In Probability and Statistics taught to mathematicians as a first introduction or to a non-mathematical audience, joint independence of events is introduced by requiring that the multiplication rule is satisfied. The following statement is usually tacitly assumed to hold (and, at best, intuitively motivated): If the n events E[subscript 1], E[subscript 2],..., E[subscript n] are jointly independent then any two events A and B built in finitely many steps from two disjoint subsets of {E[subscript 1], E[subscript 2],..., E[subscript n]} are also independent. The operations "union", "intersection" and "complementation" are permitted only when forming the events A and B. Here we examine this statement from the point of view of elementary probability theory. The approach described here is accessible also to "users" of probability theory and is believed to be novel. (Contains 4 notes and 1 figure.)