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"Combination of events" made easy

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Part of the Lecture Notes in Mathematics book series (LNM,volume 452)

Abstract

The probability that precisely r out of n overlapping events shall occur is \(S_r - \left( {\begin{array}{*{20}c}{r + 1} \\r \\\end{array} } \right)S_{r + 1} + \left( {\begin{array}{*{20}c}{r + 2} \\r \\\end{array} } \right)S_{r + 2} - \ldots \pm \left( {\begin{array}{*{20}c}n \\r \\\end{array} } \right)S_n\). This result and the related result for the ‘tail’ probability that more than r shall occur are quite standard.

Here we give a proof of this group of results which could in some cases have advantages over the currently accepted proofs. Our proofs make no explicit use of special identities relating to binomial coefficients. We go on to consider events of two kinds — type E and type F — and establish the similar results for the probability that precisely r of type E and precisely s of type F shall happen, and for the corresponding tail probabilities.

A proof in the same spirit — no technical knowledge of binomial identities — is given for Bonferroni’s inequalities.

Keywords

  • Indicator Function
  • Hybrid Probability
  • Tail Probability
  • Special Identity
  • Alternative Sequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. W. Feller, An introduction to probability theory and its applications, Vol. I (New York, John Wiley, 3rd ed. 1967)

    MATH  Google Scholar 

  2. D.A.S. Fraser, Statistics: an introduction. (New York, John Wiley, 1958)

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© 1975 Springer-Verlag

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Finucan, H.M. (1975). "Combination of events" made easy. In: Street, A.P., Wallis, W.D. (eds) Combinatorial Mathematics III. Lecture Notes in Mathematics, vol 452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069550

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  • DOI: https://doi.org/10.1007/BFb0069550

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07154-9

  • Online ISBN: 978-3-540-37482-4

  • eBook Packages: Springer Book Archive