Verified solution · Exam P

SOA Exam P Sample Question #8 Solution

This Exam P sample reference tests Inclusion–Exclusion. Writing the therapist probability as t and the chiropractor probability as t+0.14 makes the union equation linear; t=0.48, choice D.

Independent analysis by ActuaryProof · reviewed August 4, 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 Inclusion–Exclusion question

Setup

Let t=Pr(T). The chiropractor probability is then t+0.14, and the probability of at least one treatment is 1-0.12=0.88.

\Pr(C\cup T)=1-0.12=0.88

\Pr(C)=\Pr(T)+0.14

Model

Apply the two-event addition rule, subtracting the 0.22 overlap once.

0.88=(t+0.14)+t-0.22

Compute

Substituting Pr(C)=t+0.14 yields 0.88=2t-0.08, so 2t=0.96 and t=0.48.

2t=0.96,\qquad t=0.48

Answer

The therapist probability is 0.48, corresponding to choice D.

\boxed{0.48\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.

AIf t=0.26, then Pr(C union T)=(0.26+0.14)+0.26-0.22=0.44, not 0.88.
BIf t=0.38, the addition rule gives a union probability of 0.68 rather than 0.88.
CIf t=0.40, the implied union probability is 0.72, so this value does not satisfy the supplied data.
EThe value 0.62 is Pr(C)=0.48+0.14 after t has been solved. It reports the chiropractor marginal instead of Pr(T).

Original practice · fully worked

Original variant: recover an overlap from exactly-one membership

Two data-quality checks fail with probabilities 0.52 and 0.41. The probability that exactly one of the checks fails is 0.47. Determine the probability that both checks fail.

A
0.18
B
0.23
C
0.29
D
0.41
E
0.46

Variant answer in brief

Exactly-one probability equals the marginal sum minus twice the overlap. Solving 0.47=0.93-2x gives x=0.23, choice B.

Setup

Let x be the probability that both checks fail.

x=\Pr(A\cap B)

Model

The exactly-one event consists of A without B and B without A, so its probability is the sum of the two marginals minus twice their overlap.

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

Compute

Substitution gives 0.47=0.93-2x, from which x=0.23.

0.47=0.52+0.41-2x,\qquad x=0.23

Answer

Therefore both checks fail with probability 0.23, which is choice B.

\boxed{0.23\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #8 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Inclusion–Exclusion question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: recover an overlap from exactly-one membership

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution