 Hi, I'm Zor. Welcome to a new Zor education. I would like to continue talking about theory of probabilities in a little bit more formal way. This lecture is part of the whole course of advanced mathematics for teenagers, primarily high school students, which is presented on Unizor.com. And I do suggest you to first read the notes to this lecture, because every lecture actually has notes. Before you listen to this particular lecture, it will help you to get into this formality or a little bit more formal approach to theory of probabilities a little better. So anyway, I spent some time during the previous lectures explaining what actually probability is about. We have introduced a few concepts like random experiment, event, elementary event, probability. But we didn't really approach it from more rigorous standpoint. Well, the rigorous approach to theory of probabilities was initiated by mathematician Andrey Kolmogorov, who actually built the foundation as solid mathematical foundation based on a set theory, theory of measure, and certain theorems related to measuring of infinite sets. We will concentrate only on final sets and final results of any random experiment. So if we are talking about random experiments of let's say rolling a dice, we are talking about finite number of outcomes. So in this course, I'm talking only about certain finite number of elementary events or events in general, or random experiments with certain number of outcomes. Everything will be finite. It's much easier. However, the concepts are exactly the same as in real foundation of the theory of probabilities based on infinite sets. Alright, so our purpose right now is to translate the probabilistic language which we were using before into more mathematical language related to set theory and measure theory etc. Alright, so this lecture is dedicated only to random experiments with symmetrical outcomes. So all outcomes and there are only finite number, let's say n different outcomes from random experiments. And these elementary events, as we call them, all they have equal chances to occur. So in this particular case, since they all had equal chances to occur, it means that as the number of experiments tends to infinity, the frequency of occurrence of one particular outcome, elementary event out of n different, is supposed to tend to one nth. Because that's what actually means that all our elementary events, all our outcomes have equal chances to occur. Equal chances means as the number of experiments goes to infinity, and there are n different outcomes which are completely symmetrical, it means that the frequency of each one of them is tending to one over n. Now, frequency is obviously the number of cases when this particular elementary event occurs divided by the total number of experiments. Alright, what does it mean? It means actually that we can associate the number one over n with any particular event, an elementary event out of whatever the possible outcomes are, and we can call it the probability. So this is a general explanation of the probability. Now let's talk about how can we translate it into a little bit more mathematical language. Well, first of all, I should consider elementary events. Now, elementary events are those events which cannot be subdivided any further, and in this particular case we're talking about elementary events which are completely symmetrical. Examples, rolling the dice, so one particular number on the top, being one or two or three or four of our six, are elementary events. They're all symmetrical, they all have exactly the same chance to occur, and that's why we associate with each of these six different elementary events the probability of one six is rolling the dice, right? Now, flipping the coin, we have two results, tails or heads, and again they have equal chances, which means that we can associate one half as the probability of each elementary event. Now, from these elementary events we can build some other events like, for instance, what's the probability of having an even number on the top of the dice? Even means two or four or six, so it's basically a combination of certain elementary event, and that's what we call an event. So, any kind of event can be decomposed into a certain number of elementary events which are completely unrelated to each other. So, the event, even number on the top, is equivalent of two or four or six on the top, so it's basically a combination of these three different elementary events. What does it remind you? Well, I think it immediately reminds the set theory and the ability to combine certain subsets to get bigger subsets. So, if you consider a set of points, for instance, and the point can be actually our model of the elementary event, then if you would like to combine certain elementary events into a bigger event, it's like combining a certain number of elements of a set, points in this particular case, into a combination, into a bigger subsets. So, that's the key to whatever abstraction we are going to make. So, what we have actually talked about, we talked about all the probabilistic terms, and now I'm going to translate it into terms which are a little bit more mathematical, all right? So, here is what I'm proposing to do. Now, we have a random experiment and the results, the elementary, the smallest results, the outcomes of these random experiments, that's what we call elementary events. Now, I will consider a set of elements as a model of a random experiment with certain number of elementary events as outcomes. So, the set of elements is basically a set of elementary events which can occur as a result of random experiment. Okay. So, now we are talking about random experiments with finite number of elementary events. So, that's why we have to consider a set of, not just of elements, a set of finite number n elements, all right? So, the mathematical equivalent of the random experiments with n outcomes is my set of n elements. Okay. Now, we are talking that we can combine certain elementary elements into elementary events into an event, like from the different elementary events which are the outcomes of the rolling the dice, we can combine three of them, two, four, and six, and said, okay, this constitutes an event of having an even number on top, right? So, what I can talk about equivalency in the set theory is any subset is an event. So, my elements are elementary events and the subset which is a combination of elements, obviously. Any subset can be an event. Well, there are many different subsets. Actually, there was a very interesting problem of how many subsets does a set of n elements have. So, that actually was discussed before. But anyway, so, we have some kind of an equivalency, equivalency between random experiments with n outcomes with a set of n elements. Any event which is a combination of elementary events is basically a subset of this set. Now, let's talk about the probability. Well, we have actually assigned the probability of 1 over n for every elementary event in the random experiments of n different symmetrical outcomes. So, what I can say here is I can say that the measure of each element is 1 over n. So, what does it mean? I've just associated a measure. Now, what is some kind of an analogous consideration of this? Well, it's actually very simple. For instance, you have a field, let's say, and you divide it into squares. And each square has a certain area, right? Let's say it has one square meter, right? So, these are my elements, which basically are equivalent of elementary events. And any combination, let's say this one, three elements, it constitutes an event. And if the measure of one particular element is known, then the combination of these three elements can be calculated as a sum of them, right? So, in this case, it's three square meters. So, what I'm saying is that the measure of a subset, that the measure of a subset is a sum of