Let A1, A2, ..., Am and B1, B2, ..., Bn be two sequences of events on the same probability space. Let Xm and Yn, respectively, be the numbers of those Ai and Bj which occur. Improved upper bounds and lower bounds of yl,l = P(Xm(A)[greater-or-equal, slanted] 1, Yn(B) [greater-or-equal, slanted]...