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Imperfect Bernoulli trials arise when the outcome of a Bernoulli experiment is not known with certainty. In signal processing, we often need to estimate a probability of occurrence p of an event from imperfect Bernoulli trials. A typical example is the estimation of the probability of a signal being present in noisy data. In his famous essay, Bayes solved the same problem but for perfect trials. In this letter, a solution is provided for imperfect trials. It is shown that it includes Bayes' solution as a special case.