2017 AMC 10 A

Denote "winning" to mean "picking a greater number". There is a chance that Laurent chooses a number in the interval in this case, Chloé cannot possibly win, since the maximum number she can pick is . Otherwise, if Laurent picks a number in the interval , with probability then the two people are symmetric, and each has a chance of winning. Then the total probabilty is .

We can use geometric probability to solve this. Suppose a point lies in the plane. Let be Chloe's number and be Laurent's number. Then obviously we want , which basically gives us a region above a line. We know that Chloe's number is in the interval and Laurent's number is in the interval , so we can create a rectangle in the plane, whose length is and whose width is . Drawing it out, we see that it is easier to find the probability that Chloe's number is greater than Laurent's number and subtract this probability from . The probability that Chloe's number is larger than Laurent's number is simply the area of the region under the line , which is . Instead of bashing this out we know that the rectangle has area . So the probability that Laurent has a smaller number is . Simplifying the expression yields and so .

Scale down by to get that Chloe picks from and Laurent picks from . There are an infinite number of cases for the number that Chloe picks, but they are all centered around the average of . Therefore, Laurent has a range of to to pick from, on average, which is a length of out of a total length of . Therefore, the probability is .