Worked solution · Exam P

SOA Exam P Sample Question #304 Solution

This Exam P sample reference tests Sampling Without Replacement. This is a without-replacement complement calculation. The probability that both selected units come from the five non-target units is (5/9)(4/8), so the probability of selecting at least one of the four target units is 1-5/18=13/18, approximately 0.722 and choice E.

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 Sampling Without Replacement question

Setup

Separate the nine units into four target units and five non-target units.

N=9,\qquad K=4,\qquad N-K=5

Model

The complement of selecting at least one target unit is selecting two non-target units.

\Pr(X\ge1)=1-\Pr(X=0)

Compute

Because sampling is without replacement, the second non-target probability changes after the first selection.

\Pr(X=0)=\frac59\frac48=\frac5{18}

\Pr(X\ge1)=1-\frac5{18}=\frac{13}{18}=0.722222\ldots

Answer

The probability rounds to 0.72.

\boxed{0.72\quad\text{(E)}}

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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 0.28 is the complementary probability 5/18 that neither selected unit is a target.
BThe value 0.31 is approximately (5/9)². It treats the two selections as if the first unit were replaced.
CThe value 0.56 is approximately 5/9, the probability that a single selection is non-target rather than the requested two-selection event.
DThe value 0.69 is approximately 1-(5/9)². It uses the correct complement idea but incorrectly assumes sampling with replacement.

Original practice · fully worked

Original variant: expected regions in a document sample

An archive has ten sealed reports: four concern the north region, three concern the south region, and three concern the central region. Three reports are selected uniformly without replacement. Calculate the expected number of different regions represented in the selection.

A
0.75
B
1.95
C
2.00
D
2.25
E
2.30

Answer and calculation

Use one indicator for whether each region appears. The north is absent in C(6,3)/C(10,3) of selections, while each three-report region is absent in C(7,3)/C(10,3). Adding the three appearance probabilities gives 2.25, choice D.

Setup

Let I_N, I_S, and I_C indicate whether the north, south, and central regions appear, respectively.

R=I_N+I_S+I_C,\qquad \binom{10}{3}=120

Model

A region is absent exactly when all three reports are drawn from the other regions.

\Pr(I_N=0)=\frac{\binom63}{\binom{10}3}

\Pr(I_S=0)=\Pr(I_C=0)=\frac{\binom73}{\binom{10}3}

Compute

Apply linearity of expectation to the three appearance indicators.

\operatorname{E}[R]=3-\frac{\binom63+2\binom73}{\binom{10}3}

=3-\frac{20+2(35)}{120}=\frac94=2.25

Answer

The selection represents 2.25 regions on average.

\boxed{2.25\quad\text{(D)}}

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How to solve this Sampling Without Replacement question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: expected regions in a document sample

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability