Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.8

## A Probabilistic Two-Pile Game

### 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 pn, 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 pn1, n2, the probability of the second player winning the game by collecting n2 chips before the first player collects n1 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.