There are $ n$ points randomly by uniform distribution distributed in the plane. We know that no two points have the same $ (x ,y)$ . What is the probability that a point has one of these properties?

• it has maximum value of $ x$ and maximum value of $ y$ among $ n$ points.
• it has maximum value of $ x$ and minimum value of $ y$ among $ n$ points.
• it has minimum value of $ x$ and maximum value of $ y$ among $ n$ points.
• it has minimum value of $ x$ and maximum value of $ y$ among $ n$ points.

My solution:

First, I am going to answer a trivial question to clarify my solution. What is the probability that a point has minimum value of $ x$ among $ n$ points and what is the probability that this set contains a point with minimum value of $ x$ ? Since there is no bound on $ x, y$ and $ n$ points are random, I assume exactly one point has the minimum value of $ x$ . The probability that every point has the minimum value of $ x$ is $ 1/n$ , So the probability that a set of $ n$ points has a point with minimum value of $ x$ is $ \sum_{i=1}^n{1/n} = 1$ .

To answer the original question, There are $ n$ different values of $ x$ and $ n$ different values of $ y$ (I assume this is true, If not I should calculate the expected number of different $ x$ s and $ y$ s), the point must have a specific $ (x,y)$ , so the probability that a point has one of the above properties is $ 1/{n^2}$ . So the probability that a set of points have one of the above properties is $ 4 * \sum_{i=1}^n{1/{n^2}} = 4/n$ .

Is this a right answer to my question?