# A new two-urn model

May Ru Chen, Shoou Ren Hsiau, Ting Hsin Yang

Research output: Contribution to journalArticle

4 Citations (Scopus)

### 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 rA red and wA white balls and urn B contains rB red and wB white balls. We first draw m balls from urn A and note their colors, say i red and mi white balls. The balls are returned to urn A and bi red and b(mi) white balls are added to urn B. Next, we draw - balls from urn B and note their colors, say j red and -j white balls. The balls are returned to urn B and aj red and a(-j) white balls are added to urn A. Repeat the above action n times and let Xn be the fraction of red balls in urn A and Yn the fraction of red balls in urn B. We first show that the expectations of Xn and Yn have the same limit, and then use martingale theory to show that Xn and Yn converge almost surely to the same limit.

Original language English 590-597 8 Journal of Applied Probability 51 2 https://doi.org/10.1239/jap/1402578645 Published - 2014 Jun

Urn model
Ball
Martingale

### All Science Journal Classification (ASJC) codes

• Statistics and Probability
• Mathematics(all)
• Statistics, Probability and Uncertainty

### Cite this

Chen, May Ru ; Hsiau, Shoou Ren ; Yang, Ting Hsin. / A new two-urn model. In: Journal of Applied Probability. 2014 ; Vol. 51, No. 2. pp. 590-597.
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A new two-urn model. / Chen, May Ru; Hsiau, Shoou Ren; Yang, Ting Hsin.

In: Journal of Applied Probability, Vol. 51, No. 2, 06.2014, p. 590-597.

Research output: Contribution to journalArticle

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AU - Hsiau, Shoou Ren

AU - Yang, Ting Hsin

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N2 - 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 rA red and wA white balls and urn B contains rB red and wB white balls. We first draw m balls from urn A and note their colors, say i red and mi white balls. The balls are returned to urn A and bi red and b(mi) white balls are added to urn B. Next, we draw - balls from urn B and note their colors, say j red and -j white balls. The balls are returned to urn B and aj red and a(-j) white balls are added to urn A. Repeat the above action n times and let Xn be the fraction of red balls in urn A and Yn the fraction of red balls in urn B. We first show that the expectations of Xn and Yn have the same limit, and then use martingale theory to show that Xn and Yn converge almost surely to the same limit.

AB - 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 rA red and wA white balls and urn B contains rB red and wB white balls. We first draw m balls from urn A and note their colors, say i red and mi white balls. The balls are returned to urn A and bi red and b(mi) white balls are added to urn B. Next, we draw - balls from urn B and note their colors, say j red and -j white balls. The balls are returned to urn B and aj red and a(-j) white balls are added to urn A. Repeat the above action n times and let Xn be the fraction of red balls in urn A and Yn the fraction of red balls in urn B. We first show that the expectations of Xn and Yn have the same limit, and then use martingale theory to show that Xn and Yn converge almost surely to the same limit.

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