Worked solution · Exam P
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 solution
Setup
All ordered pairs of selected integers are equally likely.
Model
Winning pairs have absolute difference zero, one, two, or three. A positive difference d occurs in both orientations.
Compute
Add the four diagonal bands and divide by the full square sample space.
Answer
The winning probability is 0.32.
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Original practice · fully worked
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.
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.
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.
Compute
The counts by input are three for the first five inputs, two for input six, and zero thereafter.
Answer
Seventeen of the ninety-six setting pairs are accepted.
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