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Fingerprints have long been used for person authentication. However, there is not enough scientific research to explain the probability that two fingerprints, which are impressions of different fingers, may be taken as the same one. In this paper, we propose a formal framework to estimate the fundamental algorithm independent error rate of fingerprint matching. Unlike a previous work, which assumes that there is no overlap between any two minutiae uncertainty areas and only measures minutiae's positions and orientations. In our model, we do not make this assumption and measure the relations, i.e., ridge counts between different minutiae as well as minutiae's positions and orientations. The error rates of fingerprint matching obtained by our approach is significantly lower than that of previously published research. Results are shown using NIST-4 fingerprint database. These results contribute toward making fingerprint matching a science and settling the legal challenges to fingerprints.