Worked solution · Exam P

SOA Exam P Sample Question #619 Solution

This Exam P sample reference tests Poisson Distribution. Taking the ratio of the Poisson probabilities cancels their common exponential factor and gives λ⁴⁄⁴!=54. Thus λ=6 per minute, and linearity of expectation gives 60(6)=360 over one hour, selecting choice E.

Independent analysis by ActuaryProof · reviewed July 30, 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 Poisson Distribution question

Setup

Let the common mean count in one minute be the positive parameter λ.

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

Model

Form the stated probability ratio. The exponential terms cancel.

\frac{\Pr(N=4)}{\Pr(N=0)}=\frac{\lambda^4}{4!}=54

\lambda^4=54(24)=1296

Compute

Use the positive fourth root for a Poisson mean, then scale the expectation to sixty minutes.

\lambda=1296^{1/4}=6

\operatorname{E}[N_{60}]=60\lambda=360

Answer

The expected count in the requested period is 360.

\boxed{360\quad\text{(E)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate random variables 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.

AThis corresponds to guessing a rate of 2 per minute instead of solving the fourth-power probability-ratio equation.
BOmitting the factor 4! gives the incorrect equation λ⁴=54; rounding its fourth root to 3 produces 180 over sixty minutes.
CUsing the observed count label 4 as the mean rather than solving for λ gives 60(4)=240.
DThis corresponds to using 5 as the fourth root of 1296; checking powers shows that 5⁴ is only 625 while 6⁴ is 1296.

Original practice · fully worked

Original variant: bright-meteor thinning

Meteors reach an automated detector according to a Poisson process with mean rate 12 per hour. Each detected meteor is independently classified as bright with probability one-third and dim otherwise. During a thirty-minute interval, calculate the probability of exactly two bright meteors and at least one dim meteor.

A
0.045
B
0.266
C
0.271
D
0.982
E
2.000

Answer and calculation

Poisson thinning produces independent bright and dim counts with means 2 and 4. Multiplying the probability of two bright meteors by the probability of at least one dim meteor gives 0.265713, so choice B is correct.

Setup

Convert the hourly rate to the mean count over one half-hour.

N\sim\operatorname{Poisson}\!\left(12\cdot\frac12\right)=\operatorname{Poisson}(6)

Model

Thin the process by classification. The bright and dim counts are independent Poisson variables with their respective split means.

B\sim\operatorname{Poisson}\!\left(6\cdot\frac13\right)=\operatorname{Poisson}(2)

D\sim\operatorname{Poisson}\!\left(6\cdot\frac23\right)=\operatorname{Poisson}(4)

B\ \text{and}\ D\ \text{are independent}

Compute

Multiply the exact bright-count probability by the dim-count complement.

\Pr(B=2,D\ge1)=\left(e^{-2}\frac{2^2}{2!}\right)\left(1-e^{-4}\right)

\Pr(B=2,D\ge1)=0.2657130621\ldots

Answer

The requested joint probability rounds to 0.266.

\boxed{0.266\quad\text{(B)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

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 Univariate Random Variables practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Poisson Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: bright-meteor thinning

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables