Buffon's Needle
by aoum, Mar 30, 2025, 9:23 PM
Buffon’s Needle: A Probability Approach to 
1. Introduction to Buffon's Needle Problem
Buffon’s Needle is a famous probability problem first posed by Georges-Louis Leclerc, Comte de Buffon, in the 18th century. It is one of the earliest problems in geometric probability and provides a way to estimate the value of. The problem is as follows:
Suppose we drop a needle of length
onto a floor covered in parallel lines spaced a distance 
apart. What is the probability that the needle will cross one of the lines?
This problem has significant applications in Monte Carlo methods and stochastic geometry.
2. Mathematical Formulation of Buffon’s Needle
Let:
A crossing occurs if the endpoint of the needle extends past the nearest line, which happens when:
![\[
x \leq \frac{\ell}{2} \sin \theta.
\]](//latex.artofproblemsolving.com/f/b/2/fb299dedfffd6d356a74f815b181ccbcd0d74dcb.png)
Since
is uniformly distributed over ![$[0, d/2]$](//latex.artofproblemsolving.com/d/1/3/d13583b6eeb0199fdd2788cf889b6d9443120500.png)
, the probability of crossing a line is given by integrating over all possible needle orientations.
3. Computing the Probability
The probability
that the needle crosses a line is given by:
![\[
P = \frac{2}{\pi} \cdot \frac{\ell}{d}.
\]](//latex.artofproblemsolving.com/e/0/2/e02b335df2a3971ebce19fb3d32dc589f2037caa.png)
Proof:
The probability density function of is uniform over , so its probability distribution is:
![\[
f(x) = \frac{2}{d}, \quad 0 \leq x \leq \frac{d}{2}.
\]](//latex.artofproblemsolving.com/e/a/8/ea80e4f5f55f48c25badf91d7b953bf79c0a24cd.png)
For a fixed
, the needle crosses a line when:
The probability of this happening, given, is:
![\[
P(\text{cross} | \theta) = \frac{\ell}{d} \sin \theta.
\]](//latex.artofproblemsolving.com/2/2/0/22087d963be21997995e6f01c1e30dc12ed61e1b.png)
Since is uniformly distributed in ![$[0, \pi/2]$](//latex.artofproblemsolving.com/3/7/2/37278732e6da75cdf01fab8f86657b7b3feb2006.png)
, we integrate over all possible angles:
![\[
P = \int_0^{\pi/2} \frac{\ell}{d} \sin \theta \cdot \frac{2}{\pi} d\theta.
\]](//latex.artofproblemsolving.com/0/f/0/0f054f302d1d31a2535b766daa4238137eeb0a34.png)
Evaluating the integral:
![\[
P = \frac{\ell}{d} \cdot \frac{2}{\pi} \int_0^{\pi/2} \sin \theta \, d\theta.
\]](//latex.artofproblemsolving.com/5/7/8/578fbc71b2b1c1b7d5d85d06c04b9b154d8a0551.png)
Since
, we obtain:
4. Estimating Using Buffon's Needle
Rearranging the formula, we can estimate:
![\[
\pi \approx \frac{2\ell}{dP}.
\]](//latex.artofproblemsolving.com/a/c/7/ac72ca08410bb491cc85318c4b8441a54b319e1c.png)
To estimate, one can perform an experiment by dropping 
needles and counting the number of crossings, 
. The empirical probability is 
, giving the approximation:
![\[
\pi \approx \frac{2\ell N}{dC}.
\]](//latex.artofproblemsolving.com/d/9/b/d9b2b83489db35ea97e0a7e9ffb63d20d318817e.png)
This method is a Monte Carlo approach to approximating.
5. Generalizations of Buffon's Needle
6. Conclusion
Buffon’s Needle is an elegant probability problem connecting geometry, probability, and number theory. It provides an intuitive way to approximate and serves as an early example of stochastic methods used in modern simulations.
7. YouTube Video: Buffon's Needle
References
1. Introduction to Buffon's Needle Problem
Buffon’s Needle is a famous probability problem first posed by Georges-Louis Leclerc, Comte de Buffon, in the 18th century. It is one of the earliest problems in geometric probability and provides a way to estimate the value of
. The problem is as follows:Suppose we drop a needle of length
onto a floor covered in parallel lines spaced a distance 
apart. What is the probability that the needle will cross one of the lines?
