We present a novel technique to analyze the bounded reachability probability problem for large Markov chains. The essential idea is to incrementally search for sets of paths that lead to the goal region and to choose the sets in a way that allows us to easily determine the probability mass they represent. To effectively analyze the system dynamics using an SMT solver, we employ a finite-precision abstraction on the Markov chain and a custom quantifier elimination strategy. Through experimental evaluation on PRISM benchmark models we demonstrate the feasibility of the approach on models that are out of reach for previous methods.