Activity: Buffon's Needle
A few hundred years ago people enjoyed betting on coins tossed on to the floor ... would they cross a line or not?
A man called "Buffon" started thinking about this and worked out the probability. It is called "Buffon's Needle" in his honor.
Now it is your turn to have a go!
You will need:
A match, with the head cut off.
(You can use a needle, but be careful!)
A sheet of paper with lines 50 mm apart.
- Measure the spacing of your lines (it may not print at exactly 50mm): ____ mm
- Measure the length of your match (must be less than the line spacing): ____ mm
- Make sure your sheet of paper is on a flat surface such as a table top or the floor.
- From a height of about 5cm, drop the match onto the paper and record whether it lands:
A: Not touching a line
B: Touching or crossing a line
The exact height from which you drop the match is not important, but don't drop it so close to the paper that you are cheating!
If the match rolls completely off the paper, then do not count that turn.
Now we will drop the match 100 times, but first ...
... what percentage do you think will land A, or B?
Make a guess (estimate) before you begin the experiment:
|Your Guess for "A" (%):|
|Your Guess for "B" (%):|
OK let's begin.
Drop the match 100 times and record A (does not touch a grid line) or B (touches or crosses a grid line) using Tally Marks:
- Are the bars the same height?
- Did you expect them to be?
- How does the result compare with your guess?
We Can Calculate What It Should Be ...
Buffon used the results from his experiment with a needle to estimate the value of π (Pi). He worked out this formula:
- L is the length of the needle (or match in our case)
- x is the line spacing (50 mm for us)
- p is the proportion of needles crossing a line (case B)
But today we are going to "change the subject" of the formula to work out the "p" (the proportion of B):
|Start with:||π ≈ 2L/xp|
|multiply both sides by p:||πp ≈ 2L/x|
|divide both sides by π:||p ≈ 2L/πx|
And we get:
Example: John had a match of length 36 mm, and a 50 mm line spacing.
So John has:
- L = 36
- x = 50
Substituting these values into the formula, John got:
2 × 36
π × 50
So John should expect the match to cross the line (case B) 46 times out of 100
Fill in the following table using your own results:
|Length of match "L" (mm):|
|Line Spacing "x" (mm):|
|Estimate for p (= 2L/πx):|
How close were you?
It won't be exact (because it is a random thing) but it may be close.
Different Size of Match
Try repeating the experiment using a different sized match (but not larger then the line spacing!)
- Did you get better or worse results?
What You Have Done
You have (hopefully) had fun running an experiment.
You have had some experience with calculations.
And you have seen the relationship between theory and reality.