In this paper we point out the lack of the classical definitions of stochastical independence (particularly with respect to events of 0 and 1 probability) and then we propose a definition that agrees with all the classical ones when the probabilities of the relevant events are both different from 0 and 1, but that is able to focus the actual stochastical independence also in these extreme cases. Therefore this definition avoids inconsistencies such as the possibility that an event $A$ can be at the same time stochastically independent and logically dependent on an event $B$. In a forthcoming paper we will deepen (in this context) the concept of {\sl conditional independence} (which is just sketched in the last section of the present paper) and we will deal also with the extension of these results to the general case of any (finite) number of events.