Verified solution · Exam P

SOA Exam P Sample Question #137 Solution

This Exam P sample reference tests Poisson Mode and Tail. Two adjacent Poisson modes at 2 and 3 imply mean 3. The tails are P(N>3)=0.3528 and P(N>4)=0.1847, so the smallest qualifying K is 4, choice D.

Independent analysis by ActuaryProof · reviewed August 4, 2026

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Independent solution

How to solve this Poisson Mode and Tail question

Setup

A Poisson distribution has two adjacent modes only when its mean is the larger of those two modes. The stated modes therefore determine mean three.

\text{modes }2,3\Longrightarrow \lambda=3

Model

The requested cutoff is the first integer whose upper tail is below 0.25. The tail beyond three is still approximately 0.35277.

\Pr(N>3)=1-13e^{-3}=0.35277

Compute

The tail beyond four is approximately 0.18474, below 0.25. Comparing these adjacent cutoffs proves that four is minimal.

\Pr(N>4)=1-\left(13+\frac{81}{24}\right)e^{-3}=0.18474<0.25

Answer

The smallest cutoff with tail below 25% is 4.

\boxed{4\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.

AA cutoff of 1 is below the mode and leaves an upper tail far greater than 0.25.
BA cutoff of 2 is also below the larger mode and cannot be the first point at which the upper tail falls below 0.25.
CThe value 3 is the Poisson mean and larger mode, but its upper tail is approximately 0.35277, still above 0.25.
EA cutoff of 5 also has an upper tail below 0.25, but cutoff 4 already satisfies the condition, so 5 is not minimal.

Original practice · fully worked

Original variant: tail cutoff from two Poisson modes

A Poisson count has two modes at 4 and 5. Find the smallest integer K for which the probability that the count exceeds K is less than 0.20.

Variant answer in brief

The adjacent modes imply mean 5. The tail above 6 is 0.2378, while the tail above 7 is 0.1334, so K=7, choice C.

Setup

Adjacent Poisson modes at four and five determine a mean of five.

\text{modes }4,5\Longrightarrow \lambda=5

Model

At cutoff six, the upper tail is approximately 0.237817, which is still above 0.20.

\Pr(N>6)=0.237817>0.20

Compute

At cutoff seven, the upper tail falls to approximately 0.133372. Thus seven is the first qualifying integer.

\Pr(N>7)=0.133372<0.20

Answer

The smallest qualifying cutoff is 7.

\boxed{7\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #137 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Poisson Mode and Tail question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: tail cutoff from two Poisson modes

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution