Safety analysis of high confidence systems requires guaranteed bounds on the probability of events of interest. Establishing the correctness of algorithms that compute such bounds is challenging. We address this problem in three steps. First, we use monadic transition systems (MTS) in the category of sets as a general framework for modeling discrete time systems. MTS can capture different types of system behaviors, but here we focus on a combination of non-deterministic and probabilistic behaviors that arises often when modeling complex systems. Second, we use the category of posets and monotonic maps as general setting to define and compare approximations. In particular, for the MTS of interest, we consider approximations of their configurations based on complete lattices of interval probabilities. Third, we obtain an algorithm that computes over-approximations of system configurations after a finite number of steps, by restricting to finite lattices.