Worked solution · Exam P

SOA Exam P Sample Question #350 Solution

This Exam P sample reference tests Exponential Distribution. The capped payment equals its maximum whenever the underlying loss reaches or exceeds 3000. An exponential loss with mean 2000 has survival probability e raised to minus 1.5 at that threshold, approximately 0.223 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 Exponential Distribution question

Setup

Translate an exact maximum payment into a tail event for the uncapped loss.

Y=\min(X,3000),\qquad \{Y=3000\}=\{X\ge3000\}

Model

Use the exponential survival function with mean 2000.

S_X(x)=e^{-x/2000}

Compute

Evaluate the survival probability at the policy limit.

\Pr(Y=3000)=e^{-3000/2000}=e^{-1.5}

=0.2231301601\ldots

Answer

The maximum payment occurs with probability approximately 0.223.

\boxed{0.223\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.

AThis uses the fact that a continuous loss equals exactly 3000 with probability zero but overlooks the probability mass created at the cap by every larger loss.
CThis is approximately 0.487, the complement obtained after reversing the mean and threshold in the exponential exponent.
DThis reverses the mean and threshold in the exponential exponent, producing a survival value of approximately 0.513.
EThis is the probability that the loss remains below 3000, so the payment is less than its maximum.

Original practice · fully worked

Original variant: capped total from two gauges

Two independent runoff gauges each report an amount uniformly distributed from 0 to 10 units. A reimbursement equals their combined reading, but is capped at 12 units. Calculate the probability that the reimbursement equals exactly 12 units.

A
0.000
B
0.080
C
0.320
D
0.500
E
0.680

Answer and calculation

The cap is reached when the two readings sum to at least 12. In the 10-by-10 sample square, that event is an upper triangle with legs of length 8 and area 32, giving probability 0.32 and choice C.

Setup

Translate the exact capped value into an event for the uncapped sum.

R=\min(U+V,12),\qquad \{R=12\}=\{U+V\ge12\}

Model

The independent readings are uniform over a square of area 100.

0\le U\le10,\qquad 0\le V\le10

Compute

The favorable upper triangle has legs from 2 to 10 on each axis.

\Pr(U+V\ge12)=\frac{\tfrac12(8)(8)}{100}=0.32

Answer

The reimbursement reaches its cap with probability 0.320.

\boxed{0.320\quad\text{(C)}}

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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: capped total from two gauges

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables