Another physical device to show the laws of probability theory, it was designed specially for this. This is what is called Galton Board, and also the other name is B Machines, so if you look at Wikipedia page, it's called B Machine. What it is, it's a vertical thing. You see the picture I think of a real machine from Wikipedia. And what is happening, so the beans go here and then they fall down through this hole, and there is some kind of cylinder and beams go down. And what is random here, they can go left or right, and the idea is you can not predict for an individual beam whether it will go left or right. But half of them normally will go approximately half, will go left and then rather how go to the right. And after that they go fall down and they see another, meet another cylinder. And in this cylinder again, they can go left or right. So at the end they are collected in these boxes. And what you should, what is interesting here that the central boxes are full, and the boxes far from the center are almost empty. So there is some kind of concentration effort. Those beans have a tendency to concentrate near the center of the machine. This is the effect which should be explained. You can also look at the video at Wikipedia page, it has a nice video how it works. [SOUND] Now, the analysis of this machine. So, we assume, this is assumption about the real world, we assume that at each level beans are split evenly. So here if we take it, the total number of beans we denote as one, then half of them goes here, and half of them go here. And then they go here is also one-half and one-half, and here we get one-fourth, and one-fourth, just this one-half is split into two equal parts. And here we also have one-fourth, and one-fourth. What is new on the next level is that these two things combine. So, together we have two-fourth here, and only one-fourth and one-fourth here. On the next level, the same thing happens. This one-fourth is split into equal parts and two-fourths are also split into two-eights, and one-eighth and one-eighth. And we have a combination, let me write here, it's one-eighth, three-eighth, three-eighth, and three-eighth. So there is some general rule how we fill the table. So if, for example let's imagine we look at this place. So imagine we have some, I don't know, x, and here with some y fraction, here we have x/2 and y/2, and here we have x, the sum, and get x + y over 2. So we assume that at each splitting, the flow goes into equal parts. This is the rule, so you can easily fill the table and probably you see that this table is similar to, I think we should know, pascal triangle. So, assuming the beans are divided evenly, this assumption about real world, but then after that we have mathematical computation, and this is the rule that we used. And this is the table that we got, and this tables is just, you see this is the power of twos, one, two, four, eight. And this things are like in pascal triangle. And this is pascal triangle divided by powers of two and if you want to write a formula, it's like this. It's binomial coefficient, k is the number number, and the line, and n is the number of the line itself. And this concentration effect now can be estimated quantitatively, you can just compute and see clearly, if the model is correct, which fraction of the bean should be near the center. And you can find just the number. You can take a hundred layers, then the beans are from 0 to a 100, this k is from 0 to 100. And if you decide for example the center is between 40% and 60%, you can just compute with a easy program what fraction of beans goes into the center part. And if you compute the same for 1,000 layers, it's also not very difficult, for a computer you need an area size 1,000 you don't need to fill the entire area, you just want to go from top filling line by line. And if you do this, then you see what fraction is here, and you will see, I promise you, I don't remember the numbers, but I promise you that for this part, the concentration effort is much stronger. Almost all the beans will be near the center. So this is the same effect somehow, that we observe with frequencies, but now it's not for just one sequence. It's just we'll look on each individual B, and somehow correspond has some individual trajectory. And we count the number of trajectories going in different, ending at different places.