On Sampling without replacement and OK-Corral urn models

Abstract: In this work we discuss two urn models with general weight sequences $(A,B)$ associated to them, $A=(\alpha_n){n\in\N}$ and $B=(\beta_m){m\in\N}$, generalizing two well known P\'olya-Eggenberger urn models, namely the so-called sampling without replacement urn model and the OK Corral urn model. We derive simple explicit expressions for the distribution of the number of white balls, when all black have been drawn, and obtain as a byproduct the corresponding results for the P\'olya-Eggenberger urn models. Moreover, we show that the sampling without replacement urn models and the OK Corral urn models with general weights are dual to each other in a certain sense. We discuss extensions to higher dimensional sampling without replacement and OK Corral urn models, respectively, where we also obtain explicit results for the probability mass functions, and also an analog of the dualitiy relation. Finally, we derive limit laws for a special choice of the weight sequences.