Worked solution · Exam P

SOA Exam P Sample Question #133 Solution

This Exam P sample reference tests Bayes' Theorem. A red insured contributes qualifying claims at rate 0.10(0.90)=0.09 and a green insured at rate 0.05(0.80)=0.04. Weighting by 300 and 700 insureds gives 27 red and 28 green qualifying claims, so the red share is 27/55=0.491, choice C.

Independent analysis by ActuaryProof · reviewed October 6, 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

Combine each color group's accident probability with its conditional probability of crossing the claim threshold.

q_R=0.10(0.90)=0.09

q_G=0.05(0.80)=0.04

Model

Weight these qualifying-claim rates by the numbers insured in the two groups.

w_R=300(0.09)=27,\qquad w_G=700(0.04)=28

Compute

Condition on a claim being in the qualifying pool.

\Pr(R\mid Q)=\frac{w_R}{w_R+w_G}

\Pr(R\mid Q)=\frac{27}{27+28}=0.4909091

Answer

The qualifying claim is from a red car with probability about 0.491.

\boxed{0.491\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.

AThe value 0.300 is the red share of all insured cars. Red and green cars have different qualifying-claim rates, so the prior share is not the posterior share.
BThe value 0.462 results from conditioning on accidents alone: 0.3(0.10)/[0.3(0.10)+0.7(0.05)]. It omits the deductible-crossing probabilities.
DThe value 0.667 is 0.10/(0.10+0.05), which compares accident rates without weighting by group sizes or later claim qualification.
EThe value 0.692 is 0.09/(0.09+0.04), which compares per-car qualifying rates but fails to weight them by the 300-to-700 portfolio mix.

Original practice · fully worked

Original variant: production line given either alarm

Twenty percent of components come from a high-heat line and the rest from a low-heat line. Conditional on the line, pressure and vibration alarms are independent. Their trigger probabilities are 0.50 and 0.40 on the high-heat line, and 0.10 and 0.20 on the low-heat line. A component triggered at least one alarm. Determine the probability it came from the high-heat line.

A
0.2000
B
0.2800
C
0.3500
D
0.3846
E
0.7000

Answer and calculation

At least one alarm occurs with probability 0.70 on the high-heat line and 0.28 on the low-heat line. Bayes' rule gives 0.20(0.70)/[0.20(0.70)+0.80(0.28)]=0.3846, choice D.

Setup

Let H denote the high-heat line and A denote at least one alarm.

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

Model

Within each line, use conditional independence to complement the probability that neither alarm triggers.

\Pr(A\mid H)=1-(0.50)(0.60)=0.70

\Pr(A\mid H^c)=1-(0.90)(0.80)=0.28

Compute

Weight the alarm probabilities by the line shares and normalize.

\Pr(H\mid A)=\frac{0.20(0.70)}{0.20(0.70)+0.80(0.28)}

\Pr(H\mid A)=\frac{0.14}{0.364}=0.3846154

Answer

The posterior high-heat probability is about 0.3846.

\boxed{0.3846\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: production line given either alarm

Answer and calculation

Setup

Model

Compute

Answer

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