Exam P Poisson Distribution Practice Problems

Try this original problem before opening the solution. Then compare the method with SOA Exam P sample question 736.

Original practice · fully worked

Original variant: observatory alert count

An observatory models the number of false instrument alerts during a 12-hour shift with a Poisson random variable N. The probability of no alerts during a shift is exp(−1.6). Calculate the probability that at most two alerts occur during a shift.

A
0.202
B
0.323
C
0.525
D
0.783
E
0.921

Show the answer and worked solution

Answer and calculation

The zero-count probability identifies the rate as 1.6. Summing the masses for zero, one, and two gives approximately 0.783.

Setup

For a Poisson count, the zero-count probability is the exponential of the negative rate.

\Pr(N=0)=e^{-\lambda}

Model

Match the supplied zero-count probability to the Poisson formula.

e^{-\lambda}=e^{-1.6}\quad\Longrightarrow\quad\lambda=1.6

Compute

Add the probabilities of zero, one, and two alerts.

\Pr(N\le 2)=e^{-1.6}\left(1+1.6+\frac{1.6^2}{2}\right)

\Pr(N\le 2)=3.88e^{-1.6}=0.783358\ldots

Answer

The closest listed value is choice D.

\boxed{\Pr(N\le 2)\approx 0.783\quad\text{(D)}}

Technique orientation

What poisson distribution questions are testing

A Poisson variable counts events in a fixed exposure when events occur independently at a stable average rate. The parameter is the expected count for the exact interval or exposure under study, so the first calculation should rescale the rate before any probability mass is evaluated.

Exam P tests exact counts, tails, combined independent sources, conditional allocation of a total, and compound-count moments. A useful solution states the exposure-specific mean, decides whether the event is exact or cumulative, and keeps enough precision for the final answer-choice comparison.

Recognition audit

Signals that this is the right method

The random quantity is a count over time, distance, area, policies, or another specified exposure.
A rate per unit is supplied but the requested count covers a different-sized exposure.
Independent count streams are combined, split into categories, or observed through thinning.
An arrival-time question can be translated into the number of events observed by a deadline.

Before calculation

A disciplined first pass

Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.

Formula core

Poisson Distribution formulas to know

These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.

Poisson probability mass

P(N=k)=e^{-\mu}\frac{\mu^k}{k!},\qquad k=0,1,2,\ldots

Here μ is the expected count over the full exposure used in the question.

Mean and variance

E[N]=\mu,\qquad \operatorname{Var}(N)=\mu

The equality of mean and variance is also a quick diagnostic for the parameterization.

Independent aggregation

N_1\sim\operatorname{Poisson}(\mu_1),\ N_2\sim\operatorname{Poisson}(\mu_2)\Longrightarrow N_1+N_2\sim\operatorname{Poisson}(\mu_1+\mu_2)

Add parameters only when the component counts are independent.

Three failure modes to catch

Using a per-unit rate as the Poisson mean without multiplying by the requested exposure.
Evaluating one mass value for wording such as “at most,” “at least,” or “more than.”
Adding count means while ignoring dependence or overlapping event streams.

Verified practice ledger

Exam P poisson distribution problems and solutions

The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.

General Probability

5 verified references

SOA Exam P Sample Question #314Bayes' TheoremVerified
SOA Exam P Sample Question #455Conditional ProbabilityVerified
SOA Exam P Sample Question #572Bayes' TheoremVerified
SOA Exam P Sample Question #603Bayes' TheoremVerified
SOA Exam P Sample Question #699Bayes' TheoremVerified

Univariate Random Variables

55 verified references

Multivariate Random Variables

12 verified references

Continue the study loop

Keep every worked reference in one searchable manual

The complete Exam P manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.

View contents and sample pagesGet the free formula sheet

Related drills

Other Exam P technique collections

Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.

103 verified problems

Poisson Distribution practice problems