Buffon's needle

Given a needle of length $L$ dropped on a plane ruled with parallel lines $D$ units apart, what is the probability that the needle will cross a line? This question is first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon. The answer is $P=\frac{2L}{D\pi}$ where $D$ is the distance between two adjacent lines, and $L$ is the length of the needle.

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 $\pi$.

In the animation package, the function buffon.needle() can be used to simulate Buffon’s needle. There are three graphs made in each step: the top one is a simulation of the scenario, the bottom-left one can help us understand the connection between dropping needles and the mathematical method to estimate $\pi$, and the bottom-right one is the simulation result $\pi$ for each drop.

library(animation)
ani.options(nmax = 100, interval = 0.5)
par(mar = c(3, 2.5, 0.5, 0.2), pch = 20, mgp = c(1.5, 0.5, 0))
buffon.needle(mat = matrix(c(1, 2, 1, 3), 2))

You can use larger nmax values in the code to drop the needle for more times.

References

This article was reproduced from vistat