Urns & Tubes

TL;DR

This paper introduces an extended urn model with tubes, analyzing the probabilities of tubes filling first and the distribution of draws needed for all tubes to fill, using multisets across various drawing modes.

Contribution

It systematically describes first-full and negative distributions in urns & tubes models using multisets for multinomial, hypergeometric, and Polya drawing modes.

Findings

Formal multivariate descriptions of distributions

Application across multiple drawing modes

Enhanced understanding of urns & tubes dynamics

Abstract

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).