Probabilities of complex events and the linear programming
S. A. Pirogov
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 $k<n$ be fixed integers and $0<p<1$. 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  by a different method. We reduce both the problems to finding of extrema of some linear forms on rather simple convex polyhedrons.
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Theory of Probability and its Applications, 1968, 13:2, 329–332
S. A. Pirogov, “Probabilities of complex events and the linear programming”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 344–347; Theory Probab. Appl., 13:2 (1968), 329–332
Citation in format AMSBIB
\paper Probabilities of complex events and the linear programming
\jour Teor. Veroyatnost. i Primenen.
\jour Theory Probab. Appl.
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