### Instruction: Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives?### Response: I need to find the fraction of the time between 1:00 and 2:00 that Alex and the train are both at the station.
I can think of this as a geometric problem, where the x-axis is the time Alex arrives and the y-axis is the time the train arrives.
The region of interest is the square with vertices (1:00, 1:00), (2:00, 1:00), (2:00, 2:00), and (1:00, 2:00).
The train is there when Alex arrives if and only if the point (x,y) is within 10 minutes of the line y = x.
This means that the point (x,y) must satisfy |x - y| <= 10/60, where x and y are measured in hours.
Graphically, this is the region between the two parallel lines y = x + 10/60 and y = x - 10/60.
The area of this region is the fraction of the square that is shaded.
I can calculate this area by subtracting the areas of the two right triangles that are not shaded from the area of the whole square.
The area of the whole square is 1 hour * 1 hour = 1 square hour.
The area of each right triangle is 1/2 * base * height.
The base of each triangle is 50/60 hours, since they are cut off by the lines y = x + 10/60 and y = x - 10/60.
The height of each triangle is also 50/60 hours, since they are symmetric about the line y = x.
So the area of each triangle is 1/2 * 50/60 * 50/60 = 25/72 square hours.
Therefore, the area of the shaded region is 1 - 2 * 25/72 = 22/72 square hours.
This means that the probability that Alex and the train are both at the station is 22/72, which can be simplified to 11/36.