LetX1andX2be independent random variables with the same distribution functionFv, with the parameter v being unknown. A prediction interval forX2, given an observation ofX1, is constructed on the basis of the acceptance region of the two-sample problem with the hypothesisH0that the parameters v1and v2of the distributions ofX1andX2are the same against the one- or two-sided alternative. Although this simple idea can be found in the statistical literature, it deserves more attention in statistical textbooks. Applications are given to the Poisson and the binomial distribution. Besides the description of different procedures to compute exact prediction limits in the Poisson and binomial case, approximation formulas are given with an error term o(1) with corresponding reliable bounds for the approximation error, that seem to be new.