measures of its elements. So, we are combining certain elements which constitute our subset and their total measure, which is a sum of measures of each element, is a measure of our event. So, this measure is a probability. So, the probability is equivalent to this measure, as we say. We can measure the angles, we can measure areas, we can measure lengths, we can measure lots of different things. So, we can measure elements. How can we measure elements? Well, I'll just assign the measure, which is the numerical value of one end to every element. And I can assign the measure of any subset as a sum of the measures of all elementary events. Now, this analog with area is very, very useful in the theory of probabilities. You can always just keep it in your mind that this is a picture of the probability. Probability is a measure. All right. So, we've got the measure of every element and that's the equivalent of the probability of the elementary event. You've got the probability of any subset, which is an equivalent of the probability of any event, which is the combination of elementary events. What else? Okay. What's the probability of having no events at all? Well, that means that... What does it mean that nothing happens? Well, it's an empty subset in our set theory. In our new terminology, if we want to know what's the probability of nothing happens, well, the probability is the sum of all the measures of the elements which constitute an empty subset. Well, how many elements are an empty subset? Zero. Right? So, out of n elements which are constituting our set, zero elements combined together constitute the empty set. And obviously, since the number is zero, the measure of the empty set is zero. So, the probability of occurring nothing is zero. Now, what's the probability of occurring something? We don't really care what occurs. So, some event occurs. So, either the dice falls on one or two or three or four or five or six. So, anything goes, basically. Well, what is this? That's the probability of an entire set. So, all the elements should be combined together. The measure of each one is one over n, and there are n elements. So, we have the probability of the entire full set of the entire... I think it's called full subset, actually, is equal to one. So, the probability of nothing happens is zero. Probability that something happens is one. So, these are two extremes. Everything else is in between zero and one. So, if my subset contains m elements in it, then the probability associated with this subset, its measure, is equal to m over n. All right. So, we know how to measure the probability of every event. We just add the probabilities of elements which constitute that particular subset. So, event is a subset. It contains elements. So, we add them up together. All right. That's covered. And what's also interesting is, let's go back to this example of having an even number on top of the dice, two, four or six. Well, what's very important in this case is to understand that you have the probability you have the probability of each event, right? So, the probability of two is one, six. Probability of four is one, six. Probability of six is one, six. And the probability of two or four or six equals two, three, six, which is one over two, one half. You see, the probability is not just a measure, it's additive measure. So, the measure is additive if you have two different subsets. So, this is your set. It has elements, okay? Now, this is one subset and this is another subset. Now, you have measure of this thing and you have measure of this thing. Now, let's consider a subset which contains this and this element. So, all elements. It's basically the result of a logical or, all right? So, the probability of a point being an element of this or element of that base basically is equal to some of these probabilities because the measure of the combined area is basically a sum of these two areas. As long as they don't intersect, obviously, because if they intersect, then you count twice something. But if they don't, if these two events are completely independent, then their measure is always a sum of measures and that's why their probability is measure of these two probabilities. So, that's kind of an explanation of what actually this equivalent equivalency to measure theory is about. So, what's important is to understand that to have this parallel between theory of probabilities and more mathematically expressed set theory and measure theory is to understand that we need to have certain elements of this analogy. We should have a set of elements. Each one of them is an elementary event. We have to have measure and we really have to have this measure as additive measure. It should be additive. And we can always say that the measure of the entire set should be equals to one and that gives us the baseline. So, everything else, the subset of this set would be smaller than one down to an empty subset which has a measure of zero. So, that's basically my explanation of how to provide more mathematical background to theory of probability. And that actually allows to use the purely mathematical apparatus to theory of probabilities. Before this foundation, this mathematical foundation was developed for theory of probabilities, we could not really calculate lots of different very, very sophisticated things in the theory of probabilities. It lacked that mathematical apparatus. Well, since the foundation was basically established, since we can say that the set theory and measure theory are basically the foundations, the theory of probabilities, we can have full scale of mathematics to deal with theory of probabilities and its problems. Now, let me repeat that in this course, I'm talking only about finite sets of elements as being equivalent to certain random experiments with finite number of occurrences. Now, in most of the cases, all these occurrences are equal chance. So, the probability of every element, which is one of the elements in that set, which is an analogy of the random experiment, would be equal to one over n, where n is the number of elements or number of outcomes from these random experiments. We will deal only with these finite distributions of probabilities. Well, that's basically it for what I wanted to say. Okay, so finally, the sample space, which is a random experiment with all the outcomes, basically, is equivalent to mathematical set. The elementary event is element of this set. Any event is a subset and probability is additive measure. Editive in the sense that if you have two subsets, which do not have any intersection, then the measure of two of them is equal to the sum of measures of each one. With a condition that the entire set has a measure of one, basically. So, the probability of anything happen is one, basically. That's what it is. All right, so that's it for today. This is the lecture about certain mathematical foundations in case our experiment produces n-symmetrical outcomes. I will spend some time to talk about asymmetry if certain elements are not exactly the same as others. But the analogy is very much working quite well in this case as well. That would be probably the next lecture. So, that's it for today. I do suggest you to read again the notes. You know, when you're reading something which is written, it's completely different kind of sensing the material than you're listening. So, it will complement each other. It's basically the same material just maybe presented in a written form. I also encourage you to sign to register basically on theunisor.com because it allows you to take exams, for instance, and basically to make the whole educational process a process rather than just occasional reference points for this particular lecture, that particular lecture. I do suggest you to take an entire course as basically self-study or homeschooling or whatever else. That's it. Thank you very much, and good luck.