Verified solution · Exam P

SOA Exam P Sample Question #90 Solution

This Exam P sample reference tests Hypergeometric Distribution. The inventory contains 10 defective and 70 sound modems. A hypergeometric calculation for two defectives in five gives about 0.102, choice C.

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 Hypergeometric Distribution question

Setup

Combine the two inventory groups to obtain 10 defective and 70 sound modems among 80 total.

D=0.20(30)+0.08(50)=10

G=80-10=70

Model

For a sample of five without replacement, choose two items from the 10 defectives and three from the 70 sound items, then divide by all five-item samples.

P(K=2)=\frac{\binom{10}{2}\binom{70}{3}}{\binom{80}{5}}

Compute

The hypergeometric combination ratio evaluates to 0.10177.

P(K=2)=0.10177

Answer

The probability rounds to 0.102, corresponding to choice C.

\boxed{0.102\quad\text{(C)}}

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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.

DThe value 0.105 is the binomial-with-replacement approximation using defective rate 10/80. Sampling is without replacement.
EThe value 0.125 is the overall defective fraction 10/80, not the probability of exactly two defectives in five draws.

Original practice · fully worked

Original variant: sample rare batteries without replacement

A crate contains 12 high-capacity and 48 standard batteries. Six batteries are sampled without replacement. Find the probability exactly two sampled batteries are high-capacity.

C
0.1974
B
0.2308
A
0.2565
D
0.3265
E
0.4096

Variant answer in brief

The hypergeometric ratio C(12,2)C(48,4)/C(60,6) gives the required probability.

Setup

The crate contains 12 high-capacity and 48 standard batteries, and six are selected without replacement.

N=60,\quad K=12,\quad n=6

Model

Count samples with exactly two high-capacity batteries by choosing two of 12 and four of 48, then divide by all six-battery samples.

P(X=2)=\frac{\binom{12}{2}\binom{48}{4}}{\binom{60}{6}}

Compute

The resulting hypergeometric combination ratio is 0.256518.

P(X=2)=0.256518

Answer

The exact-two probability is 0.2565, corresponding to choice A.

\boxed{0.2565\quad\text{(A)}}

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ActuaryProof. “SOA Exam P Sample Question #90 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Hypergeometric Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: sample rare batteries without replacement

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution