What is the probability that a set of points have exactly $3$ points as bounding box?

Question

The bounding box of a set of points is the smallest rectangle that all the points lie. Like the following image

Given a set of $n$ random points uniformly distributed in the plane, I would like to know what is the probability that there are exactly $3$ points on the borders of bounding box?

I mean this image

The points on the bounding box must have minimum or maximum values of $x$ or $y$. So In fact I would like to know what is the probability that a point includes more than one minimum or maximum of $x$ or $y$.

thanks

Answer 1

If the distributions of the $x$ and $y$ coordinates are independent and continuous, then with probability $1$ every point has a unique $x$ coordinate (different from the $x$ coordinates of any of the other $n-1$ random points) and a unique $y$ coordinate. Moreover, if we list the points from left to right by $x$-coordinate, there is no reason for any of those points to have a better chance than any other to have the largest $y$-coordinate, or to have the smallest $y$-coordinate.

Now you just need to consider the probabilities of the following three events:

• The leftmost point is in a corner of the bounding box.
• The rightmost point is in a corner of the bounding box.
• Both the leftmost point and the rightmost point are in corners of the bounding box.

(Notice that these are not disjoint events, and only some of these events cause the bounding box to be determined by exactly three points.)

It does not really matter whether the distributions of the $x$ and $y$ coordinates are uniform, provided that they are continuous and independent. If the distributions are discrete, however, then the exact distribution makes a difference, and things get a lot more complicated due to the possibility that multiple points lie on the same edge of the bounding box.