 Hi, I'm Zor. Welcome to a new Zor education. We have just introduced some theoretical foundation under the theory of probabilities, which is actually based on the set theory and the measure theory. We have introduced the concept of a sample space with elementary events. Each of them is assigned a certain measure. In some simple case, all the elementary events are equally probable, so all of them have the same measure, and the sum of all the elementary events, some of the measures is equal to one. Basically, that's how the probability started. So the total probability is always one, the probability of something happening. Alright, so I would like to deal with certain operations on events in the same way as we had certain operations in the set theory, like union, intersection. So this is a continuation of the theory of probability chapter of this course of advanced mathematics, which is presented on Unizor.com. It's a pre-course for everybody. It's kind of advanced, and the purpose of this course is to basically not as much to give certain skills or certain number of facts, etc. The purpose of this course is to develop the creativity and analytical thinking in students. And that's why I'm paying a lot of attention to certain theories which are significantly deeper than is customary to study at school and also extremely important is problem solving. And I have a lot of lectures dedicated to problem solving. So anyway, so let's get into this particular lecture. It's, I call it event arithmetic. So I'm going to operate with events as with sets and subsets in the theory of sets, in the set theory. And the question is how the probability measure is dependent on what kind of operations we are performing with events. So I will start with an example which will be kind of a basis for this lecture. I will use it for all the different operations which I'm doing with events. And this example is a very simple one. Consider we are rolling two dice, and one of them has this type of outcomes. On another, we have the same outcomes. And on crossing of these columns and rows, I will put the result, what actually happens. So first, I would like to introduce the concept of a sample space. What is a sample space? A sample space is obviously all the different combinations of the first and the second dice. Now, we have six for one and six for another. And we can have obviously, for each of these, we can have each of those. So we have 36 different combinations. Now, let's assume that the experiment is ideal and all of these combinations have exactly the same probability. Now, since there are 36 of them, each one, each elementary event, which means that the first die shows, let's say, two and the second shows five, or any other combination. Each one of these combinations has the probability of 136. Now, the first thing which we were talking about when introducing the theory of probabilities was the concept of event, which consists of certain elementary events. So let's say I'm interested in an event that the sum of two dice is equal to, let's say, five. Now, this is an event. Now, how is it represented in the language of elementary events? Well, obviously, the sum can be equal to five if the first die shows one and the second four. So this is an elementary event which we are interested in. Also, two and three would show exactly the same sum of five. Three and two also, and four and one. So we have one, two, three, four. Four different elementary events. One, four, two, three, three, two, and four, one. All of them result in the sum of two equal to five, none other, by the way. So the probability is obviously some of these probabilities because they are completely exclusive from each other. If one shows three and another is two, it's completely unrelated to one shows four and another one, for instance. It's two completely unrelated elementary events. Each of them has a probability of 136. So four of them have the probability for 136, which is one lines. So that's the probability of this event. Okay, now that's one event. Now, I was talking about this lecture is dedicated to different operations on events. So we need another event. Well, let's just consider another event. Another event is when the sum is equal to eight. Let's think about what is the probability of this? Well, the probability of this is equal to, if I have a sum of eight, then the first one can be two and the second one should be six, or three and five, or four and four, or five and three, or six and two, and none others. We have one, two, three, four, five. So we have five, three, six. Now, and that's my first operation. Now, what kind of operations, by the way, we can introduce into the theory of probabilities? Well, considering that the theory of probabilities deals with events and elementary events, it's exactly the same as in the set theory, right? So we have operations of union, intersection, negation, maybe. So I'm going to expand these set theory operations into the theory of probabilities. So in our case, my first example is operation or union. So if this is, if this is event X, let's say, and this is event Y, what is X or Y, or X union Y? What is this event? Well, obviously, this event is that the sum of two dice is equal to five or eight, right? That's what or actually means. So all these elementary events which constituted my first event when sum is equal to five, which is this one, one, four, two, three, three, two, and four, one. All of these four elements satisfy this particular event because the sum is equal to five or eight. Now, in these cases, it's five. In these cases, it's eight. So these elements also correspond to the event X or Y. So what's the probability of this event X or Y? Well, it's all these plus all these should be summarized together. So it's four of these and five of these, so it's nine. So the probability of X union Y is equal to nine thirty six or one fourth. Now, what's important here that I just added these and these. When can I do it? Obviously, I can do it only in case these two subsets, these two events, have nothing in common. They are mutually exclusive, so to speak. There are no elementary events which belong here and there. If I have one event consisting of a certain number of elementary events and another consisting of other, completely other elementary events, then I can say that I