A basic experiment in probability theory is drawing without replacement from an urn filled with multiple balls of different colours. Clearly, it is physically impossible to overdraw, that is, to draw more balls from the urn than it contains. This paper demonstrates that overdrawing does make sense mathematically, once we allow signed distributions with negative probabilities. A new (conservative) extension of the familiar hypergeometric ('draw-and-delete') distribution is introduced that allows draws of arbitrary sizes, including overdraws. The underlying theory makes use of the dual basis functions of the Bernstein polynomials, which play a prominent role in computer graphics. Negative probabilities are treated systematically in the framework of categorical probability and the central role of datastructures such as multisets and monads is emphasised.

翻译：概率论中的一个基本实验是从装有多种颜色球的瓮中无放回地抽取。显然，物理上不可能出现超抽取，即从瓮中抽取的球数超过其实际容量。本文证明，一旦允许使用带负概率的符号分布，超抽取在数学上就具有合理性。本文对熟悉的超几何（"抽取-删除"）分布提出了一种新的（保守的）扩展，允许任意大小的抽取，包括超抽取。其基础理论利用了伯恩斯坦多项式的对偶基函数，这些函数在计算机图形学中扮演着重要角色。负概率在范畴概率框架下得到系统处理，并强调了多重集和单子等数据结构在其中的核心作用。