Authors: Bart Jacobs ¹

• 1 Institute for Computing and Information Sciences, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

Urn models play an important role to express various basic ideas in probability theory. Here we extend this urn model with tubes. An urn contains coloured balls, which can be drawn with probabilities proportional to the numbers of balls of each colour. For each colour a tube is assumed. These tubes have different sizes (lengths). The idea is that after drawing a ball from the urn it is dropped in the urn of the corresponding colour. We consider two associated probability distributions. The first-full distribution on colours gives for each colour the probability that the corresponding tube is full first, before any of the other tubes. The negative distribution on natural numbers captures for a number k the probability that all tubes are full for the first time after k draws. This paper uses multisets to systematically describe these first-full and negative distributions in the urns & tubes setting, in fully multivariate form, for all three standard drawing modes (multinomial, hypergeometric, and Polya).