Asymptotic probability of encountering an accidental similarity in the presence of counter examples

Abstract

Previously, we found the generating function of an accidental resemblance to the b parent examples at m counter examples [1]. In this paper, we restrict ourself to the case where b = 2 with equal success probabilities p in Bernoulli trials for all attributes of each counter example and a success probability р² for each attribute in an accidental similarity. If the number n of attributes tends to infinity, the success probability is defined as \(p = \sqrt {a/n} \), and m = b√n counter examples are considered, then the probability of the occurrence of an accidental similarity avoiding these m counter examples tends to 1 − e^−a − ae^−a [1 − e^−b√a]..