You can complain that I am telling more what you should not do after hearing this section. But it's very important because the most danger of probability theory is not that you cannot compute something. The danger happens if you believe that you have compute something and this is wrong and you make a wrong decision thinking because it's based on probability theory. Let me say something about what you should not do. The first thing you should be very careful is the probability of the past events, events in the past. To give an example, I just googled for some historical statement and that's what I found. It's from actual historical paper and it's quite probable that in 12th century that says something happened. You can say that it's quite probable, it's a mathematical probability. If the other guy asks what you think is the probability greater than two thirds or smaller than two thirds are probably, this has no meaning because two thirds mean that if we repeat the experiment many times, in two thirds of cases we will get something. But you cannot repeat the experiment. So the numerical value for this probability is something very strange you should avoid this. Another example more clear, so imagine I ask you what is the probability that I have a dollar bill in my pocket. And you may start to guess this, but it's not the correct thing to do because it's not, I either have it or I don't have it, and it's not clear what will the probability could mean there is no if I, each day I come with a dollar bill or without the dollar bill you can compute the fraction of the days when I come with it. But in one is the later case the question has no meaning, or we will say see later something when it's still can have a meaning but a different meaning. Okay. There is one very clear example of this case and this was a practical one. So let me tell you. So you know that first prime number. Prime number is a number that cannot be factored. So, I don't know, five is a prime but six is not because it's two times three. So a prime number you cannot factor and big prime numbers are rather important because we will see this later. They are used for cryptography for cryptography you take two large big prime number, you multiply them and nobody can fit them back. But anyway you need the big prime numbers and their algorithms which are used to decide whether a number is really prime or not. But the fastest algorithms are randomized. What does it mean that the algorithm used some internal coin. It's not a coin actually, it's some other physical device or some prepared random bits. But the answer of the algorithm depends on this very use of these random bits. So if we apply this algorithm many times to some number then it can give a different answer. And there is a proportion of correct answer. And if you write a good algorithm you can prove for example that for whatever number you take if you repeat the same algorithm of the same number but with different random bits you will get different answers. But most of them will be correct. The probability of wrong answer, I say is less than 1%. Okay this is perfectly good statement. What is not good if you say something like this. Imagine you tried this algorithm for some number. So I choose some number which was actually people don't know whether it's prime so they try this algorithm. So imagine you tried your good algorithm on this number, and the algorithm says this is prime as it did for this number, and then you say that look this number is prime with probability at least 99%. But this is complete nonsense. This number is either prime or composite. It's not prime with any probability. So you should avoid such a statement and you should see when somebody else makes such a statement and you should be very careful about the knowledge of probability for this guy who says this. This you should note. Now let me say a few words about what probability theory or how it can be extended. And at first the simplest thing which I already have said that is infinite probability spaces. So for example the meaningful setting if for example we want to take a random natural number and we agreed that 0 it would appear with probability one half, one with probability one fourth, and two with probability one eighth, and so on. So it's even easier to imagine an experiment which would produce this. So for example you can toss a coin and count the heads before the first tail. So you say zero if you get a tail immediately. You say one if it's head, tail, and something. You stop the coin tossing here. But anyway if it's two, this is two heads before tail and the probabilities are exactly one half, one fourth, one eighth, and so on. So it's kind of a reasonable thing to consider such a space. But if somebody says or take a random integer and make all of them equally probable then you should immediately complain. And this has no meaning because if they're equally probable then what is this probability. If it's positive then the sum becomes bigger than one for a large enough initial segment. So the probability should be zero and if all probabilities are zero, what does that mean. So equal probable integers is something impossible. Okay. This is the first remark and the second case we also mentioned. And when we take what we can find a random point in a square. If you go on the street and wait until the drop falls somewhere and it's reasonable to believe that it's a random point in the square and the probability to get it inside some zone is proportional to the area of the zone and so on. So it's infinite probability space are dangerous but not that dangerous. In many cases it's a reasonable thing. And another thing which somehow even the probability of individual event, which I told you that you should not consider, sometimes they can be considered in a different economic sense, you see. So imagine that we want to ask what is the probability of some guy, politician X, to be re-elected. And you can say the probability is 80%. What does it mean, of course you cannot repeat the experiment. You don't have many roles with the same guy in each of them and see how many of them and how in which proportion are the roles, and if he's re-elected. And so it looks like something stupid and nonexistent. But still imagine that some people don't like this guy to be re-elected and they want some insurance. The insurance is just a paper that guarantees you that if he's re-elected the insurance company will pay you, I don't know, $1 or any other amount but that would be $1. And the question is the market price of this insurance paper now. So if there is a market for such a thing and if the selling price is close to buying price then you can see that if it's, for example, around 0.8 dollars, then you can see the probability of being correct that is 80%. Of course it has no statistical meaning but still this number obeys some rules of probability theory. And you can even prove, somehow prove the same rule for disjoint events. So for example if you want to consider the probability of A being collected and B being collected and the probability for A or B being collected and there is some reason why the insurance paper for all three cases should have matching probability. The last one should be the sum of the previous two. So I will not tell you why it's the case but you can think about this and you will guess why it happened. So if not, the problem with my pocket has a dollar in it. There is no dollar actually. So the problem with this insurance is that nobody will sell or buy insurance on this, so there is no market. But for a more important event there is a market and if there is a market, actually you can bet that in some countries it's legal. You can bet from political events and so there is a market and this market can be considered the definition, the market price can be considered the definition of probability for individual events. And then we also get somehow probability space even if there is no repeatable experiment there.