A New Two-Urn Model

Abstract

We propose a two-urn model of Pólya type as follows. There are two urns, urn A and urn B. At the beginning, urn A contains r A red and w A white balls and urn B contains r B red and w B white balls. We first draw m balls from urn A and note their colors, say i red and m - i white balls. The balls are returned to urn A and bi red and b(m - i) white balls are added to urn B. Next, we draw ℓ balls from urn B and note their colors, say j red and ℓ - j white balls. The balls are returned to urn B and aj red and a(ℓ - j) white balls are added to urn A. Repeat the above action n times and let X n be the fraction of red balls in urn A and Y n the fraction of red balls in urn B. We first show that the expectations of X n and Y n have the same limit, and then use martingale theory to show that X n and Y n converge almost surely to the same limit.