SOA Exam P Sample Question #723. Official problem wording is withheld; follow the source link to read it.

SOA Exam P Sample Question #723 — Verified Solution

Independent solution

How to solve this Law of Total Probability question

Answer in brief

Express the low- and high-group death probabilities as multiples of the medium-group probability, then apply the law of total probability. Solving the weighted equation gives approximately 0.009863, which corresponds to choice B.

Setup

Let q denote the one-year death probability for a member of the medium group. Translate the two relative-risk statements into probabilities expressed in terms of q.

q_M=q,\qquad q_L=\frac{q}{2},\qquad q_H=4q

Model

The remaining population share is 0.15, so the overall probability is the weighted average of the three group probabilities.

0.15\left(\frac{q}{2}\right)+0.55q+0.30(4q)=0.018

Compute

Collect the coefficients of q and solve the single linear equation.

(0.075+0.55+1.20)q=1.825q=0.018

q=\frac{0.018}{1.825}=0.0098630137\ldots

Answer

The medium-group probability rounds to the value displayed in choice B.

\boxed{q_M\approx0.010\quad\text{(B)}}

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 low-group probability, q_L=q_M/2=0.0049315..., rounded to 0.005; it answers for the wrong category.
CUsing a high-group multiplier of 2 instead of 4 gives q_M=0.018/1.225=0.014694..., which rounds to choice C.
DThis repeats the population-wide death probability 0.018 without resolving the mixture into its three group-specific rates.
EIf q_M were 0.021, the stated weights and multipliers would produce an overall probability of 1.825(0.021)=0.038325, not 0.018.

Original practice · fully worked

Original variant: field-sensor failure classes

A sensor fleet is divided among sheltered, standard, and harsh sites in proportions 0.20, 0.50, and 0.30. A standard-site sensor is three times as likely to fail during a year as a sheltered-site sensor, while a harsh-site sensor is twice as likely to fail as a standard-site sensor. The annual failure probability for a randomly selected sensor is 0.028. Calculate the annual failure probability at a standard site.

A
0.80%
B
1.40%
C
2.40%
D
2.80%
E
4.80%

Variant answer in brief

Write the sheltered and harsh failure rates as one-third and twice the standard-site rate. The weighted fleet equation gives a standard-site probability of 0.024, so the answer is C.

Setup

Let s be the standard-site annual failure probability and express the other two rates relative to it.

q_{\mathrm{sheltered}}=\frac{s}{3},\qquad q_{\mathrm{standard}}=s,\qquad q_{\mathrm{harsh}}=2s

Model

Average the three rates with the fleet proportions.

0.20\left(\frac{s}{3}\right)+0.50s+0.30(2s)=0.028

Compute

The coefficient is 7/6, so the standard-site rate follows directly.

\left(\frac{1}{15}+\frac{1}{2}+\frac{3}{5}\right)s=\frac{7}{6}s=0.028

s=0.028\left(\frac{6}{7}\right)=0.024

Answer

The standard-site annual failure probability is listed in choice C.

\boxed{s=0.024\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #723 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.