Pi and Buffon's Matches - Numberphile


Uploaded by numberphile on 12.03.2012

Transcript:

TONY PADILLA: We're going to do Buffon's needle trick.
So this is named after Georges Louie Leclerc, and this is a
method that he proposed for working out pi.
So basically he proposed that you would chuck a bunch of
needles just randomly on a floor.
You can work out pi from that, yeah.
BRADY HARAN: Go on, then.
TONY PADILLA: Due to the random nature.
So anyway, we haven't got needles, and we're not going
to do it on a floor.
We're going to do it on one of Brady's lovely
brown pieces of paper.
And we're going to use matches instead, because we're cheap.
So what we want is we want to draw some lines on this.
So there's one line.
And what we want is we want the gaps between lines to be
two matches long.
This is all very approximate.

What we're going to do is we're going to chuck a bunch
of matches on this board.
So I'll just chuck the first two completely randomly.
We need a lot of matches for this.
If I counted these right, we should have 163.
Let's just spread them out.

So hopefully, there should be 163 matches randomly
distributed around this board.
Now, this is what I've gotta do now, is I'm going to count
how many matches cross a line.
OK, so I reckon that one does.
So you count, Brady, that was 1, 2, 3.
5, 6, 11, 12.
13, 14, 51, and I think that's it.
52.
52 matches that cross the lines.
There's 163 in total.
Now, let's see.
Let's do the following calculation.
So 163 divided by 52 is equal to 3.13.
And that's not bad.
That's not bad, I'm quite happy with that.
BRADY HARAN: That's very close.
TONY PADILLA: Yeah, yeah.
So pi is 3.141.
So this is not bad.
OK, so why isn't it perfect?
Because this is not a very large sample.
Viewers could have a little go themselves.
Just literally play the same game.
If you've got more matches, that's better.
You'll get a better accuracy.
There was a mathematician called Lazarine.
He used over 3,000 needles, and he actually managed to get
pi accurate to I think it was about six decimal places.
What we're seeing here is something is akin to a Monte
Carlo simulation.
That's what it's called.
You basically take a random sample.
The reason you get pie out of this is it's all to do with
the angle that the match takes as playing some role here.
And of course if you're working in radians, if you go
through a full 360 degrees, well, in radians that's 2 pi.
OK, so that's where pi's coming into this game, and
that's why you can get pi out of this sort of
setup, because the angle.
So we're going to try and explain the maths of it now.
Let me say, imagine that the length of the match is l, and
that means that the distance here is 2l.
There's a few things that play a role here.
There's the position of the center of the match, which
I'll call there.
We'll say that's a distance x from here.
And then the other thing that we're interested in is the
angle the match is making.
So let's just draw it, I guess, like so.
We're interested in that angle beta.
Assuming everything's completely random.

how do we measure the probability of where the
center of the match is?
Well, the probability density of the center of the match,
which is position x, is just given by essentially 1/l.
The other thing that we're interested in is
the angle it makes.
So this is sort of a measure of the probability of the
angle it's making.
And basically, the angles that we're interested in are
somewhere between 0 and pi/2 which is of course 90 degrees.
This is where the pi's coming in.
Now, if we want to work out essentially the probability
for the match to cross the line, what we do is
we take these two.
We have to do an integration.
OK, now there's an important point here.
What's the condition for the match to cross the line?
Well, the condition for the match to cross the line is
that x should be less than l/2 sine theta.
So that means when we do the x integration, we go between 0
and l/2 sine theta.
OK, and of course when we do the theta integration, we're
going between 0 and pi by 2.
So we do it, theta equals 0, this is pi/2, 2 over l pi.

2 over l pi.
When we do the x integration, we get an l over 2 sine theta.
And then we integrate sine theta between 0 and pi by 2.
That gives me 1, and I've got the 1/pi there, so
the answer is 1/pi.
This tells me the probability that you're going to cross the
line, right?
So if I've got n matches, and I want to work out how many
cross, let me call them-- so this is the n total.
And I want to work out how many cross.
Then I just multiply and total that.
1/pi.
So n total is 163.
And n crossing was 52, I think we got, didn't we?
OK.
So what you see from that if we've got 163/pi.
Let's put an approximate sign in there.
Approximately 52, or if you like, what we actually saw was
that pi, just rearrange this equation, so
approximately 163/52.
Yeah, I'm happy with that.
Yeah, yeah.
That's good.
So that's how it works.
That's the theory behind it.