Buffon's Needle - Solution

Solution

The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?

Let x be the distance from the center of the needle to the closest line, let θ be the acute angle between the needle and the lines.

The uniform probability density function of x between 0 and t /2 is

$\begin{cases} \frac{2}{t} &:\ 0 \le x \le \frac{t}{2}\\ 0 &: \text{elsewhere.} \end{cases}$

The uniform probability density function of θ between 0 and π/2 is

$\begin{cases} \frac{2}{\pi} &:\ 0 \le \theta \le \frac{\pi}{2}\\ 0 &: \text{elsewhere.} \end{cases}$

The two random variables, x and θ, are independent, so the joint probability density function is the product

$\begin{cases} \frac{4}{t\pi} &:\ 0 \le x \le \frac{t}{2}, \ 0 \le \theta \le \frac{\pi}{2}\\ 0 &: \text{elsewhere.} \end{cases}$

The needle crosses a line if

Now there are two cases.

Read more about this topic: Buffon's Needle

Famous quotes containing the word solution:

“Let us begin to understand the argument.
There is a solution to everything: Science.”
—Allen Tate (1899–1979)

“The Settlement ... is an experimental effort to aid in the solution of the social and industrial problems which are engendered by the modern conditions of life in a great city. It insists that these problems are not confined to any one portion of the city. It is an attempt to relieve, at the same time, the overaccumulation at one end of society and the destitution at the other ...”
—Jane Addams (1860–1935)

“Any solution to a problem changes the problem.”
—R.W. (Richard William)