A graph is randomly matchable if every matching of the graph is contained in a perfect matching. We generalize this notion and say that a graph G is randomly H-coverable if every set of independent subgraphs, each isomorphic to H, that does not cover the vertices of G can be extended to a larger set of independent copies of H. Various problems are considered for the situation where H is a path. In particular, we characterize the graphs that are randomly \(P_ 3\)-coverable.