Worked solution · Exam P

SOA Exam P Sample Question #183 Solution

This Exam P sample reference tests Set Operations. Removing the triple region gives 82 in life-mutual only and 77 in annuity-mutual only, leaving 45 in mutual only. The exactly-one total then yields 100 in annuity only and 145 in life only; the remaining annuity-life region has 35. All seven regions total 500, choice D.

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

Convert the two inclusive mutual-fund intersections to pair-only counts and recover the mutual-only region.

n_{LM}=98-16=82,\qquad n_{AM}=93-16=77

n_M=220-82-77-16=45

Model

Let x be the annuity-only count; the life-only count is x plus 45. Use the total owning exactly one type.

x+45+(x+45)=290

x=100,\qquad n_L=145

Compute

Use the annuity marginal to obtain the last pair-only region and sum the partition.

n_{AL}=228-100-77-16=35

N=100+45+145+77+82+35+16=500

Answer

The agent has 500 clients owning at least one listed product.

\boxed{500\quad\text{(D)}}

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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 455 is the full total with the 45-client mutual-fund-only region omitted.
BThe value 495 falls five below the exact sum of the seven disjoint regions. It cannot reproduce the stated annuity, mutual-fund, and exactly-one totals.
CThe value 496 is an off-by-four region sum. The determined counts are integers 100, 45, 145, 77, 82, 35, and 16, whose sum is exactly 500.
EThe value 516 adds the 16-client triple intersection a second time after it has already been included in the disjoint partition.

Original practice · fully worked

Original variant: exactly one feature from conditional rates

A platform has 400 accounts. Fifty-five percent enable backups. Among backup-enabled accounts, 30% also enable encryption; among accounts without backups, 20% enable encryption. Calculate the number of accounts that enable exactly one of the two features.

A
66 accounts
B
102 accounts
C
154 accounts
D
190 accounts
E
256 accounts

Answer and calculation

There are 220 backup-enabled accounts, of which 154 lack encryption. Of the 180 accounts without backups, 36 enable encryption. These two exactly-one regions total 190, choice D.

Setup

Split the account population by backup status.

n(B)=400(0.55)=220,\qquad n(B^c)=180

Model

Calculate the two disjoint regions in which exactly one feature is enabled.

n(B\cap E^c)=220(0.70)=154

n(B^c\cap E)=180(0.20)=36

Compute

Add the backup-only and encryption-only regions.

n(\text{exactly one})=154+36=190

Answer

Exactly 190 accounts enable one feature.

\boxed{190\quad\text{accounts (D)}}

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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: exactly one feature from conditional rates

Answer and calculation

Setup

Model

Compute

Answer

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