Worked solution · Exam P

SOA Exam P Sample Question #528 Solution

This Exam P sample reference tests Conditional Probability. The conditional-probability ratio implies that the fire marginal is twice the flood marginal. The outside-versus-overlap relation then makes their marginal sum 0.80, so the flood probability is 0.80/3=0.2667 and 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.

Read the solution Open official source

Independent solution

How to solve this Conditional Probability question

Setup

Let f and w be the two policy marginals and let j be their positive intersection probability.

f=P(F),\qquad w=P(W),\qquad j=P(F\cap W)>0

Model

Translate the ratio of the two reverse conditional probabilities. Positivity of j permits cancellation.

\frac{j}{w}=2\frac{j}{f}

f=2w

Compute

Express the outside cell using inclusion-exclusion and apply its stated difference from the intersection.

1-f-w+j=j+0.20\quad\Longrightarrow\quad f+w=0.80

2w+w=0.80\quad\Longrightarrow\quad w=\frac{4}{15}=0.266667

Answer

The flood-policy marginal rounds to 0.27.

\boxed{0.27\quad\text{(A)}}

Test the method on unseen problems

General Probability Practice Workbook

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

See the cover and sample pages

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

BThis forgets the 0.20 gap and sets the two marginals to sum to 1. With the 2:1 ratio, that gives w=1/3=0.33.
CThis correctly obtains f+w=0.80 but incorrectly assumes equal marginals, reporting 0.80/2=0.40.
DThis reports the other marginal: f=2w=8/15=0.5333, rather than the flood marginal.
EThis reports the sum f+w=0.80 as if it were the probability of one policy; the sum also double-counts the intersection.

Original practice · fully worked

Original variant: reverse condition for sensor alerts

On a fleet-monitoring day, event H means at least one heat alert and event V means at least one vibration alert. Suppose P(H)=0.40, P(H union V)=0.65, and P(V given H)=0.25. Calculate P(H given V).

A
0.100
B
0.250
C
0.286
D
0.350
E
0.650

Answer and calculation

The forward conditional probability gives an intersection of 0.10. Inclusion-exclusion then gives P(V)=0.35, so the reverse conditional probability is 0.10/0.35=2/7 and choice C.

Setup

Convert the supplied forward conditional probability into the joint alert probability.

P(H\cap V)=P(V\mid H)P(H)=0.25(0.40)=0.10

Model

Use inclusion-exclusion to recover the vibration-alert marginal from the union, the heat marginal, and the intersection.

P(H\cup V)=P(H)+P(V)-P(H\cap V)

Compute

Solve for P(V), then condition the joint alert probability on V.

P(V)=0.65-0.40+0.10=0.35

P(H\mid V)=\frac{P(H\cap V)}{P(V)}=\frac{0.10}{0.35}=\frac{2}{7}

Answer

The reverse conditional probability rounds to 0.286.

\boxed{0.286\quad\text{(C)}}

Keep practicing

Next steps for Exam P General Probability

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 General Probability practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Conditional Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: reverse condition for sensor alerts

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability