Worked solution · Exam P

SOA Exam P Sample Question #288 Solution

This Exam P sample reference tests Conditional Probability. This is a Bayes-theorem update after a positive test outcome. True positives contribute 0.18 of the tested population and false positives contribute 0.16, so the posterior is 0.18/0.34=0.5294 and choice D.

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 Conditional Probability question

Setup

Let P denote pregnancy and + denote a positive test outcome. Convert the false-negative rate to sensitivity.

\Pr(P)=0.20,\qquad \Pr(P^c)=0.80

\Pr(+\mid P)=1-0.10=0.90,\qquad \Pr(+\mid P^c)=0.20

Model

Bayes' theorem compares the positive mass from the target group with all positive results.

\Pr(P\mid +)=\frac{\Pr(+\mid P)\Pr(P)}{\Pr(+\mid P)\Pr(P)+\Pr(+\mid P^c)\Pr(P^c)}

Compute

Weight both test error rates by their population shares.

\Pr(P\cap +)=0.90(0.20)=0.18

\Pr(+)=0.18+0.20(0.80)=0.34

\Pr(P\mid +)=\frac{0.18}{0.34}=\frac9{17}=0.5294118\ldots

Answer

The posterior probability rounds to 0.53.

\boxed{0.53\quad\text{(D)}}

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 0.10 is the false-negative probability. It concerns a negative outcome within the pregnant group, not pregnancy within the positive-result group.
BThe value 0.20 is the prior pregnancy share and also the false-positive rate here. Reporting either number without population weighting skips the Bayes update.
CThe value 0.50 comes from normalizing the prior 0.20 against the false-positive rate 0.20 as though they were comparable positive masses. The likelihood and the non-pregnant base rate are both missing.
EThe value 0.90 is the sensitivity P(+|P). It reverses the conditioning direction and is not P(P|+).

Original practice · fully worked

Original variant: two diagnostic flags from prior odds

A service incident comes from configuration R or configuration S, with prior odds R:S of 1:4. Two diagnostic flags are conditionally independent given the configuration. The first flag has likelihood ratio 3 in favor of R, and the second has likelihood ratio 2 in favor of R. Both flags appear. Calculate the posterior probability that configuration R caused the incident.

A
0.2000
B
0.3333
C
0.4286
D
0.6000
E
1.5000

Answer and calculation

Conditional independence allows the likelihood ratios to multiply. The prior odds 1:4 become posterior odds 6:4=3:2, which convert to probability 3/5=0.6000 and choice D.

Setup

Represent the prior information as odds for R against S.

O_{\mathrm{prior}}(R:S)=\frac14

Model

For conditionally independent evidence, the combined likelihood ratio is the product of the two individual ratios.

LR_{\mathrm{both}}=3(2)=6

O_{\mathrm{post}}(R:S)=O_{\mathrm{prior}}LR_{\mathrm{both}}

Compute

Update the odds and then convert odds to a probability.

O_{\mathrm{post}}(R:S)=\frac14(6)=\frac32

\Pr(R\mid\text{both flags})=\frac{3}{3+2}=\frac35=0.6000

Answer

The posterior probability of configuration R is 0.6000.

\boxed{0.6000\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 Conditional Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: two diagnostic flags from prior odds

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability