Worked solution · Exam P

SOA Exam P Sample Question #690 Solution

This Exam P sample reference tests Discrete Random Variables. A total of two can arise as 2+0+0 or as 1+1+0. Accounting for the arrangements of both disjoint patterns gives 0.120000+0.018375=0.138375, which rounds to choice E.

Independent analysis by ActuaryProof · reviewed July 30, 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 Discrete Random Variables question

Setup

Let X1, X2, and X3 be the independent annual counts for the three selected policies.

\Pr(X_i=0)=0.8,\quad \Pr(X_i=1)=0.0875,\quad \Pr(X_i=2)=0.0625

Model

The sum equals two only through one count of two and two zeros, or through two counts of one and one zero.

\Pr(X_1+X_2+X_3=2)=3\Pr(2,0,0)+3\Pr(1,1,0)

Compute

Use independence within each pattern and include its three possible positions.

3(0.0625)(0.8)^2=0.120000

3(0.0875)^2(0.8)=0.018375

\Pr(X_1+X_2+X_3=2)=0.138375

Answer

The requested probability rounds to four decimals as 0.1384.

\boxed{0.1384\quad\text{(E)}}

Test the method on unseen problems

Probability Multivariate & Risk Models Practice Workbook

30 original multivariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

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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.

AThis is the 1+1+0 contribution alone, 3(0.0875)²(0.8)=0.018375, and omits the 2+0+0 pattern.
BThe value 0.0230 is approximately 3(0.0875)²; it counts the positions of two one-claim policies but omits the required zero-claim probability.
CThis is about twice the preceding 1+1 contribution and reflects treating the two identical one-claim outcomes as ordered, while still failing to include the 2+0+0 pattern correctly.
DThis is the 2+0+0 contribution alone, 3(0.0625)(0.8)²=0.1200, and omits the 1+1+0 pattern.

Original practice · fully worked

Original variant: community help-desk requests

For a randomly chosen community center, the number of help-desk requests received before noon is 0, 1, 2, or 3 with probabilities 0.70, 0.20, 0.08, and 0.02. Counts from different centers are independent. Four centers are monitored. Calculate the probability that they receive exactly two requests in total before noon.

A
0.1098
B
0.1176
C
0.1600
D
0.2274
E
0.2778

Answer and calculation

The only possible patterns are 2+0+0+0 and 1+1+0+0. Their probabilities are 0.10976 and 0.11760, so the total is 0.22736 and choice D.

Setup

Let R1 through R4 be the independent request counts for the four centers.

\Pr(R_i=0)=0.70,\quad\Pr(R_i=1)=0.20,\quad\Pr(R_i=2)=0.08

Model

A total of two requests consists either of one center with two requests or two centers with one request each.

\Pr\!\left(\sum_{i=1}^4R_i=2\right)=4(0.08)(0.70)^3+\binom42(0.20)^2(0.70)^2

Compute

Evaluate and add the two disjoint pattern probabilities.

4(0.08)(0.70)^3=0.10976

6(0.20)^2(0.70)^2=0.11760

0.10976+0.11760=0.22736

Answer

The probability of exactly two requests is about 0.2274.

\boxed{0.2274\quad\text{(D)}}

Keep practicing

Next steps for Exam P Multivariate Random Variables

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How to solve this Discrete Random Variables question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: community help-desk requests

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables