A calculation of the theoretical significance of matched bullets

Abstract:  The comparison and identification of bullets from the striations that appear on their surfaces, after they have been fired from a gun, have been practiced since the 1920s. Although the significance of the correspondences of these impression marks has been empirically justified, there is a conspicuous absence of any theoretical foundation for the likelihood. What is presented here is the derivation of the formulae for calculating the probability for the correspondence of the impression marks on a subject bullet to a random distribution of a similar number of impression marks on a suspect bullet of the same type. The approach to the calculation entails subdividing the impression marks into a series of individual lines having widths equal to the separation distance at which a misalignment of striations between the two bullets cannot be distinguished. This distance depends upon the resolution limit imposed by the microscope as well as by the visual acuity of the examiner. A calculation of the probabilities for finding pairs and triplets of consecutively matching lines on nonmatching bullets, by an examiner with normal perception using a microscope at 40× magnification, produces values that agree well with the empirical probabilities determined by Biasotti in the 1950s and when determined for larger consecutive sequences suggest that they are extremely unlikely to occur. The formulae can be used to determine the probabilities for the random occurrence of any sequence of striae and provide a straightforward way to quantitatively justify the significance of a specific match between any two bullets.