Symbol-Crunching With the Gambler's Ruin Problem

By Doron Zeilberger

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[Appeared in "Tapas in Experimental Mathematics", Contemporary Mathematics, v. 457 (2008), 285-292, (Tewodros Amdeberhan and Victor Moll, eds.)].
Written: Sept. 15, 2006.
Dedicated to my brother Gil Zeilberger
The Gambler's Ruin problem is an apt metaphor for life. A tiny advantage gets amplified enormously in an artificial way, and just being a tiny bit luckier (or smarter or stronger or better-looking) gets you very far in the long run. That's why my good friend Geroge appears to be a so much better backgammon player than his wife Martha, even though he is only slightly better.

But today, thanks to our silicon brethern, we can answer many more questions, exactly and symbolically, than Abraham De Moivre and James Bernoulli could answer three hundred years ago.

Important: This article is accompanied by Maple package RUIN that automatically computes, symbolically (and numerically), statistical quantities related to the Gambler's Ruin problem, and also has simulation programs.

Sample Input and Output
Added Jan. 12, 2007: This work was mentioned in Sharon Begly's Science Journal Column that was originally published, today, in The Wall Street Journal, page B1.
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