Today we're gonna talk about the Galton board invented or, at least named, after Francis Galton.
What we have is a bunch of pegs and a bunch of slots.
Now when I turn this over all of these balls,
3,000 of them, will fall through those pegs and they'll scatter
randomly (in the sense that we don't know where each individual ball will go) but
because there are so many of them, out of chaos,
comes order.
Look at all that wonderful wild randomness turning into pretty much the same exact shape every single time.
A normal distribution.
Let's begin by talking about the pegs.
The pegs on my Galton board are arranged in a
triangular pattern like this. We start with one peg. Below that peg, in the second row,
there are two pegs on either side. The third row contains pegs arranged like this. The fourth like this, and so on.
Now it's because of this arrangement of pegs that the Galton board is sometimes called a "quincunx".
A quincunx is a pattern of five
objects - a certain way of
arranging them - where four of the objects are on the corners of a square and the fifth is in the middle.
You're probably familiar with the quincunx if you've ever looked at a die before.
Now anyway, when the balls fall through the machine, they first reach peg number one in the first row. And
once they do that, they have a choice.
Well, they don't really have a choice,
but they will either go left or right and because of the way the Galton board is designed,
there's about a 50/50 chance that any given ball will go to the left or to the right. Once that's happened,
they've reached the second row of pegs where they will either go left or right,
left or right. Now because at each peg a left or right bounce is
equally likely, in the long run, across a large number of trials,
we expect balls whose complete paths contain about the same number of left and right turns to be the most common,
which means we expect more balls to wind up pretty much right below where they started and
many fewer on the edges which corresponds to having way more of one kind of bounce than the other.
We don't know where any individual ball will wind up
but we can say something about how they will all behave.
Now the actual probability that an individual ball will fall in any one of these slots follows what is called a "binomial distribution".
But there's a nice thing in mathematics called the Central Limit Theorem
which tells us that under a large number of trials with a large number of objects (like 3,000 balls), a binomial distribution
approximates a Normal Distribution, which is what you see drawn here on the board.
But this Galton board doesn't just demonstrate the Central Limit Theorem. It's got a bunch of other cool features. For instance,
look carefully at the pegs. You'll see a
hexagonal tiling with numbers. That is Pascal's triangle.
Let's build Pascal's triangle ourselves.
I happen to already have a beautiful
set-up here: a triangle with 1 2 3 4 5 rows
Now to fill in Pascal's triangle, you start at the very top and you put the number 1.
Great, now the rule for filling in the rest is that you take each
spot and you sum the two numbers that are above into the left and above and to the right.
So for this cell we have nothing plus 1 which is 1, and for this one we have nothing plus 1 which is 1
Now down here, we have nothing plus 1 which is 1 but here we have 1 plus 1 which is 2. Now you continue this pattern...
That's a 1. That's a 3. 3. 1.
Down here, a 1 again and then a 4 and then a 6 and then a 4 and then a 1.
Now you'll continue doing this for as long as you want. This is
very cool.
It shows us all kinds of neat things.
One is that the number on each cell can correspond to a peg on the Galton board and
the number in that cell tells us how many paths a ball can take to get to that peg.
So when it comes to hitting the first peg, there's only one way to hit that first peg: you fall out of the hopper.
But once you've done that you either can go left or right
Well, to hit this peg in the second row. There's only one way to do it:
you have to have gone left when you hit the first peg Same here except you had to have gone right.
But when it comes to this peg, I've numbered 2, there are two paths that can take you there:
you can either go left and then right or you can go right and then left. Two paths. Now what this means
is that here, when a ball reaches this third row,
it will have traveled along one of one plus two plus one - four - unique paths.
However, two of those paths take you to that peg. The other two take you here or here.
So if you wanted to place some bets, it'd be safest to say that more balls will be going to this peg than to the two
on the outside. Now this pattern continues on and as you can see, there are more and more paths
downward as we continue but the middle destination is always served by the largest number of paths
possible for a ball to take, which means that we should expect more balls to show up in the middle and
fewer as we go further out to the edges. Tada! A normal distribution.
But we're not done with Pascal's triangle yet. There's all kinds of other fun things you can do with it. For instance,
Let's take a look at the diagonals. Okay, I'm gonna do diagonals like this and that's why I drew hexagons.
So if I draw a diagonal out from this first number,
I get a 1 now. If I go from this number diagonally out, I get 1 again. If I go from this one,
I have a 1 and a 1 which sums to 2. If I start here and I diagonally go out
I have 1 plus 2 which is 3 and
I'm gonna erase that arrow because it's in my way. Where's my eraser?
Beautiful. You might already know the pattern if you're a big fan of
fibbing... Just kidding, Fibonacci...sequences. This is what we're going to be finding here
1 plus 3 is 4 plus 1 is 5 and sure enough, we will be building ourselves a
beautiful Fibonacci sequence in which each number is the sum of the two numbers before. So, 1 plus 1 is 2.
1 plus 2 is 3. 2 plus 3 is 5. 3 plus 5 is 8. 5 plus 8 is 13, and so on forever,
no matter how big you make your Pascal's triangle.
But even that's not it! Pascal's triangle produces something even more amazing and helpful.
This represents and gives you, row by row, the sequence of coefficients in binomial powers. Let me show you what I mean.
Now if you have a pulse and are breathing then I think we could all agree that binomial
powers are one of your favorite things. A binomial is a...
an expression with two terms in it. For instance,
"7xz", that's a monomial. But if I add in a, I don't know, let's do "7x^2" now
I've got one, two terms. I have a binomial.
But when you raise a binomial to a power,
things get a little bit...
extra fun. Let's take a very simple binomial: "x + y". Now if I square this...
what do I get?
Well first, let's use FOIL to find their product.
How beautiful! (x + y)^2 = x^2 + 2xy + y^2.
Why am I bringing this up? Look back at Pascal's triangle.
The second row -
second, SECOND, row -
contains the sequence "1, 2, 1". And what are the coefficients in our answer?
1. 2. 1.
Do we dare raise a binomial to the fourth power? Of course not.
Just kidding! Of course we do.
"x + y"
to the fourth! Alright now, this is super fun. Here's how we can break this down.
You begin with this first term "x" and you raise it to that power, 4, now
we're going to add to that a whole string of x's two powers that go down by one each time, 4 3 2 1 0 and so on
So the next term will be "x" to the third power and then
"x^2" and then "x" to the first power and then "x" to the zeroth power
which is just 1 so I don't even need to really draw it because
this is where the "y" goes and "y" times 1 is just "y". y's exponents go in the reverse order, falling down from 4
to 3, to 2,
to 1, to 0, which is "x"- which is- which is equal to 1 so I'm not going to need to write it there.
We're almost done but the problem is we don't know what
coefficients should go in front of each of these terms. But what we do know is Pascal's triangle.
Let's look at the fourth row on Pascal's triangle.
1 4 6 4 1
So all we need to do is put in "1 4 6 4 1"
Tada! We're done. Thank you, Pascal's triangle, and thank you Galton board for showing me all of these neat things and more.
So at the end of the day, the Galton board is a... almost chilling reminder that out of randomness, unpredictability,
order can come, across enough trials or involving enough objects.
And so what this means that although our universe is certainly full of unpredictable and random events, especially at the quantum level
big things like us are pretty gosh darn predictable and stable. So, thank you randomness,
thank you even more statistics,
and as always,
thanks for watching.