I want to say more things about

the division of probability theory between natural science and mathematics.

So if you ask some people on the street,

what is probability theory,

maybe some of them will tell you that the probability theory says that

the coin should get head and tail equally off.

But actually, strictly speaking, this is not the case.

Mathematical probability theory doesn't care about the coins at all,

it's just a part of mathematics.

And if somebody makes a coin which has two tails on both sides, of course,

this expression will not destroy mathematical probability theory,

and just the coin is bad,

but theory is good.

So again, let's see what is separation line.

So natural science care whether the real coin

behave according to this equal head and tail model.

And mathematics studies the implication of this model

not for one coin tossing is not much to study.

But still, the mathematics starts with the model and

then finds what are the consequences of this model.

Of course, normally mathematicians work together with some other scientists,

so if the model is not good,

they found this and they tried to suggest a different model and so on.

But purely theoretical there is a division like this.

Let's look at some special case where it illustrates this division.

So imagine we tossed two coins, for example,

or you can just take two coins and do something like this.

And you see something,

what do we see?

Oh, now we see two tails.

But the question which we to

ask is what is the probability to have one head and one tail?

And just imagine a discussion,

this question is asked and then Alice says something like that,

that there are two coins,

so the first can be head and the second can be head,

or first head, second tail,

or first tail second head or two tails.

So there are four outcomes and we are interested in which fraction

of the coin experiments we will have one head and one tail.

So if all four outcomes are equally probable,

there are two of them which corresponds to one head and one tail.

The probability is two of four,

so is one half. This is what Alice says.

But now, some other guy comes, Bob,

and says, look you see that why there are four outcomes.

You have two coins,

you tossed them and when you look you

don't know where is the first coin and where is the second coin.

So essentially, there are three outcomes,

two heads, head plus tail and two tails.

And so if we assume that these three outcomes appear equally often,

then the probability of this outcome is one of three.

And so, who is right here?

Then a pure mathematician,

Charley comes and says you are both correct,

you just have different models.

And I don't care about real coins,

but if you want you can consider this model or that model, why not?

They are just different assumptions.

And, of course, if you want to know what happens really,

it's not the mathematicians who should be asked.

We should ask some, I don't know,

natural scientists or physicists and he will tell you that indeed Alice is more right.

So for the real coins just it's an experimental observation.

These four outcomes have the same probability.

And so if we classify things according to book scheme,

this outcome will appear twice more often than this or this because in

this model there are two cases which corresponds to one case in this module.

So this is the distinction between mathematics and natural science.

Let's look from this viewpoint on Galton Board,

what we assume really.

So one can say that the assumption is at each level,

half of the beans go left and half go right.

And yeah, of course we assumed this,

but this is not all because you can remain in, for example,

if a board, just depending on the board construction,

maybe the beans actually when they make a first move they get some velocity.

And if they go to the left then in the next move they

will also go to the left because they already have this direction.

So maybe in the physical board it can happen that

some decision on that level can influence the decision,

the level K plus one is, for example, going left,

makes the next left move more probable.

But we assume our assumption,

our computation was based on the assumption this doesn't happen.

So we assume that probability theory language it's independence,

we assume that amount of the bean that go left,

if we consider only the bean that go left at level one,

half of them will go left at level two,

and half will go right at level two.

So the second decision is somehow independent from the first.

This was our assumption and we made computation based on it.