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?
To find the probability that a randomly chosen point s is contained in a randomly chosen path p, we need to understand the number of points in S and the number of paths in P.
In a 25x25 grid of city streets, there will be (25+1)x(25+1) = 26x26 = 676 points of intersection represented by S.
To calculate the number of paths in P, we need to count the number of steps required to reach the top right corner from the bottom left corner. Since we can only move right or up, we need to take a total of 25 steps to the right and 25 steps up. This can be done in any order, so we can consider this as selecting 25 steps out of 50 total steps. Therefore, the number of paths is given by the binomial coefficient C(50, 25) = 196,125.
Now, for a given point s in S, the probability that it is contained in a path p is equal to the number of paths that pass through that point divided by the total number of paths. This is equivalent to the number of paths from the bottom left corner to s multiplied by the number of paths from s to the top right corner, divided by the total number of paths.
The number of paths from the bottom left corner to s can be calculated by counting the number of steps required to reach s from the bottom left corner. If s has coordinates (x, y), then the number of steps to the right is x, and the number of steps up is y. This can be represented using the binomial coefficient C(x+y, x).
Similarly, the number of paths from s to the top right corner is given by C((25-x) + (25-y), 25-x).
Therefore, for a given point s, the probability of it being contained in a path p is:
P(s) = (C(x+y, x) * C((25-x) + (25-y), 25-x)) / C(50, 25)
Since s is chosen uniformly at random from S, the final probability is the average of P(s) over all points in S.
To calculate the overall probability, we need to sum P(s) over all points in S and divide it by the total number of points in S:
P = Σ[P(s)] / |S|
Calculating this sum manually could be time-consuming, but we can leverage symmetry to speed up the process. Since the grid is symmetric, there is an equal number of paths passing through any point and its symmetric counterpart. Therefore, we can calculate the sum over half the points in S and multiply it by 2:
P = 2 * Σ[P(s)] / |S|
With this approach, we can calculate the probability and express it as a fraction a/b.
Finally, we need to find the values of a and b to calculate a+b.