Worked solution · Exam P

SOA Exam P Sample Question #25 Solution

This Exam P sample reference tests Bayes’ Theorem. Let the smoking rate without the condition be q; then the two Bayes weights are 0.50q and 0.75q. Their ratio gives posterior 0.40, choice C.

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 Bayes’ Theorem question

Setup

Let q be the smoking probability without the condition; the smoking probability with the condition is then 2q.

\Pr(S\mid C^c)=q,\qquad \Pr(S\mid C)=2q

Model

Apply Bayes' formula using the condition and no-condition prior shares 0.25 and 0.75.

\Pr(C\mid S)=\frac{(2q)(0.25)}{(2q)(0.25)+q(0.75)}

Compute

After the common factor q cancels, the condition-and-smoking weight is 0.50 and the total smoking weight is 1.25, giving 0.40.

\Pr(C\mid S)=\frac{0.50}{1.25}=0.40

Answer

Given smoking, the probability of the condition is 2/5, corresponding to choice C.

\boxed{\frac25\quad\text{(C)}}

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.

AThe value 1/4 is the prior probability of the condition and ignores the doubled smoking likelihood.
BThe value 1/3 is the prior odds 0.25/0.75 treated as a probability; it does not normalize the two smoking weights.
DThe value 1/2 is the unnormalized condition-and-smoking weight after q is canceled, not the posterior ratio.
EThe value 2/3 normalizes the likelihood ratio 2:1 without using the unequal prior shares 0.25 and 0.75.

Original practice · fully worked

Original variant: posterior membership from a likelihood ratio

Thirty percent of devices use controller C. A warning is three times as likely on devices using C as on other devices. Given that a warning appears, find the probability that the device uses controller C.

A
0.3000
B
0.5000
C
0.5625
D
0.7000
E
0.9000

Answer and calculation

Warning weights are proportional to 3(0.30) and 1(0.70). The controller-C share is 0.90/1.60=0.5625, choice C.

Setup

Let q be the warning probability for devices not using controller C; devices using C have warning probability 3q.

\Pr(W\mid C)=3q,\qquad \Pr(W\mid C^c)=q

Model

Multiply the two warning likelihoods by the controller priors and normalize the controller-C weight.

\Pr(C\mid W)=\frac{3q(0.30)}{3q(0.30)+q(0.70)}

Compute

After q cancels, the controller-C and other-controller weights are 0.90 and 0.70, so the posterior is 0.90/1.60=0.5625.

\Pr(C\mid W)=0.90/1.60=0.5625

Answer

Given a warning, the probability that the device uses controller C is 0.5625, which is choice C.

\boxed{0.5625\quad\text{(C)}}

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 Bayes’ Theorem question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: posterior membership from a likelihood ratio

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability