Experimental evidence suggests that choice behaviour has a stochastic element. Much of this evidence is based on studying choices between lotteries – choice under risk. Binary choice probabilities admit a strong utility representation (SUR) if there is a utility function such that the probability of choosing option A over option B is a strictly increasing function of the utility difference between A and B. Debreu (1958) obtained a simple set of sufficient conditions on binary choice probabilities for the existence of a SUR. More recently, Dagsvik (2008) considered binary choices between lotteries and provided axiomatic foundations for a SUR in which the underlying utility function is linear (i.e., conforms with expected utility). Our paper strengthens and generalises Dagsvik’s result. We show that one of Dagsvik’s axioms can be weakened, and we extend his analysis to encompass choices between uncertain prospects, as well as various non-linear specifications of utility.