  We show that there is a positive constant $\delta < 1$ such that the
probability of satisfying either the $2$-Engel identity $[X_1, X_2, X_2] = 1$
or the metabelian identity $[[X_1, X_2], [X_3, X_4]] = 1$ in a finite group is
either $1$ or at most $\delta$.
