An Experimental Mathematics Perspective on the Old, and still Open, Question of When To Stop?
By Luis A. Medina and Doron Zeilberger
[Appeared in "Gems in Experimental Mathematics",
Contemporary Mathematics series (AMS) v. 517 (T. Amdeberhan, L. Medina, and V. Moll, eds.), 265274.]
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Written: June 23, 2009.
Ted Hill is not only a great
probability theorist, but also an amazing writer. We were really fascinated
by his recent American Scientist article, where he talked about the important problem, both in mathematics and in real life, of
when to stop? The present article can be viewed as a footnote to one of the problems discussed
by Hill.
Added Sept. 9, 2010: Read Julian Wiseman's
amazingly precise
estimate that explains why it is good to stop when you have 5 heads and 3 tails.
Added Jan. 23, 2012: If you are a skeptical mathematician,
who only believes in rigorous proof, you should read
Olle Haggstrom and Johan Wastlund's
beautiful article
that proves that if you have five heads and three tails, it is good
to stop, and more importantly, a completely new approach for
proving rigorously, using a computer (of course!), that
it is good to stop when it is indeed the case.
Added Aug. 2, 2012: the present article is mentioned in
Henk Tijms's delightful and insightful "semipopular"
article
Added Aug. 9, 2022: Read John Elton's beautiful article with precise asymptotics, using a "Catalan tree". It is a true masterpiece.
Maple and Mathematica Packages
Important: This article is accompanied by

Maple Packages

Mathematica Packages

Mathematica NoteBooks
Sample Input and Output
Sample Input and Output for the Maple package ChowRobbins

For the story of rewards and when to stop for all positions up to 10 cointosses
the input
gives the output.

For the (probably) sequence of stoppingtimes, β_{n} for n between 1 and 100
the input
gives the output.

For the story of comparative gambling up to 6 cointosses, with different ambitions
the input
gives the output.
Sample Input and Output for the Maple package STADJE

For the story of the probabilities of escaping the region y ≥ ax/b for a walker
in the square lattice with positive unit steps, starting at the origin, and
equally likely to go up or right, for
2 ≤ b ≤ a ≤ 5 and also the larger probabibility
of escaping y > ax/b (not counting the origin)
the input
gives the output.

For the first 20 terms of the sequences enumerating the number of walks,
from the origin to (n,n), staying in the region y ≥ ax/b ,
and their leading asymptotics,
for all 2 ≤ b ≤ a ≤ 6
the input
gives the output.

For the first 20 terms of the sequences enumerating the number of walks
from the origin to (an,bn), staying in the region y ≥ ax/b ,
and their leading asymptotics,
for all 2 ≤ b ≤ a ≤ 6
the input
gives the output.
Doron Zeilberger's List of Papers
Doron Zeilberger's Home Page