Point Path Probability

Consider a 25×2525 \times 25 grid of city streets. Let SS be the points of intersection of the streets, and let PP 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 ss is chosen uniformly at random from SS and then a path pp is chosen uniformly at random from PP. Over all (s,p)(s,p) pairs, the probability that the point ss is contained in the path pp can be expressed as ab\frac{a}{b} where aa and bb are coprime positive integers. What is the value of a+ba + b?

Details and assumptions

There are 25 streets running in each direction, so SS consists of 625 intersections.

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