Buffons needle problem for $k$ crossings

Purpose and setup

This Python script allows the user to simulate the probability of getting exactly $k$ crossings when a needle of length $l$ is dropped onto a floor with parallel horizontal lines that are equidistant by $d$. The purpose of this script is to provide us with an empirical estimate of the probability of getting $k$ crossings so that the mathematical derivations that have been derived can be upheld. To convince yourself, I highly recommend you read the mathematical derivations presented below, run the code for your choice of d and l, and lastly verify the mathematical expressions using the interactive file I created in Desmos under Additional resources.

The condition for this Python script is that $d\leq l$. In that case, the number of possible crossings $k$ is simply given by $0\leq k\leq \lfloor{l/d}\rfloor = m$. The geometrical setup is presented in the following figure.

Prerequisites

Ensure you have the following Python libraries installed:

• matplotlib
• numpy

Usage of k-crossings.py

Open the script in a Python environment. Modify the no_of_needles, l, and d parameters as needed. Run the script to visualize the needle dropping simulation and the resulting histogram of crossings.

Parameters

no_of_needles: Number of needles to drop during the simulation. l: Length of the needle. d: Distance between parallel lines.

Simulation Process

1. Parallel Lines Setup: Lines are drawn parallel at a distance d from each other.
2. Needle dropping loop: Random needles are dropped, and the script calculates the number of crossings for each needle.
3. Visualization: The script generates the calculated probabilities which are showcased with a plot, providing insights into the distribution.

Mathematical derivation for the probabilities

The needle is dropped as in the figure above, with a distributed $\theta \sim \mathcal{U}[0,\pi/2]$ and $y\sim\mathcal{U}(0,d)$. These two ranges are sufficient to cover the sample space we are interested in because of symmetry reasons. Also, we will keep $y$ as an open set to ignore the special cases when the needle precisely crosses the line and accept it as null measure.

Case $k= 0$

To derive the probabilities of getting exactly $k$ crossings, we will firstly impose a condition on our $\theta$. Note that the result we will derive is simply for $0$ < $k$ < $m$. The case $k=0$ is covered by studying the common Buffons needle problem for a long needle. In https://en.wikipedia.org/wiki/Buffon%27s_needle_problem the probability of getting at least $1$ crossing is derived, which means the probability of getting $k=0$ crossings is simply given by

$$ P(0 \ \text{crossings}) = 1 - P(\text{at least one crossing})$$

Case $0$ < $k$ < $m$

We initialize the problem by working with a simpler case, and hopefully it will become clearer on how we can extrapolate the result. Firstly study the figure below

We are interested in finding out the probability of getting exactly $1$ crossing. Note that the minimum angle that suffices this condition is given by $\theta_\text{min} = \arcsin \frac{1\cdot d - y}{l}$ simply by using trigonometry. First subtracting the left dashed white distance $1\cdot d$ with the uniformly distributed $y$ gives us the displacement in the $y$ direction, which gives us the shorter green dashed distance, and the hypotenuse is then trivially given by $l$.

Similarly we can deduce that the the maximum angle $\theta$ is given by $\theta_\text{max} = \arcsin \frac{2\cdot d - y}{l}$. Therefore we have the condition for getting $1$ crossing to be:

$$ \arcsin \frac{1\cdot d - y}{l} \leq \theta < \arcsin \frac{2\cdot d - y}{l}$$

It's very easy to extend this argument for a total of $k$ crossings, then the condition simply becomes

$$ \arcsin \frac{k\cdot d - y}{l} \leq \theta < \arcsin \frac{(k+1)\cdot d - y}{l}$$

Note that we haven't included the right end in the range of $\theta$ since then we will have $k+1$ crossings, trivially then the left end is included. To find the probability of getting $k$ crossings we have to integrate our joint probability density function, but since the ingoing random variables are independent, they are simply multipled.

$$ P(k \ \text{crossings}) = \int_0^d \int_{\arcsin \frac{k\cdot d - y}{l}}^{\arcsin \frac{(k+1)\cdot d - y}{l}} \frac{1}{\pi/2 \cdot d} d\theta dy$$

Case $k = m$

For calculating the probability of getting $m$ crossings, we utilize firstly that the probability of getting at least $1$ crossing is well documented previously, see for instance the derivation once again in the link https://en.wikipedia.org/wiki/Buffon%27s_needle_problem. This means that we can find the probability of getting $m$ crossings by

$$P(m \ \text{crossings}) = P(\text{at least one crossing}) -P(k=1,2,...,m-1)= $$

$$ = \frac{2l}{\pi d} \left(\frac{d}{l} \arccos\frac{d}{l} + 1 - \sqrt{1-\left(\frac{d}{l}\right)^2}\right) - \sum_{k=1}^{m-1} P(k \ \text{crossings})$$

Notice however that if we define

$$I(k) = \int_0^d \int_{0}^{\arcsin \frac{k\cdot d - y}{l}} \frac{1}{\pi/2 \cdot d} d\theta$$

then $P(k \ \text{crossings}) = I(k+1) - I(k)$ which means that

$$\sum_{k=1}^{m-1} P(k \ \text{crossings}) = \sum_{k=1}^{m-1} I(k+1)-I(k) = I(m) - I(1)$$

becomes a simply telescoping sum, meaning we end up with

$$P(m \ \text{crossings}) = \frac{2l}{\pi d} \left(\frac{d}{l} \arccos\frac{d}{l} + 1 - \sqrt{1-\left(\frac{d}{l}\right)^2}\right) - I(m) + I(1)$$

Additional resources

I have also created a Desmos environment where I have tried to visualize the integrals for each $k$. It can also be used for verifying the results provided by the Python simulation. Sliders are included to play around with $l$ and $d$, and all the probabilities calculated above are collected in a folder. I hope you will enjoy it!

Desmos Interactive