This paper revisits the building blocks of the most influential theories of decision under risk, and highlights the fact that the expectation criterion (together with some form of utility of money) weighs heavily on the way we study risk preferences. We propose an alternative criterion for comparing lotteries that accounts for the entire distribution function of lottery payoffs, as suggested by Allais (1953), giving birth to a new rank-dependent theory of decision under risk. Our theory is tested on experimental data and we show that the expectation assumption is indeed misleading. Risk preferences are better represented by a model in which individuals compare probability distributions rather than expectations. Furthermore, our approach allows us to build a theory or decision under risk that does not require the concept of utility of money to rationalize observed choices in risky environments.