Worked solution · Exam P

SOA Exam P Sample Question #98 Solution

This Exam P sample reference tests Expected Value. A high-risk driver has expected annual bonus 48 and a low-risk driver 54. Multiplying by the cohort sizes gives 50,400, choice B.

Independent analysis by ActuaryProof · reviewed October 6, 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 Expected Value question

Setup

A high-risk driver has expected annual bonus 12×5×0.80=48, while a low-risk driver has expected annual bonus 54.

E[B_H]=12(5)(0.80)=48

E[B_L]=12(5)(0.90)=54

Model

Weight the two annual expectations by their cohort sizes, 600 and 400, and add them.

E[T]=600E[B_H]+400E[B_L]

Compute

The high-risk cohort contributes 28,800 and the low-risk cohort contributes 21,600, totaling 50,400.

E[T]=600(48)+400(54)=28800+21600=50400

Answer

Expected total annual bonus is 50,400, corresponding to choice B.

\boxed{50{,}400\quad\text{(B)}}

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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 48,000 applies the high-risk annual expectation 48 to all 1000 drivers.
CThe value 51,000 applies the unweighted average of annual expectations 48 and 54 to all 1000 drivers, ignoring cohort sizes.
DThe value 54,000 applies the low-risk annual expectation 54 to all 1000 drivers.
EThe value 60,000 pays 5 every month to all 1000 drivers regardless of accident status.

Original practice · fully worked

Original variant: expected annual credits across service tiers

A platform has 800 basic users and 200 premium users. Each month a basic user earns a 3-dollar reliability credit with probability 0.70, while a premium user earns a 5-dollar credit with probability 0.85. Find the expected total credits over 12 months.

A
22,176 dollars
B
24,840 dollars
D
30,360 dollars
C
32,400 dollars
E
36,000 dollars

Answer and calculation

Monthly expected credits are 800(3)(0.70)+200(5)(0.85)=2,530. Multiplying by 12 gives 30,360 dollars.

Setup

The monthly expected credit is 2.10 dollars per basic user and 4.25 dollars per premium user.

E[C_B]=3(0.70)=2.10

E[C_P]=5(0.85)=4.25

Model

Multiply each monthly expectation by its tier population, add the tiers, and then multiply by 12 months.

E[T]=12\{800(2.10)+200(4.25)\}

Compute

Basic users contribute 1680 dollars per month and premium users 850, for annual total 12×2530=30,360.

E[T]=12(1680+850)=30360

Answer

Expected total annual credits are 30,360 dollars, corresponding to choice D.

\boxed{30{,}360\ \text{dollars}\quad\text{(D)}}

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How to solve this Expected Value question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: expected annual credits across service tiers

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability