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 question?
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.
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I don't want to make accounts.
To solve this problem, we need to determine the probability that a randomly chosen point s from the set of intersections S is contained in a randomly chosen path p from the set of paths P.
Let's break down the problem step by step:
Step 1: Count the total number of points and paths.
In a 25x25 grid, there are (25+1) horizontal streets and (25+1) vertical streets, resulting in a total of (25+1) * (25+1) = 26^2 = 676 points of intersection.
To count the number of paths, we need to find the number of ways to choose a sequence of 25 moves to the right and 25 moves upwards. This can be calculated using combinations, denoted by C(n, k), where n is the total number of moves and k is the number of moves to the right (or upwards).
So, the number of paths is C(50, 25).
Step 2: Count the number of points contained in a path.
For a point to be contained in a path, it must lie on one of the horizontal streets and one of the vertical streets. Since there are 25 horizontal and 25 vertical streets, the total number of points contained in a path is 25 * 25 = 625.
Step 3: Calculate the probability.
To find the probability, we divide the number of points contained in a path by the total number of points.
Probability = (Number of points contained in a path) / (Total number of points)
Probability = 625 / 676
Step 4: Simplify the fraction.
To simplify the fraction 625 / 676, we find the greatest common divisor (GCD) of the numerator and denominator, which is 25. Then we divide both the numerator and denominator by the GCD.
Probability = (625 / 25) / (676 / 25)
Probability = 25 / 26
Therefore, the probability that the point s is contained in the path p is 25/26.
Finally, the sum of the numerator and denominator is 25 + 26 = 51.
So, the value of a + b is 51.