A New Two-Urn Model. (June 2014)

• Record Type: Journal Article
• Authors: Chen, May-Ru; Hsiau, Shoou-Ren; Yang, Ting-Hsin
• Is Part Of: Journal of applied probability. Volume 51:Number 2(2014)
• Page Start: 590
• Page End: 597
• Publication Date: 2014-06
• Subjects: Two-urn model, -- martingale; 60F15, -- 60G42, -- 60E05; 519.2
• DOI: 10.1239/jap/1402578645
• Languages: English
• ISSNs: 0021-9002

Abstract

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.