 Hi, I'm Zor. Welcome to a new Zor education. This lecture begins the topic of theory of probabilities. I will try to explain certain aspects of the theory of probabilities, not on the very, very mathematically rigorous level, but on a level I believe sufficient to understand what actually probability is, how to calculate certain probabilities and how to deal with the problems. All right. Well, everybody knows what probability is, right? I mean, you flip the coin and everybody can say that, okay, the probability of the tails and heads actually is one-half or 50%. Well, yes and no. Let me tell you this. About two or three even hundred years people, mathematicians, were struggling with attempts to rigorously define the probability. And there were different approaches to this. The final mathematically rigid approach was the foundations of theory of probabilities, which were laid down by mathematician Andrey Kolmogorov in the 1936 article, which he published, the basic principles of theory of probabilities. But this is a very, very mathematically rigorous definition based on the measure theory and certain additive properties of the measure, etc. I am not going to do it. I'm not going to explain it that way. So I will attempt to do it in a way how historically it was developed by different mathematicians. And there are three lectures, actually, which I am planning to dedicate to the issue of what actually the probability is about. So this lecture is about approach to probability based on frequency of occurrence of certain things. Now, intuitively, you feel that if you will flip the coin ten times, it's not really necessary that you will get five tails and five heads, right? If you will flip it one thousand times, it's not necessary it will be 500 and 500. Well, it will be close to this particular numbers, but not exactly, right? So we all understand it. So how can I approach the probability from this frequency of occurrence standpoint? So that's basically what this lecture is all about. The first concept which I would like to talk about is a concept of random experiment. Now, when I'm talking about random experiment, let me just write it down. Random experiment. So when I'm talking about random experiment, I'm talking about certain experiment which can be repeated any number of times, ten, thousand, million, billion, whatever. I'm also very much concerned about this particular experiment being conducted under the same conditions again and again and again. So all these billions of experiments with flipping the coin must actually obey this rule. It should be the same conditions of this experiment. I mean, it should be on the planet Earth, it should be a proper, I don't know, barometrical pressure. The person who is flipping the coin is not really intentionally trying to do certain tricks or whatever else. So the repeatability and similarity of conditions of random experiments and what's also important obviously is that experiment can result in different outcomes. So there are certain deterministic experiments. I don't know, you just lift, for instance, some kind of weight and then you let it go and it will drop down and you will probably be able to calculate the speed and the place actually where it will fall down, etc. This is deterministic experiment. Flipping the coin is not. So there are certain different results of this experiment. So any kind of a result can actually be called an event. So we will talk about certain events. Now back to some kind of experiment. Let's consider you are doing the deck of cards in bridge. So you have 52 cards in the deck, you deal among four people, four players equally. So everyone has 13 cards and among 13 cards I know there are four aces. So event which I'm interested in might be that each of these four players has exactly one ace. I'm not specifying which one, but one for each. This is an event. Now our experiment is shuffling the deck of cards and dealing it among the four players. So that's an experiment. What makes this experiment repeatable? Well, there is nothing which restricts us, we are just shuffling and dealing, shuffling and dealing. If we are not, again, playing some kind of tricks with cards, we're just normally shuffling the cards or maybe putting the cards in some shuffling machine and then deal. So that actually makes our experiments, our random experiments similar in the sense of conditions they are provided under. So now if I will repeat this experiment a certain number of times, I will or will not have my event happened, occurred. So either these four players have each one ace or they don't. Now if I will conduct this experiment again and again and I will calculate the frequency of this particular event. So out of a thousand for instance experiments, certain number of these dealings were with four people having four ace one each and certain others did not. Now out of conducting another thousand experiment, I will have some other number of situations when four people have an ace each. So I can always calculate the frequency of occurring this particular event. Now, and here is an assumption. So let's assume that as the number of experiments is increasing to infinity, the frequency which is number of times this particular event occurred divided by the total number of experiments is actually changing towards certain number. Now our experience shows that if conditions are as I was explaining, so it's a repeatable random experiments under the same conditions, we do it again and again and again. So our experience shows that the frequency of this particular event or any other or not other with many other events actually tends to certain number. This is an assumption. If that is the case, then I can call this particular number the limit of the frequency of occurring of this particular event. I can call it the probability of this event. Now, other examples. Well, obviously the coin. Another example is dice, for instance. You are throwing the dice and an experiment which you can actually think about is the number on the top is even, which means two or four or six, right? There are six numbers. So one and three and five are no good. If the dice falls one of these numbers up, that's not occurrence of our event. But if it's two or four or six, it's occurrence. Well, I can calculate basically the frequency if I am throwing the dice again and again. And as this particular process continues to infinity, my frequency will be tending to certain limit. And that limit should actually be called the probability of this particular event. So this is how I introduce the probability. It's the limit of a frequency of occurrence of certain event. If this limit exists, as the number of experiments goes to infinity, again under the same conditions. So everything is the same from one random experiment to another. Okay, I have a certain plan here. Okay, another interesting concept which I will probably assume always. These random experiments are independent. So not only I'm going with the same experiments again and again and again, I'm also stating that the results of the previous experiment should not really be in any way affecting the next experiment. Let me give you an example. For instance, to throw the dice you have some kind of glass, let's say. You put the dice in, you are shaking it, and then you are throwing it away. So that's one way of doing it. Another is you throw away the dice by hand, but you don't really shake it. You take the dice as it fell whatever the side up. And without changing the position you are throwing it again with the same effort. Well, chances are that this is not exactly an independent random experiment, because the result of the previous experiments basically is considered to be the beginning condition, the position of the dice in your hands for the next experiment. So that's not independence. Independence means that there is no such effect of the result of one to the conditions of another experiment. So independent