On Buffon Machines and Numbers
Philippe Flajolet, Maryse Pelletier, Michèle Soria · HAL (Le Centre pour la Communication Scientifique Directe) · 2009
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2011 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)On Buffon Machines and NumbersPhilippe Flajolet, Maryse Pelletier, and Michèle SoriaPhilippe Flajolet, Maryse Pelletier, and Michèle Soriapp.172 - 183Chapter DOI:https://doi.org/10.1137/1.9781611973082.15PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically “computes” the number 2/π ≐ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as . Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function. Previous chapter Next chapter RelatedDetails Published:2011ISBN:978-0-89871-993-2eISBN:978-1-61197-308-2 https://doi.org/10.1137/1.9781611973082Book Series Name:ProceedingsBook Code:PR138Book Pages:xviii-1788