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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 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 $Â$ 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.

Full text: PDF file (300 kB)

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

Bibliographic databases:

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}