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What is the probability that the convex hull of $n$ randomly distributed points has length $l$?

Consider a square grid in which $n$ points are randomly distributed. Now consider the convex hull of these points. The length of the convex hull is defined as the number of points connected in the perimeter of the convex hull i.e. $n \\; -$ the number of points inside the convex hull.
I conjecture that the probability that the convex hull has length $l$ is the same for all $3 \le l \le n$ as the points are randomly arranged in the grid. However, I’m unable to come up with a proof for this claim.
Is my conjecture correct, and if not, what is the correct probability distribution?
Edit: The grid is not a lattice grid - the points can be arranged anywhere inside it, so there are infinite possibilities for the arrangement of points.

Consider a square in which $n$ points are randomly distributed. Now consider the convex hull of these points. The "length" of the convex hull is defined as the number of points connected in the perimeter of the convex hull i.e. $n \\; -$ the number of points strictly contained by the convex hull.
I conjecture that the probability that the convex hull has "length" $l$ is the same for all $3 \le l \le n$ as the points are randomly arranged in the grid. However, I’m unable to come up with a proof for this claim.
Is my conjecture correct, and if not, what is the correct probability distribution?
Note that the grid is not a lattice grid - the points can be arranged anywhere inside it, so there are infinite possibilities for the arrangement of points and there will be no identical points.

convex-hull