I can randomly realize a sequence $S_1, S_2, \cdots, S_n$ of $n$ samples from a uniform random distribution $U(0, 1)$.
Using the sequence as a $x$ axis, and assign even space $a$, where $a \in\mathbb{R}^+$ between them in the $y$ axis, we can obtain a planar set of points.
The coordinate of "1.5" dimension planar point sequence would be $P = (S_1, 0), (S_2, a), \cdots, (S_i, a*i), \cdots(S_n, a*n)$.
My interest is how to compute the probability of having $k$ point in convex position?
And what is the expected number of points that is in convex position?