In this paper we propose a model for decision making under risk that is capable of predicting empirically observed preference patterns that have been found to be incompatible with the expected utility model. The model departs from the classical expected utility model by allowing utilities to depend on the lottery. The dependence of utilities on the lottery being evaluated is achieved by restricting the utility measure to a convenient parametric family of functions. The idea then is to use each lottery to determine a specific parameter value thus characterizing the utility function for each particular lottery. The expected value of this lottery dependent utility function provides the overall measure of preference. The model retains the properties of transitivity, stochastic dominance, and continuity. It also permits types of analyses, such as exploitation of basic attitudes toward risk through risk aversion properties, that have been found useful in decision theory. The primary use of our model is in descriptive or predictive research and applications. For some decision makers who wish to retain the preference patterns that are incompatible with the substitution principle, even after the implications of their choices are made transparent, our model could be of prescriptive use as well.