Verified solution · Exam P
This Exam P sample reference tests Geometric Distribution. The trial count through the first success is geometric with success probability 0.0625. Its probability mass decreases by the factor 0.9375 at each successive trial, so the mode is 1 and choice A is correct.
Independent solution
Setup
Let T count the number of independent trials up to and including the first success.
Model
The geometric mass at trial n consists of n-1 failures followed by one success.
Compute
Compare each probability with the one immediately before it.
Answer
The mass function is strictly decreasing, so its largest value occurs at the first trial.
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Error diagnosis
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Original practice · fully worked
A decoder makes independent attempts until its first successful read. Let F be the number of failed attempts before the success, and suppose Pr(F at least 4)=0.2401. Let T=F+1 be the total number of attempts. Calculate the 75th percentile of T, defined as the smallest integer n for which Pr(T at most n) is at least 0.75.
Four initial failures have probability (1-p)⁴=0.2401=0.7⁴, so p=0.3. The geometric CDF is 1-0.7ⁿ; it is 0.657 at n=3 and 0.7599 at n=4, making 4 the 75th percentile and choice D correct.
Setup
Use the supplied tail event to recover the failure probability on one attempt.
Model
For total attempts through the first success, the cumulative probability through attempt n is one minus the probability of n consecutive failures.
Compute
Check the two integers around the target cumulative probability.
Answer
The cumulative probability first reaches 0.75 at four attempts.
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ActuaryProof. “SOA Exam P Sample Question #538 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
