder having neither policy.

Answer Step 1: Find the Union (Anyone with at least one policy) $P(A \cup H) = 0.70 + 0.40 - 0.20 = 0.90$

Step 2: Find the complement (Neither) $P(\text{Neither}) = 1 - P(A \cup H) = 1 - 0.90 = 0.10$

Common Trap When filling out a Venn Diagram, always start from the center (the intersection) and work your way out. If you just put "70" in the Auto circle, you've forgotten that 20 of those people also have Homeowners. The "Auto Only" region is actually $70 - 20 = 50$ .