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Discrete Mathematics 215 (2000) 147158 www.elsevier.com/locate/disc
 

Summary: Discrete Mathematics 215 (2000) 147­158
www.elsevier.com/locate/disc
A lower bound on the probability of a ˙nite union
of events
H. Kuai, F. Alajaji
, G. Takahara
Department of Mathematics and Statistics, Queen's University, Kingston, Ont., Canada K7L 3N6
Received 18 March 1998; revised 25 January 1999; accepted 17 May 1999
Abstract
A new lower bound on the probability P(A1 · · · AN ) is established in terms of only the
individual event probabilities P(Ai)'s and the pairwise event probabilities P(Ai Aj)'s. This
bound is shown to be always at least as good as two similar lower bounds: one by de Caen
(1997) and the other by Dawson and Sanko (1967). Numerical examples for the computation
of this inequality are also provided. c 2000 Elsevier Science B.V. All rights reserved.
1. Main results
Consider a ˙nite family of events A1; A2; : : : ; AN in a ˙nite 1
probability space ( ; P),
where N is a ˙xed positive integer. For each x , let p(x) , P({x}), and let the
degree of x -- denoted by deg(x) -- be the number of Ai's that contain x. De˙ne
Bi(k) , {x Ai: deg (x) = k}

  

Source: Alajaji, Fady - Department of Mathematics and Statistics, Queen's University (Kingston)

 

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