Worked solution · Exam P

SOA Exam P Sample Question #564 Solution

This Exam P sample reference tests Exponential Distribution. Conditioning on the call class gives interval probabilities 0.2452815 and 0.1328565. Their probability-weighted average is 0.1778265, so choice A is correct.

Independent analysis by ActuaryProof · reviewed July 30, 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 Exponential Distribution question

Setup

For an exponential variable with mean m, subtract the survival probabilities at the two endpoints.

\Pr(1<X<3\mid m)=e^{-1/m}-e^{-3/m}

Model

Evaluate the interval probability separately for the two possible call classes.

p_D=e^{-1/0.75}-e^{-3/0.75}=0.2452814992

p_I=e^{-1/0.50}-e^{-3/0.50}=0.1328565311

Compute

Average the conditional results using the class probabilities.

\Pr(1<X<3)=0.40p_D+0.60p_I

\Pr(1<X<3)=0.1778265183

Answer

The requested probability rounds to 0.178.

\boxed{0.178\quad\text{(A)}}

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.

BThis reverses the two mixture weights: 0.60(0.2452815)+0.40(0.1328565)=0.2003115.
CThis incorrectly applies the 0.60 weight to both conditional probabilities, producing 0.2268828, whose nearest listed value is 0.222.
DThis uses only the first class's interval probability, 0.2452815, instead of forming the mixture; 0.250 is the nearest listed choice.
EThis adds the two conditional probabilities without weights: 0.2452815+0.1328565=0.3781380.

Original practice · fully worked

Original variant: infer a routing fraction

A computing service routes jobs to either a fast processor, with exponential completion time of mean 0.40 minute, or a slow processor, with exponential completion time of mean 1.00 minute. The routing fraction is unknown. Across all jobs, the probability of taking longer than 0.80 minute is 0.229533. Calculate the fraction routed to the slow processor.

A
0.094
B
0.230
C
0.300
D
0.511
E
0.731

Answer and calculation

The fast and slow tail probabilities are exp(−2) and exp(−0.8). Solving 0.229533=(1-q) exp(−2)+qe⁽⁻⁰·⁸⁾ gives q=0.300, so choice C is correct.

Setup

Let q be the unknown fraction assigned to the slow processor.

q=\Pr(\text{slow route})

Model

Write the observed upper tail as a mixture of the two processor-specific tails.

s_F=e^{-0.80/0.40}=e^{-2}=0.1353352832

s_S=e^{-0.80/1.00}=e^{-0.8}=0.4493289641

0.2295333875=(1-q)s_F+qs_S

Compute

Isolate the unknown routing fraction.

q=\frac{0.2295333875-s_F}{s_S-s_F}

q=0.3000000000

Answer

Thirty percent of jobs use the slow processor.

\boxed{0.300\quad\text{(C)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

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How to solve this Exponential Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: infer a routing fraction

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables