<p>The paper considers a family of generalizations of Chebyshev’s inequality. The key focus is on asymmetric cases in which the miss region of interest is not symmetric with respect to the mathematical expectation. The well-known results of Cantelli and Selberg are examined as natural extensions of the classical Chebyshev’s bound. A generalized approach based on linear programming methods is proposed, which provides a clear geometric interpretation and allows concise derivations of the corresponding inequalities. This approach demonstrates the universality of linear programming in formulating probabilistic bounds under different types of constraints.</p>