Worked solution · Exam P

SOA Exam P Sample Question #120 Solution

This Exam P sample reference tests Event Independence. Each event has probability one half and each pair has intersection probability one quarter, but the triple intersection is empty rather than one eighth. Thus the events are pairwise but not mutually independent, choice A.

Independent analysis by ActuaryProof · reviewed July 31, 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 Event Independence question

Setup

Check pairwise independence and mutual independence separately. The three marginal probabilities are all one half.

\Pr(A)=\Pr(B)=\Pr(C)=\frac12

Model

Each pairwise intersection has probability one quarter, equal to the product of the corresponding marginal probabilities. Thus every pair is independent.

\Pr(A\cap B)=\Pr(A\cap C)=\Pr(B\cap C)=\frac14

Compute

Mutual independence would require a triple-intersection probability of one eighth, but the actual triple intersection is empty. Pairwise independence therefore holds without mutual independence.

\Pr(A\cap B\cap C)=0\ne\frac18

Answer

Every pair is independent, while the three events together are not.

\boxed{\text{pairwise independent, not mutually independent}\quad\text{(A)}}

Test the method on unseen problems

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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.

BMutual independence fails because the triple intersection has probability zero rather than the required one eighth.
CThis choice is incompatible with the three pairwise checks, each of which gives an intersection probability equal to one quarter.
DThis choice is incompatible with the fact that all three pairs pass the independence test.
EThis choice is incompatible with the fact that every pair has intersection probability equal to the product of its marginal probabilities.

Original practice · fully worked

Original variant: two coins and a matching event

Two fair coins are tossed. Let A be the event that the first coin is heads, B the event that the second coin is heads, and C the event that the two coins match. Determine the correct independence relationship among A, B, and C.

A
Pairwise independent but not mutually independent
B
Mutually independent
C
Only A and B are independent
D
Exactly two pairs are independent
E
No pair is independent

Answer and calculation

All three marginal probabilities are one half and every pairwise intersection has probability one quarter. The triple intersection has probability one quarter, not one eighth, so choice A holds.

Setup

The three events each have probability one half under the four equally likely coin outcomes.

\Pr(A)=\Pr(B)=\Pr(C)=\frac12

Model

Each pairwise intersection has probability one quarter, exactly the product of the two marginal probabilities, so all three pairs are independent.

\Pr(A\cap B)=\Pr(A\cap C)=\Pr(B\cap C)=\frac14

Compute

All three events occur together only on the two-heads outcome, with probability one quarter. This differs from the product of the three marginal probabilities, one eighth.

\Pr(A\cap B\cap C)=\frac14\ne\frac18

Answer

The events are pairwise independent but fail mutual independence.

\boxed{\text{pairwise, not mutual}\quad\text{(A)}}

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How to solve this Event Independence question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: two coins and a matching event

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability