Worked solution · Exam P

SOA Exam P Sample Question #4 Solution

This Exam P sample reference tests Independence. Equating the same-color probability to 0.44 gives a linear equation in the unknown count; its unique nonnegative solution is 4, choice A.

Independent analysis by ActuaryProof · reviewed October 6, 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 Independence question

Setup

Let x be the unknown count in the second container. Its two possible categories then have counts 16 and x.

x=\text{unknown count in the second container}

Model

Condition on the category selected from the first container. The same-category probability is the sum of the two corresponding product probabilities.

0.44=\frac4{10}\frac{16}{16+x}+\frac6{10}\frac{x}{16+x}

Compute

Clearing the denominator in the resulting mixture equation and solving the linear equation gives x=4.

4.4(16+x)=64+6x,\qquad x=4

Answer

The second container therefore has 4 items of the unknown category, which is choice A.

\boxed{4\quad\text{(A)}}

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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.

BSubstituting x=20 into the same-category formula gives (0.4)(16/36)+(0.6)(20/36)=0.5111, not 0.44.
CSubstituting x=24 gives (0.4)(16/40)+(0.6)(24/40)=0.52, so it does not satisfy the observed probability.
DSubstituting x=44 gives (0.4)(16/60)+(0.6)(44/60)=0.5467, not 0.44.
EThe value 64 is the constant term obtained before the linear equation is solved. Substitution x=64 gives a same-category probability of 0.56.

Original practice · fully worked

Original variant: infer a routing share from an aggregate success rate

A request router sends an unknown proportion p of jobs through a fast path whose success probability is 0.80; all remaining jobs use a fallback path whose success probability is 0.50. The observed overall success probability is 0.62. Determine p.

A
0.40
B
0.48
C
0.50
D
0.60
E
0.75

Answer and calculation

The aggregate rate is 0.80p+0.50(1-p). Solving 0.50+0.30p=0.62 gives p=0.40, choice A.

Setup

Let p be the proportion routed through the fast path; the remaining proportion 1-p uses the fallback path.

p=\Pr(\text{fast path})

Model

Use the law of total probability: the overall success rate is the route-share-weighted average of the two conditional success rates.

0.62=0.80p+0.50(1-p)

Compute

Expanding 0.80p+0.50(1-p) and equating it to 0.62 gives 0.30p=0.12, hence p=0.40.

0.12=0.30p,\qquad p=0.40

Answer

The fast path receives 40% of jobs, which is choice A.

\boxed{0.40\quad\text{(A)}}

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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: infer a routing share from an aggregate success rate

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability