Worked solution · Exam P

SOA Exam P Sample Question #540 Solution

This Exam P sample reference tests Bayes Theorem. Bayes' theorem weights each group's prior share by its conditional five-year rate. The target group contributes 0.4(0.2)=0.08 out of total weight 0.08+0.6(0.4)=0.32, so the posterior is 0.25 and choice B is correct.

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

Form the joint weights for the two possible groups and the observed five-year event.

w_F=0.40(0.20)=0.08

w_M=0.60(0.40)=0.24

Model

Bayes' theorem normalizes the target group's joint weight by the sum of both compatible joint weights.

\Pr(F\mid E)=\frac{w_F}{w_F+w_M}

Compute

Substitute and simplify the posterior probability.

\Pr(F\mid E)=\frac{0.08}{0.08+0.24}=\frac14=0.25

Answer

The conditional probability is one fourth.

\boxed{\frac14\quad\text{(B)}}

Test the method on unseen problems

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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.

AThis reports the target group's conditional event rate 0.20 and does not update with the other group or either prior.
CThis normalizes only the two conditional rates, 0.20/(0.20+0.40)=1/3, thereby discarding the prior group shares.
DThis assigns equal posterior weight to both groups despite their different prior shares and conditional rates.
EThis is the complementary posterior 0.24/0.32=0.75 for the other group.

Original practice · fully worked

Original variant: infer a supplier defect rate

A factory receives 70% of its components from supplier A and 30% from supplier B. The defect rate is 0.02 for A and an unknown value q for B. Among defective components, 60% came from B. Calculate q.

A
0.014
B
0.021
C
0.030
D
0.070
E
0.600

Answer and calculation

The joint defective weights are 0.70(0.02)=0.014 for A and 0.30q for B. Setting 0.30q/(0.014+0.30q)=0.60 gives q=0.07, so choice D is correct.

Setup

Write the joint probability of a defect from each supplier.

\Pr(A\cap D)=0.70(0.02)=0.014

\Pr(B\cap D)=0.30q

Model

Express the supplied posterior share of B among all defective components.

\frac{0.30q}{0.014+0.30q}=0.60

Compute

Solve the resulting linear equation for the unknown conditional defect rate.

0.30q=0.0084+0.18q

0.12q=0.0084

q=0.07

Answer

Supplier B's defect rate is 0.070.

\boxed{0.070\quad\text{(D)}}

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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: infer a supplier defect rate

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability