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A Probabilistic Two-Pile Game
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Ho-Hon Leung

Department of Mathematical Sciences

United Arab Emirates University

Al Ain, 15551

United Arab Emirates

Thotsaporn "Aek" Thanatipanonda

Science Division

Mahidol University International College

Nakornpathom

Thailand

**Abstract:**

We consider a game with two piles in which two players take turns
adding *a* or *b* chips,
randomly and independently, to their respective piles.
Here *a*, *b* are not necessarily positive.
The player
who collects at least *n* chips first wins the game. We derive general
formulas for *p*_{n}, the probability of the second player winning the game by collecting *n* chips first, and
give the calculation for the cases {*a, b*} = {-1,1} and {-1,2}.
The latter case was considered by Wong and Xu. At the end,
we derive a general formula for
*p*_{n1, n2}, the
probability of the second player winning the game by
collecting *n*_{2} chips before the
first player collects *n*_{1} chips.

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(Concerned with sequences
A000108
A001764
A006013.)

Received March 22 2019;
revised versions received May 19 2019; June 29 2019.
Published in *Journal of Integer Sequences*,
August 19 2019.

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