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This paper investigates in a stringent athematical formalism the parallel derivation of three grand probabilistic retrieval models: binary independent retrieval (BIR), Poisson model (PM), and language modelling (LM).The investigation has been motivated by a number of questions. Firstly, though sharing the same origin, namely the probability of relevance, the models differ with respect to event spaces. How can this be captured in a consistent notation, and can we relate the event spaces? Secondly, BIR and PM are closely related, but how does LM fit in? Thirdly, how are tf-idf and probabilistic models related? .The parallel investigation of the models leads to a number of formalised results: BIR and PM assume the collection to be a set of non-relevant documents, whereas LM assumes the collection to be a set of terms from relevant documents.PM can be viewed as a bridge connecting BIR and LM.A BIR-LM equivalence explains BIR as a special LM case.PM explains tf-idf, and both, BIR and LM probabilities express tf-idf in a dual way..