Worked solution · Exam P

SOA Exam P Sample Question #334 Solution

This Exam P sample reference tests Bayes' Theorem. This is an inverse Bayes calculation for an unknown within-group rate. If x is the female smoking rate, then the female and male smoker contributions are 0.4x and 0.12; requiring women to be 20% of all smokers gives x=0.075, or 7.50%, and choice A.

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.

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Independent solution

How to solve this Bayes' Theorem question

Setup

Let x denote the unknown smoking probability within the female group and form the two smoker contributions.

\Pr(F\cap S)=0.40x

\Pr(M\cap S)=0.60(0.20)=0.12

Model

Express the supplied female share among smokers as a Bayes ratio.

0.20=\Pr(F\mid S)=\frac{0.40x}{0.40x+0.12}

Compute

Solve the resulting linear equation for the within-group rate.

0.20(0.40x+0.12)=0.40x

0.08x+0.024=0.40x

x=0.075

Answer

The smoking rate among female policyholders is 7.50%.

\boxed{7.50\%\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 8.00% is 0.40(0.20), multiplying the female population share by the posterior female share among smokers. Those probabilities have incompatible conditioning.
CThe value 12.00% is the joint male-smoker probability 0.60(0.20), not the smoking rate within the female group.
DThe value 13.33% divides the erroneous 8% product by the male share 60%, compounding the conditioning error.
EThe value 20.00% simply repeats either the male smoking rate or the female share among smokers. Neither equals the requested female smoking rate.

Original practice · fully worked

Original variant: job mode behind a sampled audit record

A workflow processes 40% of jobs in standard mode, 35% in advanced mode, and 25% in diagnostic mode. A standard job produces one audit record, an advanced job produces two, and a diagnostic job produces four. One audit record is selected uniformly from all records. Calculate the probability that its job used diagnostic mode.

A
0.250
B
0.400
C
0.476
D
0.571
E
1.000

Answer and calculation

Record sampling weights each job mode by its number of records. The standard, advanced, and diagnostic contributions are 0.40, 0.70, and 1.00 records per job, so diagnostic mode supplies 1/2.1=10/21≈0.476 of all records and choice C.

Setup

Weight each job-mode frequency by the number of records that one job in that mode creates.

w_S=0.40(1)=0.40

w_A=0.35(2)=0.70

w_D=0.25(4)=1.00

Model

A uniformly sampled record comes from a mode in proportion to that mode's record contribution.

\Pr(D\mid\text{sampled record})=\frac{w_D}{w_S+w_A+w_D}

Compute

Normalize the diagnostic contribution by the total number of records generated per job.

\frac{1.00}{0.40+0.70+1.00}=\frac1{2.10}=\frac{10}{21}=0.4761904762

Answer

The sampled record has diagnostic-mode origin with probability approximately 0.476.

\boxed{0.476\quad\text{(C)}}

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How to solve this Bayes' Theorem question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: job mode behind a sampled audit record

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability