Summary: A simple coin-tossing game leads to the study of real sequences, \({\mathbf x}\) and \({\mathbf y}\), with the remarkable property that the products of their differences are majorized by the differences of their product. Such sequences are said to form a double-dipping pair. The following conjecture arises when the game is governed by sampling balls from urns: if \(A\), \(a\), \(B\), \(b\) are non-negative integers with \(A\geq a\) and \(B\geq b\), then \({\mathbf x}\) and \({\mathbf y}\) form a double-dipping pair, where \(x_k= (\begin{smallmatrix} A- k\\ a\end{smallmatrix})\), \(y_k= (\begin{smallmatrix} B- k\\ b\end{smallmatrix})\), \(k= 0,1,2,\dots\). The conjecture is proved here under the additional restriction \(- b\leq A- B\leq a\). The proof is based, in part, upon the observation that the polynomials, \(x\to \sum_k (\begin{smallmatrix} a\\ k\end{smallmatrix})(\begin{smallmatrix} b\\ n- k\end{smallmatrix})(1- x)^k\), \(n\leq a+ b\), have reciprocals, all of whose Taylor coefficients (about \(x= 0\)) are non-negative.