The probability of triviality
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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.
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