can add them to get the or condition, to get the union of two events. If there is something in common, I cannot do it so simply. I will address it a little bit later in the lecture. But meanwhile, we have just learned that if my two events are mutually exclusive, then, and that's very interesting, I can just summarize it. It's the probability of X, now it's one ninth or four thirty six plus five thirty six. It's exactly nine thirty six plus P of Y. So, what's the final formula which we have arrived with, this one? The probability of or or union of two mutually exclusive events is equal to some of their probabilities. That's very important. Now, it means that the measure, the probability measure is basically an additive function and you can really just have a very simple formula, but only in case you have a mutually exclusive events. Now, let me just bring the parallel to something like area in geometry. If you have one particular figure and it has certain area and you have another figure, it also has a certain area, then union of these, which means everything here and there together, also has a certain area and the area of this combined figure is equal to some of these areas, because the area behaves exactly like the probability. The abstract concept in mathematics is the concept of measure. This is the measure, area is a measure, same thing as length, same thing as weight. Now, in case of measure of these two figures, this one and this one, you see there is a common part. And we cannot say that the measure or the area of the combined figure is equal to some of these two separately, because there is a common place, common piece. But if there is none, if events are mutually exclusive, then the addition actually works fine. The measure of their union is equal to some of their measures. Now, to indicate that events are mutually exclusive, we sometimes use slightly different notation. Instead of union sign, we'll just use the plus. Now, what this plus means? It's not addition, because these are events. We can't really add events. It means actually two things. Number one, that X and Y are mutually exclusive, and number two, we are combining them using the logical operation of OR, or a set theory of operations of union, which is basically the same thing. So these mutually exclusive events can be combined together using the OR operation, and since the formula looks very simple in this case, it's more for aesthetics than for anything else. But in this case, plus means actually we are ORing or unionizing two events which are mutually exclusive. That's what's very important. And then this nice formula actually takes place. If they are not mutually exclusive, there is a slightly more complicated formula which I will address in a little later. Alright, so let me bypass this. So we have come up with this addition theorem. Okay, so union of mutually exclusive events is basically like addition. Now, let's talk about intersection and its connection to multiplication operation. Alright, let's consider two events now. These are our old events, and we will consider new events. Events number one, whatever the first die shows is greater or equal to five. Now, what's the probability of this event? Let's just think about it. If I'm completely disregarding what's the second die shows, and I'm talking only about the first, it means that whatever the second die show still is good enough as long as the first one is greater or equal to five. So the first one should be greater or equal to five, which means it should be here and here. No, that's the second one, sorry. The first one is this one. So the first one is five or six, this and this. Now, the second one I don't really care, which means all these guys also are part of my event. So five one, five two, five three, etc., etc., all of them are good enough for this because the first one is five, which is greater or equal to five, as well as six one, six two, etc., up to six six. Now, the probability is six and six, it's twelve, so we have twelve, thirty-six. Now, the probability of the second event, now my second event is that the die number two shows less than or equal to four. So it's four and less than four. Let's think about it. So four and three and two and one, they're all good enough. So this is good, this is good, this is good and this is good. So it doesn't matter what shows the first one as long as the second not greater than four. Notice that we have certain number of elementary events common for both. So these are elementary events which are this one, one, two, three, four, twenty-four, thirty-six, which is two-third. And these guys, these elementary events belong to this event. Now, I'm talking about their interception. Now, intersection are events which belong to both, which is this, this, this and this. One, two, three, four, eight. So, if I'm talking about intersection between, okay, this is X and this is Y. X and Y or X intersection Y. It's only these which is eight, thirty-six, right? Or two-ninths, right? And notice a very important thing. If I will multiply this by this, one-third by two-third, I will get two-ninths, exactly this one. So, it looks like for this particular case I have that the probability of intersection equals to the product of probabilities, not liabilities. It's a very interesting observation and it's not a coincidence in this particular case. Let me just explain under what conditions this actually is happening. Consider this is your total sample space. Points inside of this square, these are these points, okay? Now, certain number of events is X and X. And certain number of events is Y. So, we have two events, X and Y. The X constitutes from certain number of elementary events and Y has inside certain number of elementary events. And there is something in intersection, some common elements among them, right? Now, let me just say the following. Now, what is the probability of X? Well, it's basically the part of the entire sample space occupied by elementary events which belong to X, right? So, if we don't know anything about what's happening, we can say that statistically speaking, if we will conduct a million experiments, then a certain number of experiments would constitute our event X. And the ratio of X happening versus the total number of experiments would be close to the probability of X, right? Same