Problem:
Chloé chooses a real number uniformly at random from the interval . Independently, Laurent chooses a real number uniformly at random from the interval . What is the probability that Laurent's number is greater than Chloé's number?
Answer Choices:
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Solution:
Half of the time Laurent will pick a number between and 4034, in which case the probability that his number will be greater than Chloé's number is . The other half of the time, he will pick a number between and , and by symmetry his number will be the larger one in half of those cases. Therefore the requested probability is .
The choices of numbers can be represented in the coordinate plane by points in the rectangle with vertices at , and . The portion of the rectangle representing the event that Laurent's number is greater than Chloé's number is the portion above the line segment with endpoints and . This area is of the area of the entire rectangle, so the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions