  Let $n > m\ge 2$ be integers and let $\mathcal{A}=\{A_1,\dots,A_m\}$ be a
partition of $[n]=\{1,\dots,n\}$. For $X \subseteq [n]$, its
$\mathcal{A}$-boundary region $\mathcal{A}(X)$ is defined to be the union of
those blocks $A_i$ of $\mathcal{A}$ for which $A_i\cap X\neq \emptyset$ and
$A_i\cap ([n] \setminus X)\neq \emptyset$. For three different probability
distributions on the power set of $[n]$, partitions $\mathcal{A}$ of $[n]$ are
determined such that the expected cardinality of the $\mathcal{A}$-boundary
region of a randomly chosen subset of $[n]$ is minimal and maximal,
respectively. The problem can be reduced to an optimization problem for integer
partitions of $n$. In the most difficult case, the concave-convex shape of the
corresponding weight function as well as several other inequalities are proved
using an integral representation of the weight function. In one case, there is
an interesting analogon to the AZ-identity. The study is motivated by the rough
set theory.
