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.