The probability of triviality

Abstract.

Given a variety $\cal V$ of algebras, what is the probability that for an arbitrary identity p $\approx$ q the only algebra in $\cal V$ that satisfies p $\approx$ q is the trivial algebra? More generally, if $\cal W$ is a subvariety of $\cal V$ what is the probability that p $\approx$ q together with the identities of $\cal V$ forms an equational basis for $\cal W$? We consider these questions for various $\cal V$ and $\cal W$ and we provide criteria that allow for explicit determination of these probabilities.