  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.

Probability