Activity 1-10 : Buffon’s Needle

Take a box of needles (or matches) and measure the length of a needle (we’ll suppose the needle is 1 unit long). Now draw a series of parallel lines onto A3 paper that are 1 unit apart. Drop the needle onto the lines from a reasonable height, and then count how many needles fall within the lines, and how many cross a line. Can you now estimate roughly the probability that a needle will cross a line?

The surprising thing here is that this experiment yields an estimate for π.

The needle will cross the line if d <. The probability of this is (white area)/(total area).

So what estimate for π does your data yield? It helps, of course, to do this experiment many times to improve our estimate, and a computer is a great help with this. Try one of the simulations for the Buffon’s Needle experiment that there are on the Net; a useful address (at the time of writing) is: buffon/bufjava.html Buffon Link

You will see that this simulation will allow you to vary the length of the needle. How does this affect the probability of a crossing? The mathematics for this can get complicated, especially if the length of the needle is longer than the width of the lines. But there is an easier way to look at things…

Imagine we bend the needle. How does this change the mathematics? The new needle can cut a line twice; but on the other hand, it is more likely now not to cut a line at all. The average number of crossings will be unchanged. So what if we bend it into a circle?

Let’s suppose the length of the needle is L, and the distance between the lines is d. The circumference of our needle is L = 2πr, so

The probability of a double crossing (all crossings must be double) is and so the probability of a crossing is Notice that when L = d, we have as before the probability

With thanks to: George Reese, for an excellent applet Carom is written by Jonny Griffiths,