Buffon's noodle
Encyclopedia
In geometric probability
, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle
, named after Georges-Louis Leclerc, Comte de Buffon
who lived in the 18th century. That problem solved by Buffon was the earliest geometric probability problem to be solved.
, where D is the distance between two adjacent lines, and L is the length of the needle. See this simulation.
. We drop the assumption that the length of the noodle is no more than the distance between the parallel lines.
The probability distribution
of the number of crossings depends on the shape of the noodle, but the expected number
of crossings does not; it 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 Xi be the number of times the ith piece crosses one of the parallel lines. These random variables are not independent
, but the expectations are still additive:
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 = πD and the number of crossings is exactly 2, with probability 1. So when L = πD then the expected number of crossings is 2. Therefore the expected number of crossings must be 2L/(πD).
There is one more surprising consequence. In case the noodle is any closed curve of constant width
D the number of crossing is also exactly 2. This implies Barbier's theorem
asserting that the perimeter is the same as that of a circle.
Geometric probability
Problems of the following type, and their solution techniques, were firststudied in the 19th century, and the general topic became known as geometric probability....
, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle
Buffon's needle
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry...
, named after Georges-Louis Leclerc, Comte de Buffon
Georges-Louis Leclerc, Comte de Buffon
Georges-Louis Leclerc, Comte de Buffon was a French naturalist, mathematician, cosmologist, and encyclopedic author.His works influenced the next two generations of naturalists, including Jean-Baptiste Lamarck and Georges Cuvier...
who lived in the 18th century. That problem solved by Buffon was the earliest geometric probability problem to be solved.
Buffon's needle
Suppose there exist an infinite number of equally spaced parallel 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 cross a line? The formula is, where D is the distance between two adjacent lines, and L is the length of the needle. See this simulation.Bending the needle
The interesting thing about the formula is that it stays the same even when you bend the needle in any way you want (subject to the constraint that it must lie in a plane), making it a "noodle"—a rigid plane curvePlane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....
. We drop the assumption that the length of the noodle is no more than the distance between the parallel lines.
The probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of the number of crossings depends on the shape of the noodle, but the expected number
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of crossings does not; it 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 Xi be the number of times the ith piece crosses one of the parallel lines. These random variables are not independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
, but the expectations are still additive:
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 = πD and the number of crossings is exactly 2, with probability 1. So when L = πD then the expected number of crossings is 2. Therefore the expected number of crossings must be 2L/(πD).
There is one more surprising consequence. In case the noodle is any closed curve of constant width
Curve of constant width
In geometry, a curve of constant width is a convex planar shape whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines.More generally, any compact...
D the number of crossing is also exactly 2. This implies Barbier's theorem
Barbier's theorem
Barbier's theorem, a basic result on curves of constant width first proved by Joseph Emile Barbier, states that the perimeter of any curve of constant width w is πw....
asserting that the perimeter is the same as that of a circle.