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Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 344–347 (Mi tvp853)  

Short Communications

Probabilities of complex events and the linear programming

S. A. Pirogov

Moscow

Abstract: 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 [1] 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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English version:
Theory of Probability and its Applications, 1968, 13:2, 329–332

Bibliographic databases:

Received: 27.12.1966

Citation: 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
\Bibitem{Pir68}
\by S.~A.~Pirogov
\paper Probabilities of complex events and the linear programming
\jour Teor. Veroyatnost. i Primenen.
\yr 1968
\vol 13
\issue 2
\pages 344--347
\mathnet{http://mi.mathnet.ru/tvp853}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=252026}
\zmath{https://zbmath.org/?q=an:0167.47001|0165.20502}
\transl
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 329--332
\crossref{https://doi.org/10.1137/1113039}


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