SOA Exam P Sample Question #578. Official problem wording is withheld; follow the source link to read it.

SOA Exam P Sample Question #578 — Verified Solution

Independent solution

How to solve this Normal Distribution question

Answer in brief

The independent profit components sum to a normal variable with mean 600 and standard deviation sqrt(400^2+300^2)=500. Standardizing zero gives -1.20, so the requested probability is P(Z>-1.20) and choice B is correct.

Setup

Add the component means to obtain the center of overall profit.

E[S]=200+400=600

Model

The sum remains normal, and independence makes the variances additive.

\operatorname{Var}(S)=400^2+300^2=250000

S\sim N(600,500^2)

Compute

Standardize the zero-profit boundary.

\Pr(S>0)=\Pr\!\left(Z>\frac{0-600}{500}\right)

\Pr(S>0)=\Pr(Z>-1.20)=0.8849303298\ldots

Answer

The standardized expression matches choice B.

\boxed{\Pr(Z>-1.20)\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 uses an aggregate spread that is too small, producing an excessively large standardized mean magnitude.
CThis adds the component standard deviations to get 700, yielding 600/700 about 0.86 rather than using the root-sum-of-squares spread.
DThe displayed z-score 0.83 implies an aggregate standard deviation near 600/0.83=723, which is even larger than the unsupported arithmetic sum 400+300=700.
EThis standardizes the first profit component alone, using 200/400=0.50 and omitting the second component.

Original practice · fully worked

Original variant: inferred reading correlation

Two jointly normal calibration readings A and B have means 10 and 6 and standard deviations 3 and 4, respectively. The probability that A-B is positive is 0.8413447461. Calculate the correlation between A and B.

A
-0.375
B
0.000
C
0.375
D
0.750
E
1.000

Variant answer in brief

The difference has mean 4, and the stated probability is Phi(1), so its standard deviation must be 4. Equating Var(A-B)=25-24rho to 16 gives rho=0.375, so choice C is correct.

Setup

Let D be the difference and express its mean and variance in terms of the unknown correlation.

D=A-B

E[D]=10-6=4

\operatorname{Var}(D)=3^2+4^2-2\rho(3)(4)=25-24\rho

Model

The stated positive probability corresponds to a standard-normal score of one.

\Pr(D>0)=\Phi\!\left(\frac{4}{\operatorname{SD}(D)}\right)=\Phi(1)

\operatorname{SD}(D)=4

Compute

Set the variance expression equal to 16 and solve for the correlation.

25-24\rho=16

\rho=\frac9{24}=0.375

Answer

The readings have correlation 0.375.

\boxed{\rho=0.375\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #578 — Verified Solution.” Verified 2026-07-30.

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