Verified solution · Exam P
This Exam P sample reference tests Independence. One loss remains below its deductible with probability 0.30. Independence makes the probability of no benefit from all three policies equal to the cube of 0.30, so the probability of at least one benefit is 0.973 and choice E.
Independent solution
Setup
For one policy, identify the event that the loss does not cross the deductible.
Model
The three losses are independent, so multiply the three no-benefit probabilities.
Compute
Receiving a benefit from any policy is the complement of receiving none.
Answer
The policyholder receives at least one benefit with probability 0.973.
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Original practice · fully worked
A monitoring day is severe with probability 0.20 and ordinary otherwise. Conditional on the day's state, three pressure sensors act independently. Each sensor crosses its alert threshold with probability 0.40 on a severe day and 0.10 on an ordinary day. Calculate the probability that at least one sensor crosses its threshold.
The at-least-one probabilities are 0.784 on a severe day and 0.271 on an ordinary day. Weighting them by the weather-state probabilities gives 0.3736, which rounds to 0.374 and choice B.
Setup
Use complements within each weather state.
Model
Average the two conditional probabilities over the shared daily state.
Compute
Substitute the severe-day and ordinary-day weights.
Answer
At least one sensor crosses its threshold with probability approximately 0.374.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 192 Exam P general probability practice problems.
Drill the method: Exam P Independence problems · Exam P Uniform Distribution problems
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ActuaryProof. “SOA Exam P Sample Question #348 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
