Worked solution · Exam P

SOA Exam P Sample Question #472 Solution

This Exam P sample reference tests Binomial Distribution. For a binomial count with success probability 0.60, the ratio of the probability of three successes to that of two is (n-2)/2. Setting this ratio equal to five gives n=12, so choice D is correct.

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 Binomial Distribution question

Setup

Let W be the season success count. Independence and a constant success probability give a binomial model.

W\sim\operatorname{Binomial}(n,0.60)

Model

Form the ratio of the adjacent binomial probabilities so that the common powers and most factorials cancel.

\frac{\Pr(W=3)}{\Pr(W=2)}=\frac{\binom n3(0.60)^3(0.40)^{n-3}}{\binom n2(0.60)^2(0.40)^{n-2}}

=\frac{n-2}{3}\frac{0.60}{0.40}=\frac{n-2}{2}

Compute

Equate the simplified ratio to the stated multiple and solve for the integer number of trials.

\frac{n-2}{2}=5

n=12

Answer

The season contains 12 games.

\boxed{12\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.

AAt n=6 the adjacent-probability ratio is (6-2)/2=2, not the required five.
BAt n=8 the adjacent-probability ratio is (8-2)/2=3. This results from matching the ratio to three instead of the required five.
CAt n=10 the ratio is (10-2)/2=4, an off-by-one result from replacing the required multiplier five by four.
EAt n=14 the ratio is (14-2)/2=6, an off-by-one result from replacing the required multiplier five by six.

Original practice · fully worked

Original variant: variability from adjacent Poisson counts

The number N of photon artifacts found on a scan is Poisson distributed with an unknown mean. The probability of exactly five artifacts is three halves the probability of exactly four artifacts. Calculate the standard deviation of N.

A
1.500
B
2.449
C
2.739
D
5.000
E
7.500

Answer and calculation

For a Poisson variable, P(N=5)/P(N=4)=λ/5. The reported ratio gives λ=7.5, so SD(N)=√(7.5)=2.7386 and choice C.

Setup

Write the two adjacent Poisson masses using the unknown positive mean λ.

\Pr(N=k)=e^{-\lambda}\frac{\lambda^k}{k!}

Model

Divide the mass at five by the mass at four so that the common exponential factor and four powers cancel.

\frac{\Pr(N=5)}{\Pr(N=4)}=\frac{e^{-\lambda}\lambda^5/5!}{e^{-\lambda}\lambda^4/4!}=\frac{\lambda}{5}

Compute

Recover the Poisson mean from the ratio, then use equality of its mean and variance.

\frac{\lambda}{5}=\frac32\quad\Longrightarrow\quad\lambda=\frac{15}{2}=7.5

\operatorname{SD}(N)=\sqrt{\operatorname{Var}(N)}=\sqrt{\lambda}=\sqrt{7.5}=2.738612788

Answer

The artifact-count standard deviation is approximately 2.739.

\boxed{2.739\quad\text{(C)}}

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How to solve this Binomial Distribution question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: variability from adjacent Poisson counts

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability