Verified 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 August 4, 2026

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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)}}

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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.

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

Variant answer in brief

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)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #331 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Set Operations question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: exclusive warnings from indicator covariance

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution