Worked solution · Exam P

SOA Exam P Sample Question #331 Solution

This Exam P sample reference tests Set Operations. The overlap equals the total health share minus the health-only share, or (x-z)/100. Subtracting that overlap from the total disability share gives (y-x+z)/100, which is choice A.

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.

Read the solution Open official source

Independent solution

How to solve this Set Operations question

Setup

Let the two insurance events be H and D, and express all percentages as probabilities.

\Pr(H)=\frac{x}{100},\qquad \Pr(D)=\frac{y}{100},\qquad \Pr(H\cap D^c)=\frac{z}{100}

Model

Partition the health event into its health-only and overlap regions.

\Pr(H\cap D)=\Pr(H)-\Pr(H\cap D^c)=\frac{x-z}{100}

Compute

Remove the overlap from the disability total to isolate disability-only respondents.

\Pr(D\cap H^c)=\Pr(D)-\Pr(H\cap D)

=\frac{y}{100}-\frac{x-z}{100}=\frac{y-x+z}{100}

Answer

The required exclusive-region probability matches the first expression.

\boxed{\frac{y-x+z}{100}\quad\text{(A)}}

Test the method on unseen problems

General Probability Practice Workbook

30 original general probability problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

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

BThe expression (y-x-z)/100 treats the overlap as (x+z)/100. The health total is the sum of health-only and overlap, so the overlap is x-z, not x+z.
CThe expression (y-x-2z)/100 subtracts the health-only region twice after removing the health total. The z term must enter positively when x-z is subtracted from y.
DThe expression (y-x+2z)/200 is the average of the correct disability-only percentage y-x+z and the health-only percentage z. These are distinct regions, and the question calls for the first one rather than their average.
EThe expression (y-z)/100 subtracts the health-only share directly from the disability total. Those regions are disjoint; the quantity that must be removed from disability is the overlap x-z.

Original practice · fully worked

Original variant: exclusive warnings from indicator covariance

During a diagnostic cycle, event A is a thermal warning and event B is a vibration warning. Their probabilities are 0.40 and 0.50. The covariance of the two event indicators is -0.02. Calculate the probability that exactly one warning occurs.

A
0.18
B
0.36
C
0.50
D
0.54
E
0.72

Answer and calculation

Indicator covariance equals P(A and B)-P(A)P(B), so the overlap is -0.02+0.40(0.50)=0.18. Exactly one warning has probability P(A)+P(B)-2P(A and B)=0.54, which is choice D.

Setup

Let I_A and I_B be the two warning indicators and translate their product into the overlap event.

E[I_A]=0.40,\qquad E[I_B]=0.50

E[I_AI_B]=\Pr(A\cap B)

Model

Use the covariance to recover the otherwise unreported intersection.

-0.02=\Pr(A\cap B)-(0.40)(0.50)

\Pr(A\cap B)=0.18

Compute

The sum of the marginals counts each exclusive warning once and the overlap twice, so remove the overlap twice.

\Pr(\text{exactly one})=\Pr(A)+\Pr(B)-2\Pr(A\cap B)

=0.40+0.50-2(0.18)=0.54

Answer

Exactly one warning occurs with probability 0.54.

\boxed{0.54\quad\text{(D)}}

Keep practicing

Next steps for Exam P General Probability

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P General Probability practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Set Operations question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: exclusive warnings from indicator covariance

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability