Paired Bernoulli trials occur whenever an investigator records the presence of a particular characteristic at two sites on the same individual. While a study involving n subjects does not provide 2n independent pieces of information, neither does it provide only n pieces unless the characteristic must necessarily occur bilaterally. It is shown that the analysis of a model for sites in which the probability of occurrence at the second site given an occurrence at the first site is not functionally related to the probability of occurrence at the first site is equivalent to an analysis of counts of individuals grouped by whether the characteristic is absent, occurs unilaterally, or occurs bilaterally. It is shown that a test statistic proposed by Rosner (1982, Biometrics 38, 105-114) using a different model for such data can differ markedly from its corresponding likelihood-ratio statistic.