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Improved upper bounds of probability

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Abstract

Let A_1, A_2, cdots , A_n be a sequence of eventson a given probability space. Let m_n be the number of thoseA_j's which occur. Upper bounds of P(m_n geq 1) are obtainedby means of probability of consecutive terms which reduce thenumber of terms in binomial moments S_{2,n}, S_{3,n} andS_{4,n}.

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