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?
where did you get this problem?
I have solved the problem.
Can I answer it on brilliant?
If yes, how?
to ok. if you want to answer this on the site brilliant, you can by making an account but unfortunately, in every account, they have different sets of question basing on the level what you are in. this is such a very easy question.
49/625.
Correct right?
To approach this problem, we need to calculate the probability that a randomly chosen point s is contained in a randomly chosen path p.
First, let's analyze the possible paths from the bottom left corner to the top right corner on the grid. Since we can only move right or up, the number of steps we need to take to reach the top right corner is 25 (right) + 25 (up) = 50.
Now, let's count the number of paths. Starting from the bottom left corner, we need to take 25 right steps and 25 up steps to reach the top right corner. Essentially, this is equivalent to choosing 25 out of the 50 total steps as right steps. Therefore, there are a total of "50 choose 25" paths.
Next, for a given point s, we need to determine how many of these paths would contain s. Since s can be any point of intersection, we can count the number of paths passing through s by considering the number of right steps and up steps that occur before reaching s.
Let's say that the chosen point s is at the i-th row and j-th column on the grid. To reach s, we need to take i right steps and j up steps. After reaching s, we need to take (25 - i) right steps and (25 - j) up steps to reach the top right corner.
Thus, the number of paths passing through s is given by "i choose i + j" (for the right steps) multiplied by "(25 - i) choose (25 - j)" (for the up steps).
Now, to find the probability that a randomly chosen point s is contained in a randomly chosen path p, we divide the number of paths passing through s by the total number of paths. Let's denote this probability as P(s).
P(s) = ("i choose i + j" * "(25 - i) choose (25 - j)") / ("50 choose 25")
Since s is chosen uniformly at random from S, the probability that s is contained in the path p is the average of P(s) over all possible points s.
To find this average probability, we sum up P(s) for all valid i and j. Since i ranges from 0 to 25, and j ranges from 0 to 25, the sum becomes:
∑(for i = 0 to 25) [ ∑(for j = 0 to 25) ["i choose i + j" * "(25 - i) choose (25 - j)"] ] / "50 choose 25"
Now, we need to calculate this sum to obtain the final probability. To do this, we can use a programming language, spreadsheet software, or an online tool that can perform the required calculations. Once we have the probability, we can write it as a fraction a/b, where a and b are coprime positive integers.
Finally, we sum the numerator a and denominator b to find the value of a+b, which would be the answer to the given question.