Experimental studies have shown that the key behavioral assumption of expected utility theory, the so-called "independence axiom," tends to be systematically violated in practice. Such findings would lead us to question the empirical relevance of the large body of literature on the behavior of economic agents under uncertainty which uses expected utility analysis. The first purpose of this paper is to demonstrate that the basic concepts, tools, and results of expected utility analysis do not depend on the independence axiom, but may be derived from the much weaker assumption of smoothness of preferences over alternative probability distributions. The second purpose of the paper is to show that this approach may be used to construct a simple model of preferences which ties together a wide body of observed behavior toward risk, including the Friedman-Savage and Markowitz observations, and both the Allais and St. Petersburg Paradoxes.