Kolmogorov-Martin-Lof definition describes a random sequence as a sequence which satisfies all the laws of probability. This notion formalizes the intuitive physical idea that if an event has a probability 0, then this event cannot occur. Physicists, however, also believe that if an event has a very small probability, then it cannot occur. In our previous papers, we proposed a modification of the Kolmogorov-Martin-Lof definition which formalizes this idea as well. It turns out that our original definition is too general: e.g., it includes some clearly non-physical situations when the set of all random elements is a one-point set. In this paper, we propose a new definition which avoids such situations and is, thus, a more physically adequate description of randomness.