  Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let
$K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$
whose bounding hyperplanes are independent and identically distributed
according to a certain prescribed probability distribution. We prove an
asymptotic formula for the expectation of the difference of the volumes of
$K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume
of $K^{(n)}$. We achieve these results by first proving similar statements for
weighted mean width approximations of convex bodies that admit a rolling ball
by inscribed random polytopes and then by polarizing these results.
