Worked solution · Exam P

SOA Exam P Sample Question #188 Solution

This Exam P sample reference tests Independence. Let L mean wanting either life policy and H mean wanting the health policy. Independence and the 0.18 health-only probability give 0.18=0.45P(L-complement), so P(L)=0.60. Life-only probability is 0.60(0.55)=0.33; adding health-only gives 0.51, choice A.

Independent analysis by ActuaryProof · reviewed July 31, 2026

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Independent solution

How to solve this Independence question

Setup

Combine the mutually exclusive life-policy choices into one event L and denote the health-policy event by H.

\Pr(H)=0.45,\qquad \Pr(H\cap L^c)=0.18

Model

Use independence to recover the probability of wanting some life policy.

0.18=\Pr(H)\Pr(L^c)=0.45\Pr(L^c)

\Pr(L^c)=0.40,\qquad \Pr(L)=0.60

Compute

Exactly one policy is either health only or one life policy without health.

\Pr(L\cap H^c)=0.60(0.55)=0.33

\Pr(\text{exactly one policy})=0.33+0.18=0.51

Answer

The exactly-one probability is 0.51.

\boxed{0.51\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 value 0.60 is P(L), the probability of wanting one of the life policies. It includes people who also want health coverage.
CThe value 0.69 is 0.51+0.18, which counts the health-only region twice.
DThe value 0.73 is 0.55+0.18. It incorrectly treats every participant not wanting health coverage as wanting exactly one life policy.
EThe value 0.78 is P(L union H)=0.60+0.45-0.27, the probability of wanting at least one policy. It includes participants wanting both a life policy and health coverage.

Original practice · fully worked

Original variant: premium maintenance without monitoring

A facility chooses at most one maintenance option. The probability of choosing some maintenance option is 0.60, and premium maintenance is chosen twice as often as standard maintenance. Remote monitoring is chosen with probability 0.35 independently of which maintenance option, if any, is chosen. Calculate the probability that a facility chooses premium maintenance but not remote monitoring.

A
0.20
B
0.21
C
0.26
D
0.40
E
0.60

Answer and calculation

The 2:1 split of the 0.60 maintenance probability gives premium probability 0.40. Independence then gives 0.40(1-0.35)=0.26 for premium maintenance without monitoring, choice C.

Setup

Split the any-maintenance probability according to the stated premium-to-standard ratio.

\Pr(P)+\Pr(S)=0.60,\qquad \Pr(P)=2\Pr(S)

Model

Solve the subtype probabilities and identify the no-monitoring probability.

\Pr(S)=0.20,\qquad \Pr(P)=0.40

\Pr(R^c)=1-0.35=0.65

Compute

Use independence between monitoring and the complete maintenance choice.

\Pr(P\cap R^c)=\Pr(P)\Pr(R^c)=0.40(0.65)=0.26

Answer

The requested probability is 0.26.

\boxed{0.26\quad\text{(C)}}

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How to solve this Independence question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: premium maintenance without monitoring

Answer and calculation

Setup

Model

Compute

Answer

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