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.

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