Worked solution · Exam P

SOA Exam P Sample Question #419 Solution

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 analysis by ActuaryProof · reviewed July 31, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Uniform Distribution question

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.

\{X>4.5\}\subset\{X>2\}

\Pr(X>4.5\mid X>2)=\frac{\Pr(X>4.5)}{\Pr(X>2)}

Model

Uniform probabilities are proportional to the lengths of the corresponding subintervals.

\Pr(X>4.5)=\frac{5-4.5}{5-0}

\Pr(X>2)=\frac{5-2}{5-0}

Compute

The common support length cancels in the conditional ratio.

\Pr(X>4.5\mid X>2)=\frac{(5-4.5)/5}{(5-2)/5}

=\frac{0.5}{3}=\frac16=0.1666666667

Answer

The requested conditional probability rounds to 0.17.

\boxed{0.17\quad\text{(B)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

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Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThe value 0.10 is the unconditional tail probability (5-4.5)/5. It uses the full support rather than the shorter interval remaining after conditioning.
CThe ratio 2/4.5=0.444 compares the lower portions of two intervals and corresponds to the wrong-direction event P(X≤2 | X≤4.5), not the requested survival conditional.
DThe value 0.50 is only the raw length of the favorable interval from 4.5 to 5. A length is not yet a conditional probability until it is divided by the conditioned length 3.
EThe value 0.60 is P(X>2)=3/5, the probability of the conditioning event itself rather than the fraction of that event lying above 4.5.

Original practice · fully worked

Original variant: condition within a circular landing zone

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.

A
0.125
B
0.250
C
0.500
D
0.750
E
0.875

Answer and calculation

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.

x=r\cos\theta,\qquad y=r\sin\theta

Model

A positive horizontal coordinate selects the right semicircle. Within it, y>x selects a smaller wedge in the upper-right quadrant.

x>0\Longleftrightarrow-\frac{\pi}{2}<\theta<\frac{\pi}{2}

y>x\Longleftrightarrow\frac{\pi}{4}<\theta<\frac{\pi}{2}\quad\text{within the right semicircle}

Compute

Divide the favorable angular width by the conditioned angular width.

\Pr(y>x\mid x>0)=\frac{\pi/2-\pi/4}{\pi/2-(-\pi/2)}

=\frac{\pi/4}{\pi}=\frac14=0.25

Answer

One quarter of the retained semicircle lies above the line y=x.

\boxed{0.250\quad\text{(B)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

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How to solve this Uniform Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: condition within a circular landing zone

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables