Worked solution · Exam P

SOA Exam P Sample Question #405 Solution

This Exam P sample reference tests Expected Value. There are 142,506 possible five-number sets and five possible marked numbers within each set. Applying those two match probabilities to the base and extra awards gives an expected monthly payout of 70,172.48, so choice A.

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.

Read the solution Open official source

Independent solution

How to solve this Expected Value question

Setup

Count the possible five-number sets, before considering the distinguished position.

M=\binom{30}{5}=142{,}506

Model

A fixed entry receives the base award when its number set matches. The extra award additionally requires its distinguished number to match.

\Pr(\text{base match})=\frac1M

\Pr(\text{extra match})=\frac1{5M}

Compute

Use linearity for the two award components and then for all entries.

E[W_{\mathrm{entry}}]=\frac{50{,}000}{M}+\frac{250{,}000}{5M}

E[W_{\mathrm{month}}]=100{,}000\left(\frac{100{,}000}{142{,}506}\right)

E[W_{\mathrm{month}}]=70{,}172.48397\ldots

Answer

The expected monthly payout rounds to 70,172.

\boxed{70{,}172\quad\text{(A)}}

Test the method on unseen problems

General Probability Practice Workbook

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

See the cover and sample pages

Error diagnosis

Why the other letters tempt

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

BThis results from treating the full 300,000 award as an additional bonus on top of the 50,000 base. That gives effective conditional award 110,000 per number-set match and expected payout 77,190.
CA total of 84,731 implies per-entry expectation 0.84731 and, after the base component is removed, an implied bonus-match chance about 0.283 rather than exactly 1/5.
DThe value 100,000 repeats the number of entries as though each entry had expected payout 1; the verified per-entry expectation is about 0.701725.
EThis awards the extra amount when the distinguished number fails to match, using probability 4/5 instead of 1/5. The resulting expected total is 175,431.

Original practice · fully worked

Original variant: delivery-day net margin

The number N of deliveries completed by an autonomous cart in one day has probabilities P(N=0)=0.10, P(N=1)=0.30, P(N=2)=0.40, and P(N=4)=0.20. Each completed delivery produces an average net margin of 35 dollars, independently of N. Daily fixed overhead is 120 dollars. Calculate the expected net profit for the day.

A
-120.0
B
-85.0
C
-53.5
D
66.5
E
70.0

Answer and calculation

The expected delivery count is 1.9, so expected delivery margin is 66.5 dollars. Subtracting the fixed overhead gives expected net profit -53.5 dollars and choice C.

Setup

Compute the expected delivery count from its four-point distribution.

E[N]=0(0.10)+1(0.30)+2(0.40)+4(0.20)=1.9

Model

Conditional on the count, expected delivery margin is 35N. Subtract the fixed overhead once per day.

E[\text{net profit}\mid N]=35N-120

Compute

Apply iterated expectation to the random count.

E[\text{net profit}]=35E[N]-120

E[\text{net profit}]=35(1.9)-120=-53.5

Answer

The expected daily net profit is -53.5 dollars.

\boxed{-53.5\quad\text{(C)}}

Keep practicing

Next steps for Exam P General Probability

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P General Probability practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Expected Value question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: delivery-day net margin

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability