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布丰投针

Buffon's Needle

专题
Probability / 概率
难度
L6

题目详情

Suppose we toss a needle of length 2l2l (less than 1) on a grid with both horizontal and vertical rulings spaced one unit apart. What is the mean number of lines the needle crosses? (I have dropped the 2a2a for the spacing because we might as well think of the length of the needle as measured in units of spacing.)

解析

针长为 2l<12l<1（半长 ll），角度 θ\theta 在 [0,π)[0,\pi) 上均匀。

对水平线：针的中心到最近水平线的距离在 [0,12][0,\tfrac12] 上均匀；给定 θ\theta，穿过水平线当且仅当该距离不超过 l∣sin⁡θ∣l|\sin\theta|，因此

P(穿过水平线∣θ)=2l∣sin⁡θ∣.\mathbb{P}(\text{穿过水平线}\mid\theta)=2l|\sin\theta|.

同理穿过竖直线的条件概率为 2l∣cos⁡θ∣2l|\cos\theta|。

期望穿过总数为

E[N]=2l E[∣sin⁡θ∣+∣cos⁡θ∣].\mathbb{E}[N]=2l\,\mathbb{E}[|\sin\theta|+|\cos\theta|].

而 E[∣sin⁡θ∣]=E[∣cos⁡θ∣]=2π\mathbb{E}[|\sin\theta|]=\mathbb{E}[|\cos\theta|]=\frac{2}{\pi}，故

E[N]=8lπ.\boxed{\mathbb{E}[N]=\frac{8l}{\pi}}.