SOA Exam P Sample Question #658. Official problem wording is withheld; follow the source link to read it.

Verified solution · Exam P

SOA Exam P Sample Question #658

This problem combines conditional probability with a three-event union. Regrouping the union makes its probability 0.80-0.80(0.15r)+0.17r; setting this equal to one gives r=4, which is choice D.

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Independent solution

How to solve this Conditional Probability question

Answer in brief

This problem combines conditional probability with a three-event union. Regrouping the union makes its probability 0.80-0.80(0.15r)+0.17r; setting this equal to one gives r=4, which is choice D.

Setup

Group the full union as event B together with the union of A and C. The overlap between those two grouped events simplifies because B and C cannot occur together.

B\cap(A\cup C)=(A\cap B)\cup(B\cap C)=A\cap B

\Pr(A\cap B)=\Pr(B)\Pr(A\mid B)

Model

Apply the two-set addition rule to the grouped events and substitute the supplied expressions.

\Pr(A\cup B\cup C)=\Pr(B)+\Pr(A\cup C)-\Pr(A\cap B)

1=0.80+0.17r-0.80(0.15r)

Compute

Combine the two coefficients of r and isolate the parameter.

1=0.80+(0.17-0.12)r=0.80+0.05r

r=\frac{0.20}{0.05}=4

Answer

The parameter value satisfying all of the event relationships is four.

\boxed{r=4\quad\text{(D)}}

Error diagnosis

Why the other letters tempt

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

ASetting r to zero leaves the grouped union probability at 0.80, so it cannot satisfy the stated total probability of one.
BThe value 20/17 comes from solving 0.80+0.17r=1 and therefore treating A union C as disjoint from B; it omits the A intersection B overlap.
CThe value 100/29 comes from the incorrect equation 1=0.12r+0.17r, which drops the standalone 0.80 mass and adds the overlap rather than subtracting it.
ETreating 0.15r as P(A intersection B), instead of as the conditional probability P(A given B), produces r=10 and falsely suggests that none of A through D applies.

Original practice · fully worked

Original variant: diagnostic signals

Every diagnostic batch produces at least one of three signals: drift D, latency L, or thermal T. Records show that P(D union T)=0.58 and P(D given L)=0.25. A latency signal and a thermal signal can never occur in the same batch. Calculate P(L).

A
0.315
B
0.420
C
0.560
D
0.580
E
0.670

Variant answer in brief

Because latency and thermal signals are mutually exclusive, the overlap of L with D union T is just D intersection L. The full-union equation becomes 1=b+0.58-0.25b, giving b=0.56 and choice C.

Setup

Write b for the latency probability. The conditional drift rate among latency batches determines their intersection.

b=\Pr(L)

\Pr(D\cap L)=\Pr(L)\Pr(D\mid L)=0.25b

Model

Combine L with D union T. Since L and T are disjoint, their grouped overlap contains only D intersection L.

\Pr(L\cap(D\cup T))=\Pr(D\cap L)=0.25b

1=\Pr(L)+\Pr(D\cup T)-\Pr(L\cap(D\cup T))

Compute

Insert the known union probability and solve the one-variable equation.

1=b+0.58-0.25b=0.58+0.75b

b=\frac{0.42}{0.75}=0.56

Answer

The latency signal occurs in 56% of diagnostic batches.

\boxed{\Pr(L)=0.560\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #658 — Verified Solution.” Verified 2026-07-30.

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