Verified solution · Exam P
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 solution
Setup
Combine each color group's accident probability with its conditional probability of crossing the claim threshold.
Model
Weight these qualifying-claim rates by the numbers insured in the two groups.
Compute
Condition on a claim being in the qualifying pool.
Answer
The qualifying claim is from a red car with probability about 0.491.
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Original practice · fully worked
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.
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.
Model
Within each line, use conditional independence to complement the probability that neither alarm triggers.
Compute
Weight the alarm probabilities by the line shares and normalize.
Answer
The posterior high-heat probability is about 0.3846.
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ActuaryProof. “SOA Exam P Sample Question #133 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
