Let $A_1$,A$_2$…, Aulcorner and B$_1$,B$_2$…, Bulcorner be two sequences of events on the same probability space. Let X= Xulcorner(A) and Y-Yulcorner)(B), repectively, by the number of those Aulcorner and Bulcorner which oc-cur. We establish bivariate lower bounds on the distribution P(X$geq$1, Y, $geq$1)and P(X$geq$i , $Ygeq$j)by linear combinations of the bino-mial moments Sulcorner, ulcorner, 1$leq$i<m, 1$leq$j<n, which extend and refine bivariate Bonferroni-type lower bounds given by Chen and Seneta(1995) and Lee (1997)