About seventeen years ago A. Cherny and P. Grigoriev obtained the following striking result: for any bounded random variables X and Y with the same distribution, there exists a sequence of sigma-algebras \(\mathcal {F}_n\) such that \(||X_n-Y||_\infty <\varepsilon _n\) , where \(X_1=\mathbb {E}[X|\mathcal {F}_1], \dots , X_n=\mathbb {E}[X_{n-1}|\mathcal {F}_n]\) and \(\varepsilon _n\) tends to zero as \(n \rightarrow \infty \) . We provide a transparent and deterministic interpretation of this statement, using a problem with 2n cups, n filled with water and n empty, connected with pipes, with the goal of transferring as much water as possible from full cups to empty, using only these connections. As a result, this interpretation, as a separate problem, is accessible to high school students and allows us to present a relatively elementary probability proof of the above theorem about conditional expectations, which is accessible to undergraduate students in probability/operations research. Moreover, our approach shows the optimal selection of such a sequence of sigma-algebras and allows us to obtain the exact first term of the asymptotic behavior of \(\varepsilon _n\) , when n tends to infinity, by using moment-generating functions. This term equals the amount of water left under the optimal transfer. Finally, we discuss two other mathematical problems where the "hydrostatic" interpretation also plays an important role.
