  An $s$-subset of codewords of a binary code $X$ is said to be an {\em
$(s,\ell)$-bad} in $X$ if the code $X$ contains a subset of other $\ell$
codewords such that the conjunction of the $\ell$ codewords is covered by the
disjunctive sum of the $s$ codewords. Otherwise, the $s$-subset of codewords of
$X$ is said to be an {\em $(s,\ell)$-good} in~$X$.mA binary code $X$ is said to
be a cover-free $(s,\ell)$-code if the code $X$ does not contain $(s,\ell)$-bad
subsets. In this paper, we introduce a natural {\em probabilistic}
generalization of cover-free $(s,\ell)$-codes, namely: a binary code is said to
be an almost cover-free $(s,\ell)$-code if {\em almost all} $s$-subsets of its
codewords are $(s,\ell)$-good. We discuss the concept of almost cover-free
$(s,\ell)$-codes arising in combinatorial group testing problems connected with
the nonadaptive search of defective supersets (complexes). We develop a random
coding method based on the ensemble of binary constant weight codes to obtain
lower bounds on the capacity of such codes.
