  A set $\mathcal{A}\subset \mathbb{N}$ is called additively decomposable
(resp. asymptotically additively decomposable) if there exist sets
$\mathcal{B},\mathcal{C}\subset \mathbb{N}$ of cardinality at least two each
such that $\mathcal{A}=\mathcal{B}+\mathcal{C}$ (resp. $\mathcal{A}\Delta
(\mathcal{B}+\mathcal{C})$ is finite). If none of these properties hold, the
set $\mathcal{A}$ is called totally primitive. We define
$\mathbb{Z}$-decomposability analogously with subsets $\mathcal{A,B,C}$ of
$\mathbb{Z}$. Wirsing showed that almost all subsets of $\mathbb{N}$ are
totally primitive. In this paper, in the spirit of Wirsing, we study
decomposability from a probabilistic viewpoint. First, we show that almost all
symmetric subsets of $\mathbb{Z}$ are $\mathbb{Z}$-decomposable. Then we show
that almost all small perturbations of the set of primes yield a totally
primitive set. Further, this last result still holds when the set of primes is
replaced by the set of sums of two squares, which is by definition
decomposable.
