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arxiv: 1906.09830 · v1 · pith:RPBNZWN7new · submitted 2019-06-24 · 🧮 math.PR · hep-th· math-ph· math.AP· math.CO· math.MP

A story of balls, randomness and PDEs

Anastasios Taliotis This is my paper

Pith reviewed 2026-05-25 17:18 UTC · model grok-4.3

classification 🧮 math.PR hep-thmath-phmath.APmath.COmath.MP
keywords probabilityurn modelspartial differential equationsgenerating functionalsclosed form solutionsrecursionsball removal process
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The pith

Solutions of a 2D PDE derived from a ball-removal recursion yield closed-form probabilities for remaining balls.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers a process of random ball removals from a box with r red and w white balls under specific rules that continue until all red balls are gone, and asks for the probability that exactly k white balls remain. Using the law of total probability, a recursion for these probabilities is derived and then recast as a linear inhomogeneous 2D partial differential equation with boundary conditions. The PDE is solved analytically to produce generating functionals in r, w and k, from which explicit probability formulas are extracted. The same approach applies to the dual problem of counting remaining red balls when the process ends upon removal of all white balls. A sympathetic reader would care because the method reverses the usual direction of solving difference equations via series expansions and instead imports techniques from differential equations into probability.

Core claim

A recursion for the probability that k white balls remain after all red balls are removed is transformed into a linear inhomogeneous 2D PDE. Analytical solutions of the PDE supply generating functionals that allow the probabilities to be written in closed form for arbitrary r, w and k. The same procedure gives the probabilities for k red balls remaining when the process stops upon removal of all white balls.

What carries the argument

The linear inhomogeneous 2D PDE with boundary conditions obtained from the total probability law, solved analytically to produce generating functionals of the probabilities.

If this is right

  • The probabilities for any r, w, k are obtained in closed form.
  • The formulas reproduce existing results in the literature.
  • The method is generic and adaptable to other probability problems.
  • Generating functionals are provided as explicit functions of r, w and k.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This technique could extend to other urn models with different removal rules.
  • The PDE might correspond to equations arising in other contexts such as diffusion or growth processes.
  • Closed forms could enable computation of moments or asymptotic behaviors for large r and w.

Load-bearing premise

The recursion obtained via the law of total probability can be transformed into a linear inhomogeneous 2D PDE with suitable boundary conditions whose solutions correctly encode the original probabilities.

What would settle it

For small values such as r=2 and w=2, enumerating all possible sequences of removals and checking whether the probability for each k matches the expression derived from the PDE solution.

Figures

Figures reproduced from arXiv: 1906.09830 by Anastasios Taliotis.

Figure 1
Figure 1. Figure 1: Problem 1, Rule III. The simulated Vs the theoretical results for 1 000 000 simulations [PITH_FULL_IMAGE:figures/full_fig_p041_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Problem 1, Rule IV. The simulated Vs the theoretical results for 1 000 000 simulations [PITH_FULL_IMAGE:figures/full_fig_p043_2.png] view at source ↗
read the original abstract

Several differential equations usually appearing in mathematical physics are solved through a power series expansion, which reduces in solving difference equations. In this paper a probability problem is presented whose solution follows a completely reversed but systematic approach. Hence, this work is about illustrating how complex probability problems could be tackled with the more powerful techniques of a better studied and well understood field, that of differential equations. The problem is defined as follows: Inside a box containing r red and w white balls random removals occur. The balls are removed successively according to the three following rules. Rule I: If a white ball is chosen it is immediately discarded. If a red ball is chosen, it is placed back into the box and a new ball is randomly chosen. The second ball is then removed irrespective of the color. Rule II: Once one ball is removed, the game continues from Rule I. Rule III: The game ends once all the red balls are removed. The question posed is the determination of the probability that k white balls remain where k = 0, 1, 2, ..., w. Ending the game once all the white balls are removed, a second question is the determination of the probability that k red balls remain where k = 0, 1, 2, ..., r. While inductive solutions are possible, the current approach demonstrates a different and algorithmic route. In particular, the law of total probability yields a recursion that is transformed into a linear inhomogeneous 2D PDE, with suitable boundary conditions. The PDE solutions, which are found analytically, provide the generating functionals of the required probabilities as a function of r, w and k. Using the functionals, the probability formulas for any r, w and k are finally obtained in a closed form. Reproducing existing results of the literature this method is quite generic and adaptable to a large class of problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a probability problem involving successive random removals of red and white balls under three explicit rules, ending when all red balls are removed. Using the law of total probability, it derives a recursion for the probability that exactly k white balls remain (and symmetrically for red balls remaining). This recursion is transformed into a linear inhomogeneous 2D PDE with suitable boundary conditions; the PDE is solved analytically to produce generating functionals in r, w, and k, from which closed-form probability expressions are extracted for arbitrary parameters. The approach is claimed to reproduce existing literature results and to be adaptable to a broader class of problems.

