Computing the probability of reaching a set of goal states G in a discrete-time Markov chain (DTMC) is a core task of probabilistic model checking. We can do so by directly computing the probability mass of the set of all finite paths from the initial state to G; however, when refining counterexamples, it is also interesting to compute the probability mass of subsets of paths. This can be achieved by splitting the computation into path abstractions that calculate “local” reachability probabilities as shown by Ábrahám et al. in 2010. In this paper, we complete and extend their work: We prove that splitting the computation into path abstractions indeed yields the same result as the direct approach, and that the splitting does not need to follow the SCC structure. In particular, we prove that path abstraction can be performed along any finite sequence of sets of non-goal states. Our proofs proceed in a novel way by interpreting the DTMC as a structure on the free monoid on its state space, which makes them clean and concise. Additionally, we provide a compact reference implementation of path abstraction in PARI/GP.