We present a family of models of choice between behavioural alternatives with stochastic outcomes (risky choice) based on the effects of Weber's Law in memory. These models generalise and extend a model of risk sensitive foraging originally proposed by Reboreda & Kacelnik [(1991) Behav. Ecol. 2, 301-308], which yielded qualitative predictions (risk-aversion for amount of food and risk-proneness for delay to food). We now demonstrate how this approach can predict quantitatively the partial preferences between two alternative options with any mean and variance in their outcomes, and the certainty equivalent of an option consisting of any set of probabilistic outcomes. The approach is also relevant to the economics and psychology of risk sensitivity because it predicts risk aversion for any desirable outcome (such as monetary gains) and risk seeking for any undesirable gain (such as monetary losses). Our models are process-based rather than purely normative, and are based on linear expected utility as a function of expected outcomes. They do not account for all observed aspects of risky choice, but their descriptive performance betters that of existing functional models and requires fewer parameters.