The Rate of Convergence to the Limit of the Probability of Encountering an Accidental Similarity in the Presence of Counter Examples

Abstract

This paper refines the main result of [1], where the limit \( - {e^{ - a}} - a{e^{ - a}}\left[ {1 - {e^{ - c\sqrt a }}} \right]\) was proved for the probability of encountering an accidental similarity between two parent examples without \(m = c\sqrt n \) counter examples if each parent example and counter example is described by a series of \(\sqrt n \) independent Bernoulli trials with success probability \(p = \sqrt {a/n} \). In this paper, we prove that the rate of convergence to the limit is proportional to \({n^{\frac{1}{2}}}\).