Verified solution · Exam P
This Exam P sample reference tests Discrete Random Variables. Weight each office's relative frequencies by its share of combined sales. The combined cumulative probability is 0.41 through 160 and 0.51 through 165, so the median is 165 and choice B is correct.
Independent solution
Setup
Treat the combined sales distribution as a mixture of the two office distributions.
Model
Compute the combined masses at prices no greater than 165.
Compute
Accumulate the mixture masses and locate the first support value whose CDF reaches one-half.
Answer
The smallest combined selling price with cumulative probability at least one-half is 165 thousand.
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Original practice · fully worked
A repair center combines tickets from two service desks. Desk A handles 60% of all tickets and has wait-time probabilities 0.10, 0.20, 0.30, 0.25, and 0.15 at 4, 8, 12, 16, and 20 minutes, respectively. Desk B handles the remaining 40% and has corresponding probabilities 0.20, 0.30, 0.25, 0.15, and 0.10. Calculate the median wait time for a randomly selected ticket.
The mixed CDF is 0.38 through 8 minutes and 0.66 through 12 minutes. Its first crossing of one-half is therefore 12 minutes, choice C.
Setup
Combine the desk distributions using their ticket shares.
Model
Calculate the first three mixture masses.
Compute
Compare the cumulative probabilities immediately below and at 12 minutes.
Answer
The median wait time is the first support value at which the CDF reaches one-half.
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ActuaryProof. “SOA Exam P Sample Question #715 — Verified Solution.” Verified 2026-07-30.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.
