Buffon's needlewikipedia

In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:
his needlehis needle problem
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Georges-Louis Leclerc, Comte de Buffon

BuffonComte de BuffonLeclerc de Buffon
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:
He first made his mark in the field of mathematics and, in his Sur le jeu de franc-carreau, introduced differential and integral calculus into probability theory; the problem of Buffon's needle in probability theory is named after him.

Monte Carlo method

Monte CarloMonte Carlo simulationMonte Carlo simulations
The solution, in the case where the needle length is not greater than the width of the strips, can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question.
An early variant of the Monte Carlo method can be seen in the Buffon's needle experiment, in which can be estimated by dropping needles on a floor made of parallel and equidistant strips.

Buffon's noodle

A particularly nice argument for this result can alternatively be given using "Buffon's 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.

Pi

ππ\pi
The solution, in the case where the needle length is not greater than the width of the strips, can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question.
Buffon's needle is one such technique: If a needle of length

Geometric probability

Geometrical Probability
Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry.
(Buffon's needle) What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines?

Integral geometry

integralgeometric probability theory
Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry.
A very celebrated case is the problem of Buffon's needle: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack.

Asaph Hall

HallAsaph Hall, IIIHall, Asaph
This paper, an experiment on the Buffon's needle problem, is a very early documented use of random sampling (which Nicholas Metropolis would name the Monte Carlo method during the Manhattan Project of World War II) in scientific inquiry.

Rejection sampling

adaptive rejection samplingacceptance-rejection methodacceptance–rejection method
The algorithm (used by John von Neumann and dating back to Buffon and his needle) to obtain a sample from distribution X with density f using samples from distribution Y with density g is as follows:

Pore space in soil

soil poreporositycharacterisation of pore space in soil
The crack distribution was calculated using the principle of Buffon's needle.

Rudolf Wolf

Rudolph WolfR WolfR. Wolf
Around 1850, to study the laws of probability, Wolf performed a Buffon's needle experiment, dropping a needle on a plate 5000 times to verify the value of π, a precursor to the Monte Carlo method.

Mathematics

mathematicalmathmathematician
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:

Probability

probabilisticprobabilitieschance
What is the probability that the needle will lie across a line between two strips?

Probability density function

probability densitydensity functiondensity
The uniform probability density function of x between 0 and t /2 is

Random variable

random variablesrandom variationrandom
The two random variables, x and θ, are independent, so the joint probability density function is the product

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
The two random variables, x and θ, are independent, so the joint probability density function is the product

Approximations of π

Approximations of approximationapproximation of π
Tossing a needle 3408 times, he obtained the well-known approximation 355/113 for π, accurate to six significant digits.

Confirmation bias

selective thinkingconfirmationdisconfirmation bias
Lazzarini's "experiment" is an example of confirmation bias, as it was set up to replicate the already well-known approximation of 355/113 (in fact, there is no better rational approximation with fewer than five digits in the numerator and denominator), yielding a more accurate "prediction" of π than would be expected from the number of trials, as follows: