  We establish the following two main results on order types of points in
general position in the plane (realizable simple planar order types, realizable
uniform acyclic oriented matroids of rank $3$):
  (a) The number of extreme points in an $n$-point order type, chosen uniformly
at random from all such order types, is on average $4+o(1)$. For labeled order
types, this number has average $4- \frac{8}{n^2 - n +2}$ and variance at most
$3$.
  (b) The (labeled) order types read off a set of $n$ points sampled
independently from the uniform measure on a convex planar domain, smooth or
polygonal, or from a Gaussian distribution are concentrated, i.e. such sampling
typically encounters only a vanishingly small fraction of all order types of
the given size.
  Result (a) generalizes to arbitrary dimension $d$ for labeled order types
with the average number of extreme points $2d+o(1)$ and constant variance. We
also discuss to what extent our methods generalize to the abstract setting of
uniform acyclic oriented matroids. Moreover, our methods allow to show the
following relative of the Erd\H{o}s-Szekeres theorem: for any fixed $k$, as $n
\to \infty$, a proportion $1 - O(1/n)$ of the $n$-point simple order types
contain a triangle enclosing a convex $k$-chain over an edge.
  For the unlabeled case in (a), we prove that for any antipodal, finite subset
of the $2$-dimensional sphere, the group of orientation preserving bijections
is cyclic, dihedral or one of $A_4$, $S_4$ or $A_5$ (and each case is
possible). These are the finite subgroups of $SO(3)$ and our proof follows the
lines of their characterization by Felix Klein.
