How Many Rounds Should You Expect in Urn Solitaire?

By Shalosh B. Ekhad and Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org )

Written: Jan. 4, 2018

A certain sampling process, concerning an urn with balls of two colors, proposed in 1965 by B.E. Oakley and R.L. Perry, and discussed by Peter Winkler and Martin Gardner, that has an extremely simple answer for the probability, namely the constant function 1/2, has a far more complicated expected duration, that we discover and sketch the proof of. So unlike, for example, the classical gambler's ruin problem, for which both `probability of winning' and `expected duration' have very simple expressions, in this case the expected number of rounds is extremely complicated, and beyond the scope of humans.

# Maple packages

• UrnSolitaire.txt, a Maple package to study Urn Solitaire (with balls of two colors)

The following package is also needed in the same diretory

• GuessHolo2.txt, a Maple package needed for the previous package. Should be in the same directory

• Urn.txt, another package to study a simple drawing of balls from an urn with two colors.

# Sample Input and Output Files for the Maple package UrnSolitaire.txt,

• If you want to see linear recurrences satisfied by the discrete function E(m,n), the expected number of rounds in Oakley-Perry, Urn Solitaire with m White balls and n Black balls, in both m and n directions, as well as the linear recurrence satisfied by the diagonal, E(n,n),

the input file generates the output file.

• If you want to see linear recurrences for the diagonal E(n,n), for the expected number of rounds in Simple Urn (with never replacing balls) Solitaire with n White balls and n Black balls, as well as a recurrence for the prob. generating function of the random variable "number of rounds" and a linear recurrence for the variance

the input file generates the output file.

# Sample Input and Output Files for the Maple package Urn.txt

• If you want to see the holonomic representation of the probability generating function for the random variable "number of color changes" when you draw balls from an urn with m White balls and n Black Balls until it is empty

the input file generates the output file.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger