We consider in this paper an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn; a ball which has been drawn is immediately replaced into the urn. The numbers of balls with a given color are i.i.d. random variables with a specific probability distribution, for instance a Pareto or a Weibull distribution. We draw a small fraction $p\ll 1$ of the total number of balls. The basic problem addressed in this paper is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and Chen-Stein method, we establish bounds in total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean. We then show, under the assumption of heavy tailed distributions, that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. We finally establish bounds between the two distributions when each ball is drawn with probability $p$.