Consider a 25×25 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 a/b where a and b are coprime positive integers. What is the value of a+b?
Most probably it should be 49/625
that is correct!
To solve this problem, we need to find the probability that a randomly chosen point s lies on a randomly chosen path p.
First, let's determine the total number of points of intersection (S) in the grid. In a 25×25 grid, there are (25+1) horizontal lines and (25+1) vertical lines. Therefore, the total number of points of intersection is (25+1) * (25+1) = 26 * 26 = 676.
Next, let's consider the total number of paths (P) from the bottom left corner to the top right corner, consisting of only walking to the right and up. We can represent each path as a sequence of moves, where moving right is represented by "R" and moving up is represented by "U". In a 25×25 grid, to reach the top right corner from the bottom left corner, we need to make a total of 25 moves to the right and 25 moves up. Thus, the total number of paths is the number of ways to arrange these 50 moves, which is given by the binomial coefficient (50 choose 25). Using the formula, this is evaluated as (50!)/(25! * 25!) = 126410606437752.
Now, let's consider the number of points s that lie on the paths p. Since s is chosen uniformly at random from all the points of intersection, each s can be chosen with equal probability. Also, since p is chosen uniformly at random from all the paths, each path can be chosen with equal probability.
To calculate the number of (s,p) pairs such that s lies on p, we can count the number of paths that pass through each point s and then sum up these counts for all points s. For each point s, there are (24 choose x)(24 choose y) paths that pass through it, where x is the number of moves right of s and y is the number of moves up from s. This is because there are 24 moves remaining in the rightward direction and 24 moves remaining in the upward direction after reaching s. By summing up these counts for all points s, we obtain the total number of (s,p) pairs such that s lies on p.
To find the probability that s lies on p, we divide this total number of (s,p) pairs by the total number of possible (s,p) pairs, which is equal to the product of the total number of points of intersection (676) and the total number of paths (126410606437752).
Therefore, the probability that s lies on p is (total number of (s,p) pairs)/(total number of possible (s,p) pairs) = (sum of (24 choose x)(24 choose y) for all s)/(676 * 126410606437752).
Finally, we need to express this probability as a fraction a/b where a and b are coprime positive integers.
Now that we have all the required information, we can calculate the value of a/b and find a+b.