Is the property of being positively correlated transitive?

Suppose that X, Y, and Z are random variables and that X and Y are positively correlated and that Y and Z are likewise positively correlated. Does it follow that X and Z must be positively correlated? As we shall see by example, the answer is (perhaps surprisingly) no, We prove, though, that if the correlations are sufficiently close to 1, then X and Z must be positively correlated. We also prove a general inequality that relates the three correlations. The ideas should be accessible to, students in a first (postcalculus) course in probability and statistics.