Point Path Probability

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

Details and assumptions

There are 25 streets running in each direction, so \(S\) consists of 625 intersections.

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