For a finite poset (X, R) and elements x, y of X, there is a well-established notion of the probability P(x<yR), namely the proportion of linear extensions of R in which x lies below y. We show how to extend this idea to a certain class of infinite posets, and show that fundamental correlation inequalities for finite posets hold in the infinite case. A well-known conjecture of Fredman for finite posets is that, if (X,R) is not a chain, there are elements x, y of X such that P(x<yR) . We show that this is false in the infinite case. Kahn and Saks proved that there are elements x, y of X such that <P(x<yR)<. We show that this result does hold in the infinite case, (except that we do not obtain strict inequalities), and is, numerically speaking, close to the best possible result.