Significance. If the recursion-to-PDE step and boundary conditions are valid, the work supplies a systematic, analytic route to closed-form solutions for this class of combinatorial probabilities by importing techniques from PDEs, reversing the more common direction of reducing PDEs to recursions via series. The explicit construction of generating functionals and the reproduction of known results are concrete strengths that would support the claimed generality.

major comments (2)
  1. [PDE transformation and solution section] The central claim that the PDE solutions 'correctly encode the original probabilities' rests on the unverified assertion that the linear inhomogeneous 2D PDE with the chosen boundary conditions reproduces the discrete base cases (r=0, w=0, or k larger than remaining balls) and satisfies the original recursion. No explicit check—e.g., substitution of the extracted coefficients back into the recursion or direct comparison for small integer values—is provided.
  2. [Derivation of the PDE from the recursion] The manuscript treats r and w as continuous variables inside the PDE while the original process is strictly discrete; the change-of-variables step that justifies this extension and the precise form of the inhomogeneous term are not derived in sufficient detail to confirm that probability mass is preserved and no extraneous solutions are admitted.
minor comments (2)
  1. Notation for the generating functionals and the two probability problems (white balls remaining vs. red balls remaining) should be introduced with explicit symbols early in the text to improve readability.
  2. [Abstract] The abstract states that the method 'reproduces existing results of the literature' but does not cite the specific prior formulas or papers being matched; adding these references would strengthen the validation claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will incorporate additional verification and derivation details in the revised manuscript.

read point-by-point responses

  1. Referee: [PDE transformation and solution section] The central claim that the PDE solutions 'correctly encode the original probabilities' rests on the unverified assertion that the linear inhomogeneous 2D PDE with the chosen boundary conditions reproduces the discrete base cases (r=0, w=0, or k larger than remaining balls) and satisfies the original recursion. No explicit check—e.g., substitution of the extracted coefficients back into the recursion or direct comparison for small integer values—is provided.

    Authors: We agree that an explicit verification step would strengthen the presentation. Although the manuscript already notes that the derived closed forms reproduce known results from the literature, this provides only indirect confirmation. In the revision we will add a dedicated subsection performing direct substitution of the extracted probability expressions back into the original recursion for several small integer triples (r,w,k) and confirming agreement with the base cases r=0, w=0 and k>w. revision: yes

  2. Referee: [Derivation of the PDE from the recursion] The manuscript treats r and w as continuous variables inside the PDE while the original process is strictly discrete; the change-of-variables step that justifies this extension and the precise form of the inhomogeneous term are not derived in sufficient detail to confirm that probability mass is preserved and no extraneous solutions are admitted.

    Authors: We acknowledge that the transition from the discrete recursion to the continuous PDE is presented too concisely. The revision will expand the relevant section to include: (i) the explicit generating-function or interpolation argument that extends the recursion to real r,w; (ii) the precise origin of the inhomogeneous term; and (iii) a short argument showing that the analytic solution, when restricted to positive integers, recovers a probability distribution (i.e., sums to one and matches the recursion). revision: yes

Circularity Check

0 steps flagged

No circularity; derivation proceeds from recursion to independent PDE solution

full rationale

The paper begins with the explicit removal rules and applies the law of total probability to obtain a recursion relating the probabilities for different (r, w, k). This recursion is then converted into a linear inhomogeneous 2D PDE together with boundary conditions that encode the discrete base cases (r=0, w=0, k exceeding remaining balls). The PDE is solved analytically to produce generating functionals, from which closed-form probabilities are extracted by coefficient extraction. The resulting formulas are stated to reproduce known results in the literature, providing external verification. No step equates a derived quantity to a fitted parameter by construction, renames an input as a prediction, or relies on a load-bearing self-citation whose content is itself unverified. The central claim therefore retains independent mathematical content outside its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; the ledger records the two central modeling steps named in the abstract. No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption The law of total probability yields a recursion that captures the removal rules.
    Invoked to obtain the recursion from the three rules.
  • ad hoc to paper The recursion can be rewritten as a linear inhomogeneous 2D PDE with suitable boundary conditions.
    The key modeling step that enables the analytic solution.

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    Mullican, M

    R. Mullican, M. Leeuwen https://math.stackexchange.com/questions/153611/ probability-that-the-last-ball-is-white

    work page

  2. [2]

    A Sampling Process

    B. E. Oakley and R. L. Perry “A Sampling Process”. The Mathematical Gazette, Vol. 49, No. 367 (Feb., 1965), pp. 42-44

    work page 1965

  3. [3]

    Wolfram Research http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric3F2/03/ 02/04/0001/ 41

    work page