# 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  $(\Omega,\mathcal{F},P)$. 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. 1.

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

2. 2.

The probability of an event $B$ is denoted by $P[B]$ is equal to $\frac{area\,\ of\,\ B}{\frac{\pi}{2}}$

Now we denote by $A$ the event that the needle hits a horizontal line. It is easily seen that this happens when $\sin\theta\geq\frac{h}{1/2}$. Consequently $A=\{(\theta,h)\in\Omega:h\leq\frac{\sin\theta}{2}\}$ and then we get $P[A]=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\frac{1}{2}\sin\theta d\theta=\frac{% 1}{\pi}\square$

In general case, when the length of needle is $l$ and the distance of parallel lines is $d$ provided that $l, 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 BuffonsNeedle 2013-03-22 16:09:28 2013-03-22 16:09:28 georgiosl (7242) georgiosl (7242) 8 georgiosl (7242) Definition msc 60D05 msc 60-00