Estimating Pi using Buffon’s Needle

A few hundred years ago people enjoyed betting on coins tossed on to a wooden floor … would they cross a line or not?

A French mathematician called Georges-Louis Leclerc, Comte de Buffon (1707-1788) started thinking about this and worked out the probability.

This probability is called “Buffon’s Needle” in his honor.

Buffon then used the results from his experiment with a needle to estimate the value of π (Pi). He worked out this formula:

Where

• L is the length of the needle
• N is the total number of needles
• C is the total number of needles crossing a line
• W is the line spacing (Width of the wooden boards on the floor)

We have decided to simulate this experiment using a Python script (using Python Turtle).

Task 1

Run the above script and count how many needles (out of 50) are crossing a line. Apply Buffon’s formula to estimate the value of Pi using:
π ≈ 2LN / CW

• L is the length of the needle (L = 30 pixels
• N is the total number of needles (N = 50 needles)
• C is the total number of needles crossing a line
• W is the line spacing (Width of the wooden boards on the floor W = 40 pixels)

Task 2

Adapt this Python script to automatically detect if a needle is crossing a line.

Your Python script should then count how many needles are crossing a line and use this to estimate a value of Pi.