Finite algebras whose equational probability spectrum contains a big gap

We classify the finite algebras $\mathbf{A}$ for which there is no value $p$ satisfying $\frac{1}{|A|}<p<1$ in the equational probability spectrum of $\mathbf{A}$. We will use the classification to determine which finite algebras have minimal equational probability spectrum. In our final section, we explore the relationship, for finite algebras, between the properties of (i)~having a finite equational probability spectrum and (ii)~being an abelian algebra.