Problem:
Select numbers and between 0 and 1 independently and at random, and let be their sum. Let , and be the results when , and , respectively, are rounded to the nearest integer. What is the probability that ?
Answer Choices:
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B.
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E.
Solution:
The conditions under which are as follows.
(i) If , then .
(ii) If and , then and .
(iii) If and , then and .
(iv) If , then and .
These conditions correspond to the shaded regions of the graph shown. The combined area of those regions is , and the area of the entire square is 1 , so the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions