Buffon's noodlewikipedia

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.
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Buffon's needle

his needlehis needle problem
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.
A particularly nice argument for this result can alternatively be given using "Buffon's noodle".

Barbier's theorem

This implies Barbier's theorem asserting that the perimeter is the same as that of a circle.
An elementary probabilistic proof of the theorem can be found at Buffon's noodle.

Crofton formula

classic theorem of CroftonCauchy-Crofton theorem
Buffon's noodle

Geometric probability

Geometrical Probability
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.

Georges-Louis Leclerc, Comte de Buffon

BuffonComte de BuffonLeclerc de Buffon
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.

Plane curve

complex plane curvecurvecurve in a plane
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 curve.

Probability distribution

distributionprobability distributionscontinuous probability distribution
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).

Expected value

expectationexpectedmean
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). These random variables are not independent, but the expectations are still additive due to the linearity of expectation:

Polygonal chain

polylinepolygonal curvepolygonal path
First suppose the noodle is piecewise linear, i.e. consists of n straight pieces.

Independence (probability theory)

independentstatistically independentindependence
These random variables are not independent, but the expectations are still additive due to the linearity of expectation:

Curve of constant width

curves of constant widthconstant diameterequilateral curve
In case the noodle is any closed curve of constant width D the number of crossings is also exactly 2.