Worked solution · Exam P

SOA Exam P Sample Question #657 Solution

This Exam P sample reference tests Inclusion–Exclusion. This is an inclusion–exclusion bounds problem. Writing the unknown overlap as q makes the probability of neither event 0.1+q; the feasible overlap runs from 0 to 0.3, so the requested range is 0.1 to 0.4 and choice C is correct.

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 Inclusion–Exclusion question

Setup

Let the two indicated actions be events A and B, and denote their unknown overlap probability by q.

\Pr(A)=0.30,\qquad \Pr(B)=0.60,\qquad q=\Pr(A\cap B)

Model

Use inclusion–exclusion for the union, then take its complement to express the probability p of neither action.

\Pr(A\cup B)=0.30+0.60-q

p=1-\Pr(A\cup B)=0.10+q

Compute

The Fréchet bounds restrict the intersection to the interval from zero through the smaller marginal probability. Substituting those endpoints gives the full feasible range for p.

\max(0,0.30+0.60-1)\le q\le \min(0.30,0.60)

0\le q\le 0.30\quad\Longrightarrow\quad 0.10\le p\le 0.40

Answer

Every intermediate overlap is feasible, so the probability of neither action can be any value in the closed interval shown.

\boxed{0.10\le p\le 0.40\quad\text{(C)}}

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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 interval can arise from giving the overlap the wrong sign in the complement, writing 0.10-q, and then truncating the resulting negative values at zero.
BThe value 0.10 is only the endpoint obtained when the two events are disjoint; the information does not force their intersection to be zero.
DThe value 0.30 corresponds to selecting q=0.20 in p=0.10+q, but no condition singles out that particular overlap.
EThe range 0.70 to 1.00 belongs to the complement of the intersection, 1-q, which means 'not both' rather than 'neither.'

Original practice · fully worked

Original variant: overnight processing features

For an overnight data-processing batch, event A means that edge compression is used and event B means that redundant storage is used. Historical records give P(A)=0.55 and P(B)=0.35, but they do not specify the dependence between the events. Determine the full possible range of the probability that exactly one of the two features is used.

A
0.000 to 0.350
B
0.200 to 0.550
C
0.200 to 0.900
D
0.550 to 0.900
E
0.650

Answer and calculation

If q is the probability that both features are used, the exactly-one probability is 0.90-2q. Since q can range from 0 to 0.35, the answer ranges from 0.20 to 0.90, selecting choice C.

Setup

Represent the unspecified dependence by the intersection probability q.

q=\Pr(A\cap B)

0\le q\le \min(0.55,0.35)=0.35

Model

Exactly one feature is used in either A without B or B without A. Each marginal contains the intersection once.

\Pr(\text{exactly one})=\Pr(A)+\Pr(B)-2\Pr(A\cap B)

\Pr(\text{exactly one})=0.90-2q

Compute

The expression decreases with q, so its endpoints occur at the two feasible endpoint overlaps.

q=0\quad\Longrightarrow\quad \Pr(\text{exactly one})=0.90

q=0.35\quad\Longrightarrow\quad \Pr(\text{exactly one})=0.20

Answer

The possible exactly-one probability fills the interval from 0.20 through 0.90.

\boxed{0.20\le \Pr(\text{exactly one})\le 0.90\quad\text{(C)}}

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How to solve this Inclusion–Exclusion question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: overnight processing features

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability