Probabilities of Independent Choices Being Ordered

A simple proof of the following result of Baumert et al. is given. If A and B are chosen independently according to a probability distribution defined on subsets of an n element set, then the probability that A contains B is at least . It's shown to lead to a stronger conclusion, and to suggest definition of concepts of “partition number” and of “orthogonality of partitions into chains” as well as several conjectures.