The aim of the paper is to extend the first author's earlier results, and to complement and reformulate closely related work, by analyzing the relationship between three fundamental ways of mathematically representing judgements and preferences. The paper significantly extends the following three approaches considered in the paper to modelling the preference, judgement, and choice behaviour of a population: (1) The development of the random junction approach, which has often been used implicitly in the random utility literature, and is now made explicit. This is an effective extension to the existing literature on probabilistic choice models in general, and can be viewed as very similar to the random relation approach. Another useful feature of the random function approach is that it enables to state probabilistic versions of some results from the (deterministic) representational theory of measurement in a natural way. (2) The second approach involves the development of extended results from valued \(m\)-ary relations to relational structures and finite-valued relations. (3) The third important extension consists of the two illustrative examples that are exposed in detail. The obtained theoretical results are comprehensibly applied to probabilistic models of magnitude estimation, probabilistic extensive measurement, probabilistic metric spaces, and (binary) subjective expected utility.