Worked solution · Exam P

SOA Exam P Sample Question #369 Solution

This Exam P sample reference tests Independence. The husband's two-year survival probability is 0.90. Independence makes the joint survival probability the product of the two marginal survival probabilities, so the wife's survival probability is 7/9 and her death probability is 2/9, approximately 0.222. This is choice E.

Independent analysis by ActuaryProof · reviewed July 31, 2026

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Independent solution

How to solve this Independence question

Setup

Write the husband's survival probability as the complement of his death probability.

\Pr(H)=1-0.10=0.90

Model

For independent survival events, the probability that both spouses survive is the product of their marginal survival probabilities.

\Pr(H\cap W)=\Pr(H)\Pr(W)=0.70

Compute

Recover the wife's survival probability, then take its complement.

\Pr(W)=\frac{0.70}{0.90}=\frac79

\Pr(W^c)=1-\frac79=\frac29=0.222222\ldots

Answer

The wife's two-year death probability is approximately 0.222.

\boxed{0.222\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.

AA death probability of 0.100 simply copies the husband's rate. It would produce joint survival of 0.81, not the stated 0.70.
BA death probability of 0.118 would leave wife survival of 0.882 and hence joint survival of about 0.794. That fails the supplied joint probability.
CA death probability of 0.143 is the ratio of 0.10 to 0.70. It would imply joint survival of about 0.771 rather than 0.70.
DA death probability of 0.200 treats the two death events as though their probabilities could be removed directly from one. It implies joint survival of 0.72.

Original practice · fully worked

Original variant: simultaneous backup-link failures

Two backup communication links fail independently during a stress test. Their failure probabilities are 0.08 and 0.15. Given that at least one link fails, calculate the probability that both links fail.

A
0.012
B
0.055
C
0.080
D
0.150
E
0.218

Answer and calculation

Independence gives a double-failure probability of 0.012, while inclusion–exclusion gives 0.218 for at least one failure. Their ratio is approximately 0.055, choice B.

Setup

Multiply the marginal failure probabilities to obtain the double-failure probability.

\Pr(A\cap B)=(0.08)(0.15)=0.012

Model

Use inclusion–exclusion for the event that at least one link fails.

\Pr(A\cup B)=0.08+0.15-0.012=0.218

Compute

Condition the double-failure event on the union of the two failure events.

\Pr(A\cap B\mid A\cup B)=\frac{0.012}{0.218}=0.05504587\ldots

Answer

Given at least one failure, the probability that both links fail is approximately 0.055.

\boxed{0.055\quad\text{(B)}}

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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: simultaneous backup-link failures

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability