From: T.Moore@massey.ac.nz (Terry Moore)
Newsgroups: sci.math
Subject: Re: The Buffon Needle Problem
Date: Mon, 09 Feb 1998 13:47:17 +1200

In article <34db11ed.2884421@news.telepac.pt>, rveloso@mail.telepac.pt
(Ricardo V. Oliveira) wrote:

> Can anyone help me with this one?:
> "What are the chances of a needle,thrown randomly into an infinite
> plane with parallel lines marked on it,cross one of those lines?
> Consider the needle length to be A,the distance between the parallel
> lines to be D and A<D.
> Answer:p=2*A/(D*Pi)."

Bill Taylor once posted the most beautiful proof of this.
Instead of calculating the probability, calculate the
expected number of crossings. With A < D, that's the
same thing, but we can get the expected number without
that restriction.
We use the fact that the expected value of a sum is the
sum of the expected values. This holds even without
independence. Imagine the needle divided into pieces.
The expected number of crossings of the whole needle
is the sum of the expected numbers for the parts. If
the parts are identical, they have the same expected number,
so the expected number is proportional to the length of the
needle. We want the know the constant of proportionality.
The argument even applies to a curved needle (or even a
bootlace, or a handful of needles), so imagine a needle bent
into a circle of length A = pi D. (This is greater than D, but
that doesn't matter as stated above). Such a needle always
has exactly two crossings apart from when it is tangential to
two lines, an event that has zero probability. The expected
number is therefore 2 giving a constant of proportionality of
2/(pi D). So for a needle of length A we get 2A/(pi D).

-- 

Terry Moore, Statistics Department, Massey University, New Zealand.

Theorems! I need theorems. Give me the theorems and I shall find the
proofs easily enough. Bernard Riemann