Worked 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 July 31, 2026

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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)}}

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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.

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

Answer and calculation

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)}}

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How to solve this Independence question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: competing relay incidents

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability