Consider a grid of city streets. Let be the points of intersection of the streets, and let be the set of paths from the bottom left corner to the top right corner of which consist of only walking to the right and up. A point is chosen uniformly at random from and then a path is chosen uniformly at random from . Over all pairs, the probability that the point is contained in the path can be expressed as where and are coprime positive integers. What is the value of ?
Details and assumptions
There are 25 streets running in each direction, so consists of 625 intersections.