Probabilities of complex events and the linear programming

S. A. Pirogov

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 344-347

Résumé

The following two extremal problems are solved in the paper by methods of the linear programming. A. Let $\varepsilon\le1$ be a fixed positive number. Call the distance $\rho(A,B)$ between two events $A$ and $В$ the measure of their symmetrical difference. How many events with mutual distances not less than $\varepsilon$ can be constructed? B. Let $k$ be fixed integers and $0$. For what $c$ is it possible to choose $k$ events with the probability of their intersection not less than $c$ from every $n$ events with the probabilities not less than $p$? The second problem was investigated in [1] by a different method. We reduce both the problems to finding of extrema of some linear forms on rather simple convex polyhedrons.