Consider the probability space $(S,F,P)$ and the sequence of events Aᵢ ∈ F,i = 1,2, ⋯ ,n; the problem is to evaluate, under limited information, the probability of an event A defined through a finite number of set operations among A₁ ,A₂ , ⋯ ,Aₙ. In this paper P( ∪ i = 1ⁿ Aᵢ ) and P( ∩ i = 1ⁿ Aᵢ ) are considered. In the case where P(Aᵢ ), and P(Aᵢ ∩ Aⱼ ),1 i j n, are assumed known, then upper and lower bounds are given as functions of the known probabilities. Indicator random variables are used and linear programming techniques are employed, numerical examples are presented and applications in probability and statistics are given.
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Kounias et al. (1976) studied this question.
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