Verified solution · Exam P
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 solution
Setup
Convert the supplied no-event probability to the probability of the triggering event.
Model
A response in one year requires both the trigger and the conditional response. The three yearly response indicators are independent.
Compute
Evaluate the complement of no responses in all three years.
Answer
The probability rounds to 0.240.
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Original practice · fully worked
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.
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.
Model
If q is the one-shift escalation probability, independence gives a fourth power for four escalation-free shifts.
Compute
Recover the one-shift joint probability, then divide by the audit-selection probability.
Answer
A selected audit escalates with probability 0.100.
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ActuaryProof. “SOA Exam P Sample Question #537 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
