Worked solution · Exam P

SOA Exam P Sample Question #579 Solution

This Exam P sample reference tests Conditional Probability. Integrating the density gives survival S(t)=exp(−t)(1+t). Conditional on lasting beyond one year, the probability of ending by year two is 1-S(2)/S(1)=1-3/(2e)=0.448181, so 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 Conditional Probability question

Setup

Integrate the density tail to obtain the survival function.

S(t)=\int_t^\infty ue^{-u}\,du=e^{-t}(1+t)

Model

Express the conditional interval probability using survival at the two endpoints.

\Pr(T\le2\mid T\ge1)=\frac{S(1)-S(2)}{S(1)}

Compute

Evaluate and simplify the survival ratio.

S(1)=2e^{-1},\qquad S(2)=3e^{-2}

\Pr(T\le2\mid T\ge1)=1-\frac{3e^{-2}}{2e^{-1}}

=1-\frac{3}{2e}=0.4481808382\ldots

Answer

The conditional probability rounds to 0.448.

\boxed{0.448\quad\text{(C)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

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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 comes from an incorrect antiderivative or normalization of the density and is not a valid ratio of the two relevant probabilities.
BThis is the unconditional probability of lasting between one and two years, S(1)-S(2), without division by P(T≥1).
DThis is P(T≥1)=2/e, the probability of the conditioning event rather than the conditional interval probability.
EThis uses F(2)/S(1), putting the unconditional event T≤2 in the numerator without intersecting it with the conditioning event T≥1.

Original practice · fully worked

Original variant: remaining filter service

A two-stage filter has total service time T with a gamma distribution of shape 2 and rate 0.5 per hour. The filter is still operating after 4 hours. Calculate its expected additional operating time.

A
2.000 hours
B
2.667 hours
C
4.000 hours
D
6.667 hours
E
8.000 hours

Answer and calculation

For a shape-2 gamma lifetime with rate 0.5, survival is exp(−0.5t)(1+0.5t). Integrating the conditional survival beyond hour 4 gives mean residual life 8/3=2.6667 hours, so choice B is correct.

Setup

Write the survival function for the shape-two gamma lifetime.

S(t)=e^{-0.5t}(1+0.5t)

S(4)=3e^{-2}

Model

The expected remaining life after age four is the future survival area divided by survival to age four.

E[T-4\mid T>4]=\frac{\int_4^\infty S(t)\,dt}{S(4)}

Compute

Evaluate the survival integral and simplify the ratio.

\int_4^\infty e^{-0.5t}(1+0.5t)\,dt=8e^{-2}

E[T-4\mid T>4]=\frac{8e^{-2}}{3e^{-2}}=\frac83

Answer

The expected additional operating time is approximately 2.667 hours.

\boxed{2.667\text{ hours}\quad\text{(B)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P Univariate Random Variables practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Conditional Probability question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: remaining filter service

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables