Buffon's Noodle
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In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of
Buffon's needle In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon: :Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the ...
, named after
Georges-Louis Leclerc, Comte de Buffon Georges-Louis Leclerc, Comte de Buffon (; 7 September 1707 – 16 April 1788) was a French Natural history, naturalist, mathematician, and cosmology, cosmologist. He held the position of ''intendant'' (director) at the ''Jardin du Roi'', now ca ...
who lived in the 18th century. This approach to the problem was published by
Joseph-Émile Barbier Joseph-Émile Barbier (1839–1889) was a French astronomer and mathematician, known for Barbier's theorem on the perimeter of curves of constant width. Barbier was born on 18 March 1839 in Saint-Hilaire-Cottes, Pas-de-Calais, in the north of ...
in 1860.


Buffon's needle

Suppose there exist infinitely many equally spaced parallel, horizontal lines, and we were to randomly toss a needle whose length is less than or equal to the distance between adjacent lines. What is the probability that the needle will lie across a line upon landing? To solve this problem, let \ell be the length of the needle and D be the distance between two adjacent lines. Then, let \theta be the acute angle the needle makes with the horizontal, and let x be the distance from the center of the needle to the nearest line. The needle lies across the nearest line if and only if x \le \frac . We see this condition from the right triangle formed by the needle, the nearest line, and the line of length x when the needle lies across the nearest line. Now, we assume that the values of x, \theta are randomly determined when they land, where 0 < x < \frac , since 0 < \ell < D , and 0 < \theta < \frac . The
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
for x, \theta is thus a rectangle of side lengths \frac and \frac . The
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
of the
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...
that the needle lies across the nearest line is the fraction of the sample space that intersects with x \le \frac \ell 2 \sin \theta . Since 0 < \ell < D , the area of this intersection is given by : \text = \int^_0 \frac\ell2 \sin \theta \, d \theta = -\frac\ell2 \cos \frac\pi2 + \frac\ell 2 \cos 0 = \frac\ell 2. Now, the area of the sample space is : \text = \frac \times \frac = \frac. Hence, the probability P of the event is : P = \frac = \frac\ell 2 \cdot \frac 4 = \frac.


Bending the needle

The formula \tfrac stays the same even when the needle is bent in any way (subject to the constraint that it must lie in a plane), making it a "noodle"—a rigid
plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
, though now gives the
expected number In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected val ...
of crossings of the parallel lines rather than a probability. We can drop the assumption that the length of the noodle is no more than the distance between the parallel lines. The shape of the noodle affects the
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
of the number of crossings, but not the expected number of crossings which only depends only on the length ''L'' of the noodle and the distance ''D'' between the parallel lines (observe that a curved noodle may cross a single line multiple times). This fact may be proved as follows (see Klain and Rota). First suppose the noodle is piecewise linear, i.e. consists of ''n'' straight pieces. Let ''X''''i'' be the number of times the ''i''th piece crosses one of the parallel lines. These random variables are not
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
, but the expectations are still additive due to the
linearity of expectation In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected va ...
: : E(X_1+\cdots+X_n) = E(X_1)+\cdots+E(X_n). Regarding a curved noodle as the limit of a sequence of piecewise linear noodles, we conclude that the expected number of crossings per toss is proportional to the length; it is some constant times the length ''L''. Then the problem is to find the constant. In case the noodle is a circle of diameter equal to the distance ''D'' between the parallel lines, then ''L'' = ''D'' and the number of crossings is exactly 2, with probability 1. So when ''L'' = ''D'' then the expected number of crossings is 2. Therefore, the expected number of crossings must be 2''L''/(''D'').


Barbier's theorem

Extending this argument slightly, if C is a convex compact subset of \R^2, then the expected number of lines intersecting C is equal to half the expected number of lines intersecting the perimeter of C, which is \frac. In particular, if the noodle is any closed
curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane (geometry), plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of ...
D, then the number of crossings is also exactly 2. This means the perimeter has length \pi D, the same as that of a circle, proving
Barbier's theorem In geometry, Barbier's theorem states that every curve of constant width has perimeter times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860. Examples The most familiar examples o ...
.


Notes


References

* * {{Cite book , year=1997 , title = Introduction to geometric probability , author1=Daniel A. Klain , author2=Gian-Carlo Rota , publisher=
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, isbn=978-0-521-59654-1 , url=https://archive.org/details/introductiontoge0000klai , url-access=registration , pag
1
, authorlink2=Gian-Carlo Rota


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