Buffon’s needle
The plane is ruled by parallel lines^{} $2$ inches apart and a $1$inch long needle is dropped at random on the
plane. What is the probability that it hits parallel lines?
Solution.
The first issue is to find some appropriate probability space^{} $(\mathrm{\Omega},\mathcal{F},P)$. For this,

•
$h=$ distance from the center of the needle to the nearest line

•
$\theta =$ the angle that the needle makes with the horizontal ranging from $0$ to $\frac{\pi}{2}$.
These fully determine the position of the needle. Let us next take the

1.
The probability space is $\mathrm{\Omega}=[0,1]\times [0,\frac{\pi}{2})$

2.
The probability of an event $B$ is denoted by $P[B]$ is equal to $\frac{areaofB}{\frac{\pi}{2}}$
Now we denote by $A$ the event that the needle hits a horizontal line. It is easily seen that this happens when $\mathrm{sin}\theta \ge \frac{h}{1/2}$. Consequently $A=\{(\theta ,h)\in \mathrm{\Omega}:h\le \frac{\mathrm{sin}\theta}{2}\}$ and then we get $P[A]=\frac{2}{\pi}{\int}_{0}^{\frac{\pi}{2}}\frac{1}{2}\mathrm{sin}\theta d\theta =\frac{1}{\pi}\mathrm{\square}$
In general case, when the length of needle is $l$ and the distance of parallel lines is $d$ provided that $$, the probability we want is $\frac{2l}{\pi d}$. This is obvious just taking the $l/d$point from one edge instead of the center of the needle.
Title  Buffon’s needle 

Canonical name  BuffonsNeedle 
Date of creation  20130322 16:09:28 
Last modified on  20130322 16:09:28 
Owner  georgiosl (7242) 
Last modified by  georgiosl (7242) 
Numerical id  8 
Author  georgiosl (7242) 
Entry type  Definition 
Classification  msc 60D05 
Classification  msc 6000 