Worked solution · Exam P

SOA Exam P Sample Question #691 Solution

This Exam P sample reference tests Set Probability. Complementing the two supplied percentages gives probabilities 0.60 for the three-year event and 0.30 for combined coverage. The availability restrictions make these events disjoint, so their union has probability 0.90 and choice E.

Independent analysis by ActuaryProof · reviewed July 30, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Set Probability question

Setup

Let T be the event of a three-year contract and B the event of combined parts-and-labor coverage.

\Pr(T)=1-0.40=0.60

\Pr(B)=1-0.70=0.30

Model

The availability table permits no three-year contract with combined coverage, so the two events have empty intersection.

\Pr(T\cap B)=0

Compute

Apply the addition rule for the union and use the structural zero for the intersection.

\Pr(T\cup B)=\Pr(T)+\Pr(B)-\Pr(T\cap B)

\Pr(T\cup B)=0.60+0.30-0=0.90

Answer

Nine tenths of the contracts satisfy at least one of the two conditions.

\boxed{0.90\quad\text{(E)}}

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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 is the probability of the impossible intersection T cap B, but the question asks for the union rather than the intersection.
BThe value 0.18 is 0.60(0.30), obtained by assuming independence and calculating an intersection; the table instead establishes mutual exclusivity.
CThe value 0.28 is (1-0.60)(1-0.30), the probability of neither event under an unjustified independence assumption.
DThe value 0.72 is 0.60+0.30-0.60(0.30), which applies the union formula after incorrectly assuming that the two events are independent.

Original practice · fully worked

Original variant: observatory instrument scheduling

Every observation block at a research observatory uses an optical instrument, a radio instrument, or both. Sixty-five percent of blocks use exactly one instrument. Also, 52 percent occur during daylight or twilight. A safety rule prohibits blocks using both instruments from being scheduled at night. Calculate the probability that a randomly selected block is at night or uses both instruments.

A
0.000
B
0.168
C
0.338
D
0.662
E
0.830

Answer and calculation

Night blocks have probability 0.48 and dual-instrument blocks have probability 0.35. The safety rule makes those events disjoint, so their union is 0.83 and choice E.

Setup

Let N denote a night block and D a block using both instruments. Convert the supplied complementary percentages.

\Pr(N)=1-0.52=0.48

\Pr(D)=1-0.65=0.35

Model

The scheduling rule excludes every dual-instrument block from the night period.

\Pr(N\cap D)=0

Compute

Add the probabilities of the two disjoint events.

\Pr(N\cup D)=0.48+0.35=0.83

Answer

The requested union contains 83 percent of the scheduled blocks.

\boxed{0.830\quad\text{(E)}}

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How to solve this Set Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: observatory instrument scheduling

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability