Worked solution · Exam P

SOA Exam P Sample Question #182 Solution

This Exam P sample reference tests Set Operations. Resolving the inclusive intersections gives 23 in life-home only, 64 in auto-home only, and 88 in home only. The only-one total then gives 53 in life only and 129 in auto only; the remaining auto region has 18. Summing all seven regions gives 407, choice B.

Independent analysis by ActuaryProof · reviewed July 31, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Set Operations question

Setup

Remove the triple intersection from the two supplied inclusive pair intersections and recover the homeowner-only region.

n_{LH}=55-32=23,\qquad n_{AH}=96-32=64

n_H=207-23-64-32=88

Model

Let x be the life-only count; the auto-only count is x plus 76. Use the total belonging to exactly one product.

x+(x+76)+88=270

x=53,\qquad n_A=129

Compute

Use the auto marginal to find the last pair-only region, then add the disjoint regions.

n_{LA}=243-129-64-32=18

N=53+129+88+23+64+18+32=407

Answer

There are 407 policyholders owning at least one listed product.

\boxed{407\quad\text{(B)}}

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Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThe value 389 is the disjoint total with the 18-person life-auto-only region omitted.
CThe value 423 exceeds the sum of the seven determined regions by 16. It necessarily double-counts part of an overlap because every Venn region is already fixed.
DThe value 448 adds 41 people beyond the complete disjoint partition. It is incompatible with the stated auto and homeowner marginal totals.
EThe value 483 is 407+76. It treats the 76-person difference between two only-regions as an additional group, even though that difference is already built into their counts.

Original practice · fully worked

Original variant: recover a two-tag overlap

An archive contains 300 records. An urgent tag appears on 180 records, an external tag appears on 150 records, and exactly 170 records carry one of these tags but not both. Calculate the number of records carrying both tags.

A
30 records
B
80 records
C
150 records
D
170 records
E
250 records

Answer and calculation

If x records carry both tags, then the urgent-only and external-only counts are 180-x and 150-x. Their sum is 170, giving x=80 and choice B.

Setup

Let x denote the intersection of the urgent and external tag sets.

|U\cap E|=x

Model

Express the exactly-one count as the sum of the two disjoint one-tag regions.

|U\setminus E|+|E\setminus U|=(180-x)+(150-x)=170

Compute

Solve the resulting linear equation for the overlap.

330-2x=170

2x=160,\qquad x=80

Answer

Exactly 80 records carry both tags.

\boxed{80\quad\text{records (B)}}

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How to solve this Set Operations question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: recover a two-tag overlap

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability