# 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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