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.