We propose an original use of techniques from random graph theory to find a Monadic $\sum_{1}^{1}$ (Minimal Scott without equality) sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics $\sum_{1}^{1}(\text{FO}^{2})$ and $\sum_{1}^{1}$ (Minimal Gödel without equality). Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the corresponding $\sum_{1}^{1}$ fragment. In addition, our counterexample can be viewed as a single explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known.