Verified solution · Exam P
This Exam P sample reference tests Uniform Distribution. This problem reduces a conditional probability for a uniform lifetime to a ratio of interval lengths. The favorable tail has length 0.5 within a conditioned interval of length 3, so the probability is 1/6=0.166667 and choice B.
Independent solution
Setup
Let X have the stated uniform distribution. The later survival event is contained in the earlier survival event, so their intersection is simply the later event.
Model
Uniform probabilities are proportional to the lengths of the corresponding subintervals.
Compute
The common support length cancels in the conditional ratio.
Answer
The requested conditional probability rounds to 0.17.
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Original practice · fully worked
A robotic probe lands uniformly within a circular testing plate centered at the origin. A reading is retained only when the horizontal coordinate is positive. Given that a reading is retained, calculate the probability that its vertical coordinate exceeds its horizontal coordinate.
The positive-horizontal condition selects a semicircle with angular width pi. Requiring the vertical coordinate to exceed the horizontal one leaves the wedge from angle pi/4 to pi/2, with width pi/4, so the conditional probability is 1/4=0.250 and choice B.
Setup
Write a landing location in polar coordinates. Uniformity over the disk makes the probability of a full-radius sector proportional to its angle.
Model
A positive horizontal coordinate selects the right semicircle. Within it, y>x selects a smaller wedge in the upper-right quadrant.
Compute
Divide the favorable angular width by the conditioned angular width.
Answer
One quarter of the retained semicircle lies above the line y=x.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 388 Exam P univariate random variables practice problems.
Drill the method: Exam P Conditional Probability problems · Exam P Uniform Distribution problems
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ActuaryProof. “SOA Exam P Sample Question #419 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
