## 31. Paul and Carolina

Paul and Carolina have arranged a date under the Triumphal Arch between 11 o'clock and noon. Each of them will arrive at some (random) moment during this time interval; according to their agreement, if the other does not appear within a quarter of an hour, the one who arrived first will not wait any longer.

What is the probability that Paul and Carolina's date will take place?
31. Consider a Cartesian coordinate system on a plane. We will mark Paul's arrival time on the x-axis and Carolina's arrival time on the y-axis. Then each point on the plane corresponds to a specific arrival time of one and the other at the agreed place. Let the origin of coordinates on each axis correspond to 11 o'clock. Then all possible events of Paul and Carolina's arrivals at the meeting place between 11 and 12 o'clock are represented by points of the square with vertices \((0,0), (0,1), (1,0), (1,1)\). Since we have no reason to prefer one of these events over another, we will consider that all of them are equally probable.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-117.jpg?height=545&width=583&top_left_y=1121&top_left_x=742)

Suppose Paul arrived at the meeting place at some moment. Then he will meet Carolina if she arrived there within the previous quarter of an hour or will arrive within the next quarter of an hour. Therefore, the desired probability is equal to the area of the region shaded in the figure. This region is bounded by lines parallel to the diagonal of the square and cutting off segments of length \(1/4\) on the coordinate axes, i.e., it constitutes \(1 - 9/16 = 7/16\) of the area of the entire square.

Thus, there are 7 chances out of 16 that Paul's meeting with Carolina will take place.