Measuring the associativity of loops

In group theory folklore, there is a well-known theorem that the probability that two randomly uniformly chosen elements commute in a non-abelian group G cannot exceed 5/8. The bound is attained by the quaternion group Q_8. In this talk, we shall discuss a non-associative analog of the theorem. Namely, the probability of three randomly chosen elements associating is bounded by 43/64 in Moufang loops with nuclear commutators, with the bound attained by the Octonion loop O_16.