Verified solution · Exam P

SOA Exam P Sample Question #308 Solution

This Exam P sample reference tests Variance. The payment occurs with probability 0.48-0.05=0.43, so it is 1000 times a Bernoulli variable with parameter 0.43. Its standard deviation is 1000 times the square root of 0.43(0.57), or 495.076, which gives choice D.

Independent analysis by ActuaryProof · reviewed August 4, 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 Variance question

Setup

Use the two cumulative probabilities to isolate the survival interval that triggers payment.

p=\Pr(1\le X<5)=\Pr(X<5)-\Pr(X<1)=0.48-0.05=0.43

Model

The contract payment has two possible values.

Y=1000I,\qquad I\sim\operatorname{Bernoulli}(0.43)

Compute

Scale the Bernoulli variance and take its square root.

\operatorname{Var}(Y)=1000^2(0.43)(0.57)=245100

\operatorname{SD}(Y)=\sqrt{245100}=495.0757518\ldots

Answer

The standard deviation rounds to 495.

\boxed{495\quad\text{(D)}}

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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 218 is approximately 1000sqrt(0.05(0.95)). It uses the probability of survival below one year as the payment probability instead of excluding that group.
BThe value 430 is E[Y]=1000(0.43), the expected payment rather than its standard deviation.
CThe value 480 is 1000P(X<5). It treats everyone below five years as payable and reports an expected amount, ignoring both the one-year exclusion and variance.
EThe value 500 is the largest possible standard deviation of a 0-or-1000 payment, attained only when its payment probability is one half. Here the probability is 0.43.

Original practice · fully worked

Original variant: dispersion of a three-level delivery credit

A delivery contract pays a credit of 0 dollars for an early arrival, 400 dollars for an arrival inside the target window, and 900 dollars for a late arrival. The three outcomes have probabilities 0.18, 0.47, and 0.35, respectively. Calculate the standard deviation of the credit.

A
315.0
B
325.1
C
503.0
D
598.9
E
105691

Variant answer in brief

The first two credit moments are 503 and 358700. Subtracting the squared mean gives variance 105691, whose square root is 325.102, so choice B.

Setup

Calculate the mean across the three mutually exclusive credit levels.

\operatorname{E}[C]=0(0.18)+400(0.47)+900(0.35)=503

Model

Use the second-moment identity for variance.

\operatorname{E}[C^2]=0^2(0.18)+400^2(0.47)+900^2(0.35)=358700

Compute

Subtract the squared mean and take the square root.

\operatorname{Var}(C)=358700-503^2=105691

\operatorname{SD}(C)=\sqrt{105691}=325.1015226\ldots

Answer

The credit standard deviation rounds to 325.1 dollars.

\boxed{325.1\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #308 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Variance question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: dispersion of a three-level delivery credit

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution