Haenni et al. (2011) motivate imprecise probabilities by showing how they can arise from precise probability judgements. That is, if you have a precise probability for \(X\) and a precise probability for \(Y\), then you can put bounds on \(p(X\cap Y)\) and \(p(X \cup Y)\), even if you don’t know how \(X\) and \(Y\) are related. These bounds give you intervals of possible probability values for the compound events.