In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon :
The a needle lies across a line, while the b needle does not.
Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?
Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is p = 2/π * l/t .
This 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. The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.
https://en.wikipedia.org/wiki/Buffon%27s_needle_problem
Needle : A really weird way to find Pi
Discovery of Calculus in the 17th century opened a new window for estimating π more precisely. Since then, we've seen plenty of methods to do it. Some of them are pretty weird but simple enough to understand. Buffon's Needle Problem is one of them. This problem was first posed by the French Naturalist Georges-Louis Leclerc, Comte de Buffon in 1733. It goes like this:
Suppose you are given a floor with equally spaced parallel lines on it. If we drop a needle on the floor, what is the probability that the needle will lie across a line?
Let's say you've thrown n needles (or a single needle for n times or whatever). m number of needles cross/intersect with a line (or m times if you have one needle). The probability P that a needle crosses the line:
P = m/n
Buffon's Needle Problem. The green needles are the ones crossing the parallel lines
It may seem a bit odd how π is related to this problem. Hang on tight. I'll explain.
To solve this problem, we'll require some basic knowledge of Probability and Integral Calculus. Let's assume that the spacing between two consecutive parallel lines is D, the length of the needle is L. Now, let the distance from the middle point of a needle on the floor to its closest line be x and the acute angle between the line and needle be θ (See figure above). All needles are of equal length.
Since we are considering x to be the smallest distance from the center of the needle to any one of the lines, it can vary only within 0 and D/2. And since θ is an acute angle, it can take any value between 0 and π/2. Using trigonometry, we can find out that the vertical component of length L is L/2 * sin(θ). The needle will cross the line if x is less than L/2 * sin(θ).
Okay. Now let's visualize possible outcomes in a graph. Let the horizontal axis be θ and the vertical axis be x. So a rectangle with sides π/2 and D/2 represents all possible outcomes in this experiment.
Horizontal axis refers to θ and vertical axis refers to x. The rectangle represents all possible outcomes
Now I am going to shade all those points in the rectangle that represent the events where a needle crosses a line according to the two conditions above. Think about it. It's just the area under the curve D/2 * sin(θ) where 0 ≤ θ ≤ π .
The shaded area indicates the probability that a needle will cross a line
The probability we are looking for would be the ratio of the area under the sine curve and the rectangle.
Area of the rectangle:
Area under the sine curve:
Ratio of the areas:
Rearranging, we get
In 1901, Italian mathematician Mario Lazzarini performed Buffon's needle experiment and, tossing a needle 3408 times, obtained π up to six decimal points correctly!
https://tahsin314.github.io/writings/maths/buffons_needle.html
Finally, since Buffon’s needle problem is interested in estimating π – and since I have mostly focused on the error in this method instead – I thought it might be fun to take a look at the the empirical distribution of $\hat{\pi}$ for one of my experimental setups. The following figure shows how this distribution varies as a function of the number of needles thrown for the l = h case. The data are plotted using a dot-plot, while box-plots have been overlaid on top.
