Verified solution · Exam P
This Exam P sample reference tests Law of Total Probability. The high-risk share is 0.20, and the three conditional death rates are in the ratio 6:3:1. Calibrating their weighted average to 0.009 gives high-risk rate 0.0200, choice B.
Independent solution
Setup
Let q be the low-risk death probability and express the other rates in terms of q.
Model
Apply total probability across the three mutually exclusive risk groups.
Compute
Collect the weighted multipliers and solve for the baseline rate.
Answer
The high-risk conditional death probability is 0.0200.
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Original practice · fully worked
At dawn a controller chooses one of three operating modes with probabilities 1/6, 1/3, and 1/2. Within a chosen mode, successive components pass independently, with respective pass probabilities 0.90, 0.75, and 0.60. The first component passed. Find the conditional probability that the next component passes.
The first-pass probability is 0.70, while the probability that both successive components pass is 0.5025. Their ratio is 0.71786, choice D.
Setup
Let S1 and S2 be the pass events for the first and second components. Average the first pass over the persistent mode.
Model
Conditional independence within a mode makes each two-pass contribution the mode weight times the squared pass rate.
Compute
Evaluate the joint probability and normalize by the observed first pass.
Answer
The first success shifts weight toward the more reliable modes, raising the next-pass probability to 0.71786.
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ActuaryProof. “SOA Exam P Sample Question #154 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
