  This article presents an algebraic topology perspective on the problem of
finding a complete coverage probability of a one dimensional domain $X$ by a
random covering, and develops techniques applicable to the problem beyond the
one dimensional case. In particular we obtain a general formula for the chance
that a collection of finitely many compact connected random sets placed on $X$
has a union equal to $X$. The result is derived under certain topological
assumptions on the shape of the covering sets (the covering ought to be {\em
good}, which holds if the diameter of the covering elements does not exceed a
certain size), but no a priori requirements on their distribution. An upper
bound for the coverage probability is also obtained as a consequence of the
concentration inequality. The techniques rely on a formulation of the coverage
criteria in terms of the Euler characteristic of the nerve complex associated
to the random covering.
