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This article considers exact and approximate confidence intervals for a binomial parameter p. Specifically, we will deal with the problem of determination of sample sizes which guarantee the probability that 100(1-α)% confidence intervals for p do not include pre-specified constants is greater than 1-β. It is shown here that the coverage probability of confidence intervals is not an increasing function of the sample size, which is a consequence of the discreteness of binomial distributions. Illustrative numerical examples and some theoretical consideration for such anomalous behavior are given. We also briefly treat the case that the initial guess of the true binomial parameter is vague.