The 
needle lies across a line, while the 
needle does not.
needle lies across a line, while the 
needle does not.
This problem has significant applications in Monte Carlo methods and stochastic geometry.
2. Mathematical Formulation of Buffon’s Needle
Let:
- be the length of the needle.
- be the distance between the parallel lines (
). - be the acute angle between the needle and the vertical axis.
- be the perpendicular distance from the center of the needle to the nearest line.
)A crossing occurs if the endpoint of the needle extends past the nearest line, which happens when:
![\[
x \leq \frac{\ell}{2} \sin \theta.
\]](http://latex.artofproblemsolving.com/f/b/2/fb299dedfffd6d356a74f815b181ccbcd0d74dcb.png)
Since
is uniformly distributed over ![$[0, d/2]$](http://latex.artofproblemsolving.com/d/1/3/d13583b6eeb0199fdd2788cf889b6d9443120500.png)
, the probability of crossing a line is given by integrating over all possible needle orientations.3. Computing the Probability
The probability
that the needle crosses a line is given by:![\[
P = \frac{2}{\pi} \cdot \frac{\ell}{d}.
\]](http://latex.artofproblemsolving.com/e/0/2/e02b335df2a3971ebce19fb3d32dc589f2037caa.png)
Proof:
The probability density function of
is uniform over , so its probability distribution is:![\[
f(x) = \frac{2}{d}, \quad 0 \leq x \leq \frac{d}{2}.
\]](http://latex.artofproblemsolving.com/e/a/8/ea80e4f5f55f48c25badf91d7b953bf79c0a24cd.png)
For a fixed
, the needle crosses a line when:The probability of this happening, given
, is:![\[
P(\text{cross} | \theta) = \frac{\ell}{d} \sin \theta.
\]](http://latex.artofproblemsolving.com/2/2/0/22087d963be21997995e6f01c1e30dc12ed61e1b.png)
Since
is uniformly distributed in ![$[0, \pi/2]$](http://latex.artofproblemsolving.com/3/7/2/37278732e6da75cdf01fab8f86657b7b3feb2006.png)
, we integrate over all possible angles:![\[
P = \int_0^{\pi/2} \frac{\ell}{d} \sin \theta \cdot \frac{2}{\pi} d\theta.
\]](http://latex.artofproblemsolving.com/0/f/0/0f054f302d1d31a2535b766daa4238137eeb0a34.png)
Evaluating the integral:
![\[
P = \frac{\ell}{d} \cdot \frac{2}{\pi} \int_0^{\pi/2} \sin \theta \, d\theta.
\]](http://latex.artofproblemsolving.com/5/7/8/578fbc71b2b1c1b7d5d85d06c04b9b154d8a0551.png)
Since
, we obtain:4. Estimating
Using Buffon's NeedleRearranging the formula, we can estimate
:![\[
\pi \approx \frac{2\ell}{dP}.
\]](http://latex.artofproblemsolving.com/a/c/7/ac72ca08410bb491cc85318c4b8441a54b319e1c.png)
To estimate
, one can perform an experiment by dropping 
needles and counting the number of crossings, 
. The empirical probability is 
, giving the approximation:![\[
\pi \approx \frac{2\ell N}{dC}.
\]](http://latex.artofproblemsolving.com/d/9/b/d9b2b83489db35ea97e0a7e9ffb63d20d318817e.png)
This method is a Monte Carlo approach to approximating
.5. Generalizations of Buffon's Needle
- Randomly Oriented Needle on a Grid: If the floor has perpendicular grid lines spaced apart, the probability of intersection increases and leads to more accurate estimations.
- Buffon's Noodle: A curved object is dropped onto a plane with parallel lines. The expected number of crossings depends on the total arc length of the noodle.
- Higher-Dimensional Analogues: Buffon’s problem extends to three dimensions, where it relates to integral geometry.
6. Conclusion
Buffon’s Needle is an elegant probability problem connecting geometry, probability, and number theory. It provides an intuitive way to approximate
and serves as an early example of stochastic methods used in modern simulations.7. YouTube Video: Buffon's Needle
References
- Wikipedia: Buffon's Needle
- Monte Carlo Methods in Estimating , Statistical Science Journal
- AoPS Wiki: Buffon's Needle Problem
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