Worked solution · Exam P

SOA Exam P Sample Question #733 Solution

This Exam P sample reference tests Conditional Probability. Translate both conditional percentages into equations for the same intersection, which makes the dental-insurance probability 0.625 times the medical-insurance probability. Inclusion–exclusion then makes the requested union-to-medical ratio 1.375, corresponding to A.

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 Conditional Probability question

Setup

Let M and D denote the two insurance events and express every probability relative to the size of M.

\Pr(D\mid M)=0.25,\qquad \Pr(M\mid D)=0.40

Model

Both conditional statements describe the same overlap, so equate their two representations.

\Pr(M\cap D)=0.25\Pr(M)=0.40\Pr(D)

\frac{\Pr(D)}{\Pr(M)}=\frac{0.25}{0.40}=0.625

Compute

Apply inclusion–exclusion and divide through by the medical-insurance probability.

\frac{\Pr(M\cup D)}{\Pr(M)}=1+\frac{\Pr(D)}{\Pr(M)}-\frac{\Pr(M\cap D)}{\Pr(M)}

=1+0.625-0.25=1.375

Answer

The ratio is the value shown in choice A.

\boxed{1.375\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.

BThis adds the relative size of the dental group to one but forgets to subtract the shared members.
CThis treats the overlap as an additional group instead of a group already counted in both event totals.
DThis can arise after reversing one conditional denominator and then combining incompatible bases.
EThis exceeds even the result of adding the two relative group sizes without a valid overlap correction, signaling that the conditional percentages were inverted.

Original practice · fully worked

Original variant: media subscriptions

In a reader panel, 30% of audiobook subscribers also subscribe to a documentary service, while 45% of documentary-service subscribers also subscribe to audiobooks. Let U be the probability that a randomly chosen panelist uses at least one of the two services, and let A be the probability that the panelist subscribes to audiobooks. Calculate U/A.

A
0.300
B
0.667
C
1.067
D
1.367
E
1.667

Answer and calculation

The common overlap is 0.30A and also 0.45 times the documentary probability, so the documentary group is two-thirds the size of A. Inclusion–exclusion gives U/A = 1 + 2/3 − 0.30 = 1.3667.

Setup

Use A and D for the two subscription events and place both conditional facts on a common scale.

\Pr(A\cap D)=0.30\Pr(A)=0.45\Pr(D)

Model

Solve for the documentary probability as a multiple of the audiobook probability.

\frac{\Pr(D)}{\Pr(A)}=\frac{0.30}{0.45}=\frac{2}{3}

Compute

Divide the inclusion–exclusion identity by the audiobook probability.

\frac{U}{\Pr(A)}=1+\frac{2}{3}-0.30

\frac{U}{\Pr(A)}=1.366666\ldots

Answer

The ratio rounds to choice D.

\boxed{1.367\quad\text{(D)}}

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How to solve this Conditional Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: media subscriptions

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability