Verified solution · Exam P

SOA Exam P Sample Question #531 Solution

This Exam P sample reference tests Independence. Under independence, the no-claim probability is 0.9(0.7)=0.63. Under mutual exclusivity, the union probability is 0.10+0.30, leaving 0.60 for neither. Their difference is 0.03, so choice D is correct.

Independent analysis by ActuaryProof · reviewed August 4, 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 Independence question

Setup

Write the complement probabilities for the two claim types.

\Pr(F^c)=0.90

\Pr(T^c)=0.70

Model

Independence multiplies the complements. Mutual exclusivity instead makes the two claim events a disjoint union.

p=\Pr(F^c\cap T^c)=0.90(0.70)

r=1-\Pr(F\cup T)=1-(0.10+0.30)

Compute

Evaluate the two no-claim probabilities and subtract in the stated order.

p=0.63,\qquad r=0.60

p-r=0.63-0.60=0.03

Answer

The difference is positive three percentage points.

\boxed{0.03\quad\text{(D)}}

Continue without hunting through PDFs

All 718 Exam P sample solutions in syllabus order

The searchable 3108-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam P study manual PDF

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis subtracts the independent overlap 0.10(0.30)=0.03 twice and attaches a negative sign, producing -0.06 instead of one overlap correction.
BThis is r-p=-0.03, so it reverses the requested subtraction.
CThis treats independence and mutual exclusivity as producing the same intersection structure; independence permits a 0.03 joint-claim probability.
EThis adds the 0.03 overlap correction twice. Only one inclusion-exclusion term separates the two no-claim probabilities.

Original practice · fully worked

Original variant: competing relay incidents

A data relay has probability 0.25 of an overload incident and probability 0.10 of an automatic restart during a shift. Under model I, the two events are independent. Under model II, every restart occurs during an overload. Calculate the probability that exactly one of these event types occurs under model I minus the corresponding probability under model II.

A
-0.150
B
0.025
C
0.150
D
0.250
E
0.300

Variant answer in brief

Independence gives exactly-one probability 0.25(0.90)+0.75(0.10)=0.30. Under the nested model, exactly one means overload without restart, with probability 0.15. The difference is 0.15, so choice C is correct.

Setup

Let O denote overload and R denote restart.

\Pr(O)=0.25,\qquad \Pr(R)=0.10

Model

Under independence, exactly one incident has two disjoint cases. Under the nested model, R is a subset of O, so only O without R qualifies.

q_I=\Pr(O\cap R^c)+\Pr(O^c\cap R)

q_{II}=\Pr(O)-\Pr(R)

Compute

Evaluate both model-specific probabilities.

q_I=0.25(0.90)+0.75(0.10)=0.300

q_{II}=0.25-0.10=0.150

q_I-q_{II}=0.150

Answer

The independence model has fifteen percentage points more exactly-one probability.

\boxed{0.150\quad\text{(C)}}

Keep practicing

Turn this solution into a complete study sequence

The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

See the Exam P study manual PDF — $49 More Exam P General Probability practice problems Free Exam P formula sheet

Cite this solution

ActuaryProof. “SOA Exam P Sample Question #531 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Independence question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: competing relay incidents

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution