An Experimental Mathematics Perspective on the Old, and still Open, Question of When To Stop?

  title={An Experimental Mathematics Perspective on the Old, and still Open, Question of When To Stop?},
  author={Luis A. Medina and Doron Zeilberger},
  journal={arXiv: Probability},
In a recent article in American Scientist, Theodore Hill described a coin-tossing game whose pay-off is the number of heads over the total number of throws. Suppose that at a given point during the game you have 5 heads and 3 tails, should you stop and get 5/8, or should you keep playing, hoping to get a better score? This is still an open problem. In the present article, we explore different strategies to this game from the Experimental Mathematics perspective. 
Rigorous Computer Analysis of the Chow–Robbins Game
A simple upper bound on the expected payoff in a given position is established, allowing efficient and rigorous computer analysis of positions early in the game and confirming that with 5 heads and 3 tails, stopping is optimal.
The Chow-Robbins Game and the Catalan Triangle
The payoff in the Chow-Robbins coin-flipping game is the proportion of heads when you stop, and the goal is to maximize its expected value, by knowing when to stop. We first establish the exact
On the Sn/n problem
Abstract The Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you
In this paper we revisit the stopping problem of Shepp’s Urn, extending the body of knowledge concerning moments of urn random variables by adapting a methodology encountered in Medina and
Stochastic Games a d Dynamic Programming
This note presents some historical examples of the link between the main areas of mathematics and the statistical theory. The research in statistics has an impact on algebra and analysis as much as
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always


Explicit Solutions to Some Problems of Optimal Stopping
Suppose we are allowed to view successively as many terms as we please of a sequence Xi, X2, * * * of independent random variables with common distribution F. We can decide to stop viewing at any
The Automatic Central Limit Theorems Generator (and Much More
Why I hate the Continuous and Love the Discrete I have always loved the discrete and hated the continuous. Perhaps it was the trauma of having to go through the usual curriculum of “rigorous” ,
Existence and properties of certain optimal stopping rules
The main purpose of this note is to prove the existence of optimal stopping rules for certain problems involving sums of independent, identically distributed random variables. A special case was
Disturbing the Dyson Conjecture (in a GOOD Way)
A case study in experimental yet rigorous mathematics by describing an algorithm that automatically conjectures and then automatically proves, closed-form expressions extending Dyson's celebrated constant-term conjecture.
Two Dimensional Directed Lattice Walks with Boundaries
We present general algorithms (fully implemented in Maple) for calculations of various quantities related to constrained directed walks for a general set of steps on the square lattice in two
Knowing When to Stop:
  • T. Hill
  • Physics
    The Best Writing on Mathematics 2010
  • 2021
Knowing when to stop
Walks reaching a line
  • DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05
Doron Zeilberger's Maple Packages and Programs
  • Doron Zeilberger's Maple Packages and Programs