thing with Y. But what if I will tell you the following? You know what, guys? Event Y I know definitely is happening. In this case, under this condition, what would be the probability of X? Well, you think about this way. You know that event Y is happening. It means that the probability measure is distributed not among all the different elementary events inside the entire sample space, but only within the boundaries of event Y. So, we know for sure that one of these is definitely happening, which means that the probability measure should not be one, like in this case 136, but only one divided by the number of elementary events within the boundaries of Y. Now, if I want to know what's the probability of X under this condition, I should probably have a ratio of elementary events which are here in the common part relative to this number of elementary events inside the Y, right? So, let me just put some numbers in it. Let's consider that the probability of X is number of elementary events inside the X, which is lowercase X, divided by total number of events in an entire sample space. Now, probability of event Y means lowercase Y divided by N. This is number of elementary events here instead of Y. Now, what I'm saying is that considering the number of elementary events in their intersection is Z, which means the probability is Z over N, what I'm saying is that the conditional probability of event X under condition, notice this vertical line, under condition that event Y is happening, is basically only these common parts, which is Z, should be divided by number of elementary events inside the Y. That would be a reasonable definition of conditional probability, right? We will talk about conditional probability in another set of lectures. This is just a very brief look into it. But now, look at it this way. You can represent it as Z divided by N over Y divided by N, which is probability of intersection divided by probability of Y, right? So, what do we have now? Now, we have a very important consideration to make. Let me just introduce a new concept, independence. Events are independent from each other. Well, from the common standpoint, let me just address it this way. If you think that event X is completely independent of event Y, it means that no matter whether Y is or is not happening, conditional probability of X should be the same. So, conditional probability of X if Y happens, should be equal to unconditional probability of X if we don't really know whether Y happens or not happens. So, if my conditional probability equals to unconditional probability, then I can call events independent. And it's kind of a reasonable definition, which again we will address in some other lecture more precisely. But in this case what it means is, look at this, P of X is equal to this. So, for independent events X and Y, I can definitely say that P of X times P of Y equals P of X intersection Y, right? I just multiply this times this, should be equal to this. And this looks like our multiplication theorem. Now, remember if this is the union and events are mutually exclusive, then this is the plus. Now, if it's intersection and events are independent of each other, then this is an operation of multiplication. And that's why sometimes, again, we can use this notation. Just multiplication sign. It means events are independent of each other and we can use this theorem. So, the probability of end between two events or intersection between two events, in case they are independent of each other, is equal to the product of the probabilities. So, we have couple of theorems here. One was this, and another is this one. Let me write it down this way. Again, this is not a product in as much as this is not an addition. This is operation of union or operation or between mutually exclusive events. And then we can have this formula. Not this formula, obviously, but this formula. And in case of independent events, we have the multiplications theorem. And this is not a product. This is basically an intersection of independent events. Now, the last thing which I wanted to address in this lecture is basically, maybe I will use the same picture. And it's related to union of non-mutually exclusive events. So, not like this. This is a simple formula which works only for mutually exclusive events. Now, what if events are not mutually exclusive? Like in this particular case, we have something in common. Well, here is very easy thing to do. If I will have union y in this case, now let's think about what happens if I do this. Well, I'm counting all these elementary events and all these elementary events. What's important is I counted these twice. So, to make the count correct, I have to subtract once the intersection. So, number of events which are inside the intersection should be subtracted because it went into this number and went into this number so it went twice. Now, and obviously if there is no intersection, if events are mutually exclusive, then this member is not participating. But this is the formula for the probability of or condition or union between two events in case they are not mutually exclusive. This is a mutually exclusive case. This is not mutually exclusive case which adds this component into the equation. So, it's more general, but it's not as beautiful if you wish as this one. All right. Now, I will definitely address certain concepts which I am touching right now, especially conditional probabilities and independence in further lectures in more details. So, this is just the first view and my purpose was to introduce certain operations on events. The arithmetic, if you wish, I call it arithmetic. The operation which looks like a plus, but it's not really addition, it's actually a unionization of two mutually exclusive or not mutually exclusive in this case, events. And the theorem of multiplication, again, this is not a multiplication, this is just an intersection or end operation which kind of produced this nice looking formula. Well, that's it for today. Thanks very much. I recommend you to go through notes for this lecture, present it on Unisor.com. As you read the notes, you might just, you know, extra time reflect on all these concepts. And again, further lectures will be in more details devoted to concepts of independence and conditional probabilities. Thanks very much and good luck.