Solving the Original Buffon Needle Problem By The Method of Area Ratios

According to G. Timothy Hushion, founder of Cracking Pi, L.L.C., at crackingpi.com, the original Buffon Needle Problem (or Buffon Needle Experiment) was:

Je suppose que dans une chambre, dont le parquet est simplement divise par des joints paralleles, on jette en l’air une baguette, & que l’un des joucurs parie que la baguette ne croisera aucune des paralleles du parquet, & que l’autre au contraire parie que la baguette croisera quelques-unes de ces paralleles; on demande le sort de ces deux joucurs.

Translated into English:

With a series of equidistant parallel lines, if two players want an even chance a randomly dropped needle will cross a line, how long must the needle be?

(Compare the above to the modern version of the problem as presented today: Wikipedia: Buffon Needle Problem.)

Mr. Hushion believes the solution to the problem as originally expressed by Buffon is the key to understanding randomness.  At his website, he writes extensively about the history and implications of the original problem, as well as his experiences applying it to solve problems involving probability, including games of chance, such as roulette and baccarat, as well as the stock market and random number generators.

While G. T. Hushion states the solution to the problem is that the length is “one fourth of the circle described between and just touching two adjacent lines, that is, 1/4 C”, nowhere on his site does he present Buffon’s original deductive proof.

In the following, I offer a proof for the original Buffon Needle Problem using the method of the ratio of phase space areas.  (Using probability densities is another way to solve the problem, but using area ratios seems more intuitive.)

First, we define the following geometry of the problem:

Geometry of the Buffon Needle Problem

D = distance between two parallel lines;

l = the length of the needle, where the origin point of the needle is taken to be that of the end closer to the lower line;

d = the vertical coordinate of  the origin point of the needle, where d=0 is the vertical coordinate of the parallel line below the needle, and d=D is that of the line above it;

θ = the angle the longitudinal axis of the needle makes with the axis of the parallel lines, where θ=0 is defined to be when the needle and lines are pointing in the same direction, and θ=π/2 (radians, one quarter turn counter-clockwise from θ=0), when they are perpendicular.

It clearly follows from the geometry of the problem that the criteria for the needle to touch a line is:

                         d ≥ D - l sinθ

That is, if the needle’s vertical coordinate equals or exceeds the vertical position with the value of (D – l sinθ), then the needle must cross a line, as can be clearly seen from the geometry of the problem.

With the criteria for the solution clearly defined, we can next draw the phase space of all possible coordinates d vs. θ of the needle to determine the relevant areas needed to calculate the desired probability.

Phase Space of the Buffon Needle Problem

The total area A1 of the phase space defined by d vs. θ is simply that of the area of the rectangular region bounded by sides of length D and π, and so straightforwardly,

                         A1 = πD

Since the needle crosses a line whenever d ≥ D – l sinθ, the area A2 of the above phase space which represents the coordinates where the needle crosses a line is the area which is bounded by the curve
(l sinθ) from θ = 0 to θ = π.  A2 is the cyan-colored region in the above graph of the phase space.

Integrating over the two limits, we determine this area is:

                         A2 = ∫(l sinθ) dθ
                            = - l cosθ (from θ =0 to θ =π)
                            = - l { (-1) - (1) }
                            = 2 l

Thus, the probability P of the needle crossing a line is simply the ratio of A2/A1:

                          P = A2/A1 
                            = (2 l)/(πD)

Setting P = 1/2 to satisfy the equal-chances condition of Buffon’s original question, we thus find:

                          P = 1/2 
                            = (2 l)/(πD)

Solving for the desired length of the needle l, we find:

                          l = (πD)/4

Since the circumference C of the circle with diameter D equal to the distance between two parallel lines is C = πD, we thus finally arrive at the desired answer:

                          l = C/4
                           Q.E.D.

Stated explicitly, then, the solution to the original Buffon Needle Problem is that, in order for it to have equal chances of hitting a line, the needle’s length should be one-quarter of the circumference of the circle which is contained by and tangent to two adjacent lines.