Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A story of balls, randomness and PDEs   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.

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The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th