On Buffon Machines and Numbers

TL;DR

This paper introduces Buffon machines, simple probabilistic devices using coin flips, capable of generating various complex distributions and mathematical constants through elegant, human-accessible experiments.

Contribution

It develops a framework for constructing Buffon machines that produce a wide range of distributions and mathematical constants using only basic probabilistic mechanisms.

Findings

Buffon machines can generate geometric, Poisson, and logarithmic-series distributions.

They can produce probabilities involving constants like exp(-1), log 2, and sqrt(3).

Experiments require a dozen coin flips or less on average.

Abstract

The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 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, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). 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.