Verified solution · Exam P

SOA Exam P Sample Question #125 Solution

This Exam P sample reference tests Binomial Tail Probability. With three independent components and failure probability 0.05, overheating means two or three failures. The binomial tail is 3(0.05²)(0.95)+0.05³=0.00725, choice A.

Independent analysis by ActuaryProof · reviewed August 4, 2026

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Independent solution

How to solve this Binomial Tail Probability question

Setup

Let the number of failed components be binomial with three trials and failure probability 0.05.

K\sim\operatorname{Binomial}(3,0.05)

Model

Overheating requires at least two failures, so include the disjoint cases of exactly two and exactly three failures.

\Pr(K\ge2)=\binom32(0.05)^2(0.95)+(0.05)^3

Compute

The two probabilities are 0.007125 and 0.000125. Their sum is 0.00725.

\Pr(K\ge2)=0.007125+0.000125=0.00725

Answer

The overheat probability rounds to 0.007.

\boxed{0.007\quad\text{(A)}}

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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.

DThe value 0.135 is the probability of exactly one component failure, not at least two failures.
EThe value 0.143 is the probability of at least one component failure, which includes the non-overheating one-failure case.

Original practice · fully worked

Original variant: failure of a four-module controller

A controller has four independent modules, each failing with probability 0.10. The controller shuts down if at least three modules fail. Find the shutdown probability.

A
0.0010
B
0.0028
C
0.0037
D
0.0100
E
0.0523

Variant answer in brief

The probabilities of exactly three and exactly four failures are 4(0.1³)(0.9)=0.0036 and 0.0001. Their sum is 0.0037, choice C.

Setup

Let the number of failed modules be binomial with four trials and failure probability 0.10.

K\sim\operatorname{Binomial}(4,0.10)

Model

Shutdown requires exactly three or exactly four failures.

\Pr(K\ge3)=\binom43(0.1)^3(0.9)+(0.1)^4

Compute

The two probabilities are 0.0036 and 0.0001, giving a shutdown probability of 0.0037.

\Pr(K\ge3)=0.0036+0.0001=0.0037

Answer

The shutdown probability is 0.0037.

\boxed{0.0037\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #125 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Binomial Tail Probability question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: failure of a four-module controller

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution