### 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 |
---|---|

Pages (from-to) | 590-597 |

Number of pages | 8 |

Journal | Journal of Applied Probability |

Volume | 51 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Jun |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of Applied Probability*,

*51*(2), 590-597. https://doi.org/10.1239/jap/1402578645

}

*Journal of Applied Probability*, vol. 51, no. 2, pp. 590-597. https://doi.org/10.1239/jap/1402578645

**A new two-urn model.** / Chen, May Ru; Hsiau, Shoou Ren; Yang, Ting Hsin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A new two-urn model

AU - Chen, May Ru

AU - Hsiau, Shoou Ren

AU - Yang, Ting Hsin

PY - 2014/6

Y1 - 2014/6

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.

UR - http://www.scopus.com/inward/record.url?scp=84903991030&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903991030&partnerID=8YFLogxK

U2 - 10.1239/jap/1402578645

DO - 10.1239/jap/1402578645

M3 - Article

AN - SCOPUS:84903991030

VL - 51

SP - 590

EP - 597

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 2

ER -