The scenario optimization method developed in [5] is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design, that would otherwise be hard to tackle via standard deterministic techniques. In this note, we explore some further aspects of the scenario methodology, and present two results pertaining to the tightness of the sample complexity bounds. We also state a new theorem that enables the user to make a-priori probabilistic claims on the scenario solution, with one level of probability only.