Event-B is a proof-based formal method used for discrete systems modelling. Several works have previously focused on the extension of Event-B for the description of probabilistic systems. In this paper, we propose an extension of Event-B that allows designing fully probabilistic systems as well as systems containing both probabilistic and non-deterministic choices. Compared to existing approaches which only focus on probabilistic assignments, our approach allows expressing probabilistic choices in all places where non-deterministic choices originally appear in a standard Event-B model: in the choice between enabled events, event-parameter values and in probabilistic assignments. Furthermore, we introduce novel and adapted proof-obligations for the consistency of such systems and introduce two key aspects to incremental design: probabilisation of existing events and refinement through the addition of new probabilistic events. In particular, we provide proof-obligations for the almost-certain convergence of a set of new events, which is a required property in order to prove standard refinement in this context. Finally, we propose a fully detailed case study, which we use throughout the paper to illustrate our new constructions.