Verified solution · Exam P

SOA Exam P Sample Question #134 Solution

This Exam P sample reference tests Discrete Sample Spaces. There are 400 ordered number pairs. The diagonal contributes 20 pairs and absolute differences one through three contribute 2(19+18+17)=108 more, giving probability 128/400=0.32, choice B.

Independent analysis by ActuaryProof · reviewed August 4, 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 Sample Spaces question

Setup

All ordered pairs of selected integers are equally likely.

N_{\mathrm{all}}=20^2=400

Model

Winning pairs have absolute difference zero, one, two, or three. A positive difference d occurs in both orientations.

N_0=20,\qquad N_d=2(20-d)\ \text{for }d=1,2,3

Compute

Add the four diagonal bands and divide by the full square sample space.

N_{\lvert X-Y\rvert\le3}=20+2(19+18+17)=128

\Pr(\lvert X-Y\rvert\le3)=\frac{128}{400}=0.32

Answer

The winning probability is 0.32.

\boxed{0.32\quad\text{(B)}}

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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.

AThe value 0.27 equals 2(19+18+17)/400. It counts differences one through three but omits the 20 tied pairs, which also satisfy the winning condition.
CThe value 0.40 would require 160 favorable pairs, an average of eight admissible second choices per first choice. No interior number has more than seven choices within distance three.
DThe value 0.48 would require 192 favorable pairs and substantially overcounts the diagonal band. The direct band count gives only 128.
EThe complementary event has probability 1-0.32=0.68. A value near 0.66 comes from counting the losing region with an endpoint error and then reporting the wrong side.

Original practice · fully worked

Original variant: asymmetric calibration settings

A calibrator independently selects an input dial X uniformly from {1,2,...,8} and an output setting Y uniformly from {1,2,...,12}. A trial is accepted when |2X-Y|≤1. Determine the probability that a trial is accepted.

A
1/8
B
1/6
C
17/96
D
3/16
E
1/4

Variant answer in brief

For X=1 through 5 there are three acceptable Y values, for X=6 there are two, and for X=7 or 8 there are none. Thus 17 of the 96 ordered pairs are accepted, choice C.

Setup

The rectangular sample space has eight times twelve equally likely ordered settings.

N_{\mathrm{all}}=8(12)=96

Model

For fixed x, the admissible integer outputs lie in the three-point band from 2x-1 through 2x+1, clipped to the output range.

2x-1\le Y\le2x+1

Compute

The counts by input are three for the first five inputs, two for input six, and zero thereafter.

N_{\mathrm{accept}}=3+3+3+3+3+2+0+0=17

\Pr(\mathrm{accept})=\frac{17}{96}=0.1770833

Answer

Seventeen of the ninety-six setting pairs are accepted.

\boxed{\frac{17}{96}\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #134 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Discrete Sample Spaces question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: asymmetric calibration settings

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution