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The distribution of the number of successes in independent trials is shown to be bell-shaped of every order. The most likely number of successes is "almost uniquely" determined from the mean number and from the mean plus the largest and smallest probability of success on any trial. Bounds on the distribution function of the number of successes are obtained and extended to an infinite number of trials, including the Poisson distribution.
Stephen M. Samuels (Sun,) studied this question.