Buffon's noodle

noodle
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.wikipedia
16 Related Articles

Buffon's needle problem

his needlehis needle problemneedle
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.

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.

Random variable

random variablesrandom variationrandom
Now, we assume that the values of x, \theta are randomly determined when they land, where, since 0 < l < D, and.

Sample space

event spacespacesample spaces
The sample space for x, \theta is thus a rectangle of side lengths \frac{D}{2} and.

Probability

probabilisticprobabilitieschance
The probability of the event that the needle lies across the nearest line is the fraction of the sample space that intersects with.

Event (probability theory)

eventeventsrandom event
The probability of the event that the needle lies across the nearest line is the fraction of the sample space that intersects with.

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

distributioncontinuous probability distributiondiscrete 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 diameterconstant width
In case the noodle is any closed curve of constant width D the number of crossings is also exactly 2.

How Not to Be Wrong

Ellenberg also talks about the Law of Large numbers again, as well as introducing the Additivity of expected value and the games of Franc-Carreau or the “needle/noodle problem”.

Crofton formula

Cauchy-Crofton theoremclassic theorem of Crofton
Buffon's noodle