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Buffon’s Needle Problem

1777

Georges-Louis Leclerc, Comte de Buffon

One of the earliest problems in geometric probability, it is considered a precursor to the Monte Carlo method. It involves dropping a needle of length [latex]l[/latex] onto a floor with parallel lines a distance [latex]t[/latex] apart. The probability that the needle will cross a line is [latex]P = \frac{2l}{\pi t}[/latex] (for [latex]l \le t[/latex]). This provides a physical experiment to estimate [latex]\pi[/latex].

In 1733, Georges-Louis Leclerc, Comte de Buffon, posed the question: what is the probability that a needle, when dropped randomly on a ruled surface, will intersect one of the lines? The solution, published in 1777, is a classic result in geometric probability. To solve it, let the needle have length [latex]l[/latex] and the parallel lines be separated by a distance [latex]t \ge l[/latex]. The position of the needle can be described by two variables: the distance [latex]x[/latex] from the center of the needle to the nearest line, and the angle [latex]\theta[/latex] the needle makes with the lines. The variable [latex]x[/latex] is uniformly distributed in [latex][0, t/2][/latex], and [latex]\theta[/latex] is uniformly distributed in [latex][0, \pi/2][/latex].

The needle crosses a line if [latex]x \le \frac{l}{2}\sin\theta[/latex]. The problem is to find the area of this region in the [latex](x, \theta)[/latex] parameter space and divide it by the total area of the parameter space, which is [latex]\frac{t}{2} \times \frac{\pi}{2} = \frac{\pi t}{4}[/latex]. The area of the “favorable” region (where a crossing occurs) is given by the integral [latex]\int_0^{\pi/2} \frac{l}{2}\sin\theta \,d\theta = \frac{l}{2}[-\cos\theta]_0^{\pi/2} = \frac{l}{2}[/latex]. The probability is the ratio of these areas: [latex]P = \frac{l/2}{\pi t/4} = \frac{2l}{\pi t}[/latex]. By performing the experiment many times and observing the frequency of crossings, one can rearrange the formula to estimate [latex]\pi[/latex]: [latex]\pi \approx \frac{2l}{tP}[/latex]. This physical simulation to solve a mathematical problem is a direct intellectual ancestor of modern Monte Carlo methods.

实验设计 (DOE), 几何尺寸和公差, 数学, 蒙特卡罗模拟, 工艺优化, 质量管理, 模拟, 统计分析

UNESCO Nomenclature: 1209

- 统计资料

类型

抽象系统

中断

基础

使用方法

广泛使用

前体

development of probability theory (Bernoulli, De Moivre)
invention of integral calculus (Newton, Leibniz)
早期关于几何图形及其属性的研究（欧几里得）

应用

几何概率的早期例子
积分和概率的教学工具
随机模拟方法的历史基础

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Related to: Buffon’s needle, geometric probability, Pi, Monte Carlo, stochastic geometry, integral calculus, simulation, probability theory, needle problem, estimation.