A novel approach for establishing acceptability of risk is presented and illustrated by an application to the case of the light water reactors. The advantage of the method is that it takes into consideration the shape of the probability distribution function over consequences, instead of simply using the expected value of this distribution. An individual's attitude toward a certain consequence, such as loss of life, is described by a preference index under certainty, separately from his attitude toward uncertainty. The latter is quantified by the position of the individual in a risk space whose coordinates are related to the individual's attitude toward gambling and his degree of dislike of improbable but serious consequences.