SOA Exam P Sample Question #653. Official problem wording is withheld; follow the source link to read it.

SOA Exam P Sample Question #653 — Verified Solution

Independent solution

How to solve this Independence question

Answer in brief

On one trial the three possible totals have probabilities 1/9, 4/9, and 4/9. The two independent totals agree with probability 33/81, so they differ with probability 48/81 = 16/27 and choice C.

Setup

Determine the distribution of the total on a single trial from the two face-value probabilities.

\Pr(D=1)=\frac13,\qquad \Pr(D=2)=\frac23

\Pr(S=2)=\frac19,\qquad \Pr(S=3)=\frac49,\qquad \Pr(S=4)=\frac49

Model

It is shorter to calculate the probability that the two independent trial totals match, then take its complement.

\Pr(S_1=S_2)=\sum_{s=2}^{4}\Pr(S=s)^2

Compute

Square and add the three single-trial probabilities.

\Pr(S_1=S_2)=\left(\frac19\right)^2+\left(\frac49\right)^2+\left(\frac49\right)^2=\frac{33}{81}

\Pr(S_1\ne S_2)=1-\frac{33}{81}=\frac{48}{81}=\frac{16}{27}

Answer

The probability of different totals is sixteen twenty-sevenths.

\boxed{\frac{16}{27}\quad\text{(C)}}

Error diagnosis

Why the other letters tempt

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

AThis undercounts the ordered pairs of unequal totals; all six off-diagonal pairs must be included.
BThis also omits some off-diagonal probability mass and does not complement the exact match probability 33/81.
DThis is 2/3, the answer obtained by incorrectly treating the three possible totals as equally likely.
EThis overstates the result by assigning too little probability to a repeated total, especially the two totals that each occur with probability 4/9.

Original practice · fully worked

Original variant: diagnostic status changes

A diagnostic run independently reports Low, Medium, or High status with probabilities 0.20, 0.30, and 0.50. Two runs are made under unchanged conditions. Calculate the probability that the reported status differs between the two runs.

A
0.38
B
0.50
C
0.58
D
0.62
E
0.70

Variant answer in brief

The two statuses match with probability 0.20² + 0.30² + 0.50² = 0.38. The complement is 0.62, so choice D.

Setup

List the three single-run probabilities.

p_L=0.20,\qquad p_M=0.30,\qquad p_H=0.50

Model

Independence makes the probability of matching in a category the square of that category probability.

\Pr(\text{same})=p_L^2+p_M^2+p_H^2

Compute

Calculate the matching probability and take its complement.

\Pr(\text{same})=0.20^2+0.30^2+0.50^2=0.38

\Pr(\text{different})=1-0.38=0.62

Answer

The two diagnostic statuses differ with probability 0.62.

\boxed{0.62\quad\text{(D)}}

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ActuaryProof. “SOA Exam P Sample Question #653 — Verified Solution.” Verified 2026-07-30.

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