Can You Play a Fair Game of Craps with a Loaded Pair of Dice?

Abstract

The parts-to-totals map sends the distributions of a set of independent random variables on a finite set of probability spaces to the total distribution of their sum on the product space. We study, in special cases modeled by dice, the geometry of the extension to complex pseudoprobabilities of this map, arithmetic questions about the existence of real points in certain fibers, and, when these exist, of strict points, having all coordinates in the unit interval.