Overdrawing Urns using Categories of Signed Probabilities

For drawing (multiple) coloured balls from a statistical urn, we distinguish three well-known modes: 1. hypergeometric or draw-and-delete, which is drawing a ball from the urn without replacement, so that the urn shrinks; 2. multinomial or draw-and-replace: drawing with replacement, so that the urn remains the same; 3. Pólya or draw-and-duplicate, which is drawing a ball from the urn and replacing it together with an additional ball of the same colour, so that the urn grows. Multinomial and Pólya draws may be of arbitrary size, but hypergeometric draws are limited in size by the number of balls in the urn. In this paper we lift this limitation and allow hypergeometric draws of arbitrary size, including'overdraws', containing more balls than in the urn. Physically this is strange, but, as will show, mathematically it makes sense once we allow negative probabilities. Negative probabilities have emerged in quantum physics (e.g. in double slit experiments) and have been discussed in the work of famous physicists like Wigner, Dirac, and Feynman (see e.g.