Verified solution · Exam P
This Exam P sample reference tests Sums of Independent Random Variables. A total below two consists of no losses, one hail loss, or one theft loss. The three independent-event probabilities sum to 0.1726357, so choice A is correct.
Independent solution
Setup
Let H and T be the ten-year counts for the two independent loss types.
Model
The total is below two in exactly three disjoint count combinations.
Compute
Use independence to multiply the component probabilities within each combination.
Answer
The probability rounds to 0.17.
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Original practice · fully worked
Two alarm channels are observed for six shifts. Channel A triggers independently on each shift with probability 0.10. Channel B is independent of A and across shifts, but its trigger probability q is unknown. The probability that neither channel triggers during all six shifts is 0.0945851. Calculate the expected total number of triggers from both channels over the six shifts.
The no-trigger probability is (0.9)⁶(1-q)⁶=0.0945851, so 0.9(1-q)=0.675 and q=0.25. Linearity gives expected total 6(0.10+0.25)=2.10, selecting E.
Setup
Write the six-shift probability of no trigger from either independent channel.
Model
Take the sixth root to recover the per-shift no-trigger factor.
Compute
Solve for q and add the expected trigger counts from the two channels.
Answer
The two channels produce 2.10 triggers on average.
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ActuaryProof. “SOA Exam P Sample Question #569 — Verified Solution.” Verified 2026-07-30.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.
