Worked solution · Exam P

SOA Exam P Sample Question #537 Solution

This Exam P sample reference tests Conditional Probability. First combine the event probability and its conditional response probability to get a one-year response chance of 0.0875. Independence across three years makes the at-least-once probability 1-(1-0.0875)³=0.240201, so choice C 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 Conditional Probability question

Setup

Convert the supplied no-event probability to the probability of the triggering event.

h=1-0.75=0.25

Model

A response in one year requires both the trigger and the conditional response. The three yearly response indicators are independent.

q=0.25(0.35)=0.0875

\Pr(N\ge 1)=1-\Pr(N=0)=1-(1-q)^3

Compute

Evaluate the complement of no responses in all three years.

\Pr(N\ge 1)=1-(0.9125)^3

\Pr(N\ge 1)=0.240201171875

Answer

The probability rounds to 0.240.

\boxed{0.240\quad\text{(C)}}

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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 single-year joint probability 0.25(0.35)=0.0875 and never accounts for three opportunities.
BThis computes exactly one response, 3(0.0875)(0.9125)²=0.218572, rather than at least one response.
DThis is 0.75³=0.421875, the probability that the triggering event is absent in all three years.
EThis is 1-0.75³=0.578125, the probability of at least one trigger; it omits the conditional response requirement.

Original practice · fully worked

Original variant: quality-audit escalation rate

A plant runs four independent quality shifts. On each shift, an audit is selected with probability 0.40. If selected, the audit logs an escalation with an unknown probability r. The probability of at least one logged escalation over the four shifts is 0.15065344. Calculate r.

A
0.040
B
0.100
C
0.151
D
0.377
E
0.960

Answer and calculation

The probability of no escalation on every shift is 0.84934656=0.96⁴, so the one-shift escalation probability is 0.04. Since that joint probability is 0.40r, r=0.10 and choice B is correct.

Setup

Take the complement to obtain the probability that all four shifts finish without an escalation.

\Pr(\text{no escalation in four shifts})=1-0.15065344=0.84934656

Model

If q is the one-shift escalation probability, independence gives a fourth power for four escalation-free shifts.

(1-q)^4=0.84934656

Compute

Recover the one-shift joint probability, then divide by the audit-selection probability.

1-q=(0.84934656)^{1/4}=0.96

q=0.04=0.40r

r=0.10

Answer

A selected audit escalates with probability 0.100.

\boxed{0.100\quad\text{(B)}}

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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: quality-audit escalation rate

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability