  A two-step model for generating random polytopes is considered. For
parameters $d$, $m$, and $p$, the first step is to generate a simple polytope
$P$ whose facets are given by $m$ uniform random hyperplanes tangent to the
unit sphere in $\mathbb{R}^d$, and the second step is to sample each vertex of
$P$ independently with probability $p$ and let $Q$ be the convex hull of the
sampled vertices. We establish results on how well $Q$ approximates the unit
sphere in terms of $m$ and $p$ as well as asymptotics on the combinatorial
complexity of $Q$ for certain regimes of $p$.
