Worked solution · Exam P

SOA Exam P Sample Question #663 Solution

This Exam P sample reference tests Conditional Probability. Condition on the completion time being above its normal mean, an event with probability one-half. Standardizing the upper endpoint gives a conditional probability of 0.261117, which rounds to choice B.

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 T be the total completion time. Standardize T using the given mean and standard deviation.

T\sim N(7.5,1.5^2),\qquad Z=\frac{T-7.5}{1.5}

Model

The requested event lies inside the conditioning event, so conditional probability is the interval probability divided by the upper-half probability.

\Pr(T\le8\mid T>7.5)=\frac{\Pr(7.5<T\le8)}{\Pr(T>7.5)}

\Pr(T>7.5)=\frac12

Compute

The upper endpoint is one-third of a standard deviation above the mean.

\frac{8-7.5}{1.5}=\frac13

\Pr(T\le8\mid T>7.5)=\frac{\Phi(1/3)-\Phi(0)}{1/2}

=2\Phi(1/3)-1=0.2611173196\ldots

Answer

The conditional probability is approximately 0.26.

\boxed{0.26\quad\text{(B)}}

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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 is the unconditional interval probability between 7.5 and 8 hours. It omits division by the conditioning probability of one-half.
CThis is approximately the unconditional upper-tail probability beyond 8 hours, rather than the probability of finishing by 8 after the stated condition.
DThis is approximately the unconditional probability of completion by 8 hours, so it includes all times below 7.5 that the condition rules out.
EThis is the complement within the conditioned upper half: the probability of still working after 8 given a completion time above 7.5.

Original practice · fully worked

Original variant: laboratory batch runtime

The runtime of an automated laboratory batch is normally distributed with mean 42 minutes and standard deviation 6 minutes. A dashboard shows that a batch is still running at minute 42. Calculate the conditional probability that it completes by minute 48.

A
0.1587
B
0.3173
C
0.5000
D
0.6827
E
0.8413

Answer and calculation

Given that runtime has passed its mean, the relevant normal mass is the interval from zero to one standard deviation divided by one-half. The result is 0.682689, so choice D is correct.

Setup

Let R denote runtime and convert minutes to standard normal units.

R\sim N(42,6^2),\qquad Z=\frac{R-42}{6}

Model

The dashboard condition restricts R to the upper half of its symmetric distribution.

\Pr(R\le48\mid R>42)=\frac{\Pr(42<R\le48)}{\Pr(R>42)}

Compute

Minute 48 is one standard deviation above the mean, and the conditioning event has probability one-half.

\Pr(R\le48\mid R>42)=\frac{\Phi(1)-\Phi(0)}{1/2}

=2\Phi(1)-1=0.6826894921\ldots

Answer

The conditioned batch has probability about 0.6827 of finishing by minute 48.

\boxed{0.6827\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: laboratory batch runtime

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability