Worked solution · Exam P

SOA Exam P Sample Question #354 Solution

This Exam P sample reference tests Conditional Probability. This problem recovers a group proportion from a joint probability and a within-group conditional probability. Dividing 0.30 by 0.50 gives a male proportion of 0.60, so the complementary female proportion is 0.40 and choice B.

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 Conditional Probability question

Setup

Relate the supplied intersection to the conditional accident rate within the male group.

\Pr(M\cap A)=\Pr(A\mid M)\Pr(M)

Model

Solve the product identity for the male marginal probability.

\Pr(M)=\frac{\Pr(M\cap A)}{\Pr(A\mid M)}=\frac{0.30}{0.50}=0.60

Compute

The two driver categories are complementary.

\Pr(F)=1-\Pr(M)=1-0.60=0.40

Answer

The female proportion is 0.40.

\boxed{0.40\quad\text{(B)}}

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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.

AMultiplying 0.30 by 0.50 gives 0.15. The intersection is already supplied; the conditional relation must be inverted to recover the group proportion.
CThe value 0.50 is the accident probability within the male group, not the female marginal probability.
DThe value 0.60 is the recovered male proportion. The question asks for its complement.
EThe value 0.85 is the complement of the erroneous product 0.15, carrying the initial multiplication error into the final step.

Original practice · fully worked

Original variant: encrypted records without signatures

Seventy-five percent of archived records are encrypted. Among encrypted records, 40% carry a digital signature. Calculate the probability that a randomly selected record is encrypted but unsigned.

A
0.250
B
0.300
C
0.400
D
0.450
E
0.600

Answer and calculation

An encrypted record is unsigned with conditional probability 0.60. Multiplying by the encrypted share 0.75 gives 0.45 and choice D.

Setup

Complement the signature rate inside the encrypted group.

\Pr(S^c\mid E)=1-0.40=0.60

Model

Convert the within-group rate to a joint probability.

\Pr(E\cap S^c)=\Pr(E)\Pr(S^c\mid E)

Compute

Multiply the group share by its unsigned fraction.

\Pr(E\cap S^c)=0.75(0.60)=0.45

Answer

Forty-five percent of all records are both encrypted and unsigned.

\boxed{0.450\quad\text{(D)}}

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How to solve this Conditional Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: encrypted records without signatures

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability