Many biochemical systems at the cellular level can be described by Markov jump processes on large discrete state spaces, with the time evolution of the associated probability distribution given by the chemical master equation (CME). The solution of the CME provides an accurate model for the dynamics of such systems, and additional features can be extracted by investigating the transition mechanisms between certain states of interest. This is particularly true for systems exhibiting metastable dynamics, where rare events induce transitions between subsets of the state space. However, computing approximations of the CME for high-dimensional spaces or gathering sufficient statistics on rare events are both non trivial problems. Here, we present a numerical approach based on wavelet compression that can efficiently compute the stationary probability distribution and committor probabilities - objects that give a measure of the progress of the transition between two chosen sub-states. The methods are illustrated on multi-dimensional models of genetic toggle switches defined on large state spaces.