events are very important and most likely we will mostly continue with independent events. Well, until we will consider something else. So we talked about independence, reliability. We talked about examples. Okay, so next topic is we considered an event of having, for instance, four players to have one ace each, or another event, an even number on the top of a dice, right? Well, there is a concept of elementary event in most of the cases. Now elementary event is such an event which is in some way the smallest. And out of these smallest independent events you can basically construct the event you would like to receive. For instance, if you are talking about an event in throwing the dice to have even number on top, this actually an event which can be constructed from three independent events. Two on top, four on top, or six on top. So let me just tell you that one particular number to be on the top can be actually considered as elementary event. Why? Because most of the events which I would like actually to talk about, they can be constructed from this one. If I want to construct, for instance, an event even on the top, I will basically consider these three elementary events. So either this or this or this event, elementary event, would constitute this one. An event, for instance, with card of deck, dealt among four players, well, any individual distribution of 52 cards among four players, 13 each, actually can be considered as such an independent elementary event. And from these elementary events, I can actually construct any event which I need, which deals with distribution of cards. Now, in the first case, in the case of a dice, I have six independent events, right? Six elementary events. Now, in case of dealing the cards, well, the prerequisite for this course is combinatorics. So I will just use the combinatorics. Basically, 52 factorial divided by 13 factorial to the power of four is the number of elementary events. That's the number of different distributions of the deck of 52 cards among four players, 13 each. We actually solved the problem in one of the combinatorics course lectures. Okay, so there are different elementary events. Now, what's important is elementary events are not really overlapping each other. For instance, what is an overlapping event? Okay, one event is, on the top of the dice, I have number which is greater than four. Another event is, I have number which is greater than five. Now, these are two events, and I can actually check using the frequency how often this happens and how often this happens. But think about it. If this happens, then this definitely happens as well, which means that every event which falls into this category falls into this category as well. They are overlapping, so to speak. Now, if I would basically break it down into events greater than four means five. Sorry, it's greater. So it's five and six on top. This is six on top. Now, I was just telling that elementary events with the dice throwing are very conveniently can be characterized by any number on the top. So one, two, three, four, five, six, these are elementary events. So these are not intersecting in any way, and obviously this. And from these we construct more complex events which might or might not have this overlapping. You really have to remember the set theory. You remember there is an intersection, there is a union of the sets. That's basically what it is. Now, this event is a union of these two events, elementary events from the set theory. Now, this is, well, basically there is only one, so there is only one event here. But in any case, constructing an event from elementary events basically is unionizing in the sense of the set theory of these elementary events. It's like basically having some big area and you divide it into small areas and you're unionizing this, this, and this, and this to get this area. So these are elementary events and this is an event. It's like smallest subsets out of which we can create any other subset. Smallest subsets of the natural numbers are natural numbers. One, two, three, etc. Now, if I want numbers, natural numbers which are from five to seven, that's the union of five, six, and seven. So basically that's what it is. So we introduced a concept of event and elementary event. So let me write down these two words. So we have random experiment, we have independence, independent experiment, we have event and we have elementary event. Out of these elementary events we can construct any event. All right. Now back to our dice. Let me just give you an example. Something on the top, some number. Is it an event? Well, that's an event. What's interesting is it's a combination of all the elementary events, one on top, two on top, three, four, and five, and six. So basically all the elementary events combined together make a full event. And by the way, what's the probability of this full event? Well, the frequency of this is obviously one, because any throwing of the dice results in something on the top. So every number of experiments, whatever I do, ten or a hundred or a billion, will end up with ten or a hundred and a billion occurrences of my event that something is on the top. Because something is always on the top. Well, except those cases when the dice goes on some kind of a vertex, which hopefully doesn't happen. All right. Now, let me just repeat once more. It's very important to have repeatability of these random experiments under the same conditions. It's very important to realize what are the elementary events. And in many cases, elementary events, if chosen correctly, have a very important quality. They are equally, they have equal rights. They are symmetrical in some way. So if my cube is really symmetrical, then throwing it would actually give no advantage of one particular number on the top over another. So if dice is not loaded with some kind of an extra weight lead at that one particular site, then you can definitely say that all different sites have the same chance to be on the top. If experiment is really random. So the random, this word random is very, very important. So in this particular case, it means that no site of the dice has any advantage. In case of dealing the deck of cards, what's important is that any set of 13 cards for one particular person, for a player, is equally probable as any other. So occurrence of every elementary event is really the same as any other. The frequency of occurrence. What does it mean? It's very important actually. It means that if you know the number of elementary events and you know that these elementary events are symmetrical, they have equal probability of occurrence, the frequency, let's not use the word probability yet, so the frequency of occurrence of number two on the top. If the number of dice throwing goes to infinity would be closer and closer to the frequency of occurrence of number, let's say five on the top, right? So that's what makes them symmetrical. The frequency of two and frequency of five and frequency of six and any other number from one to six will be tending to exactly the same number. And since the total probability is always one, we are talking about frequency, of occurrence divided by total number of experiments. So if you will combine all these elementary events, they don't intersect with each other, right? So the combination of them must always be equal to one. So as we increase the number of experiments with the dice, then every frequency of one or two or three or six or whatever should actually tend to one-six. To the number one divided by the total number of elementary events. Now, in case of shuffling and dealing the deck of 52 cards, well, I told you how many different distributions of 52 cards among four players are, right? So it's 52 factorial divided by 13 factorial to the power of four. So the probability of each particular distribution of 52 cards among four players would be inverse, which is 13 factorial to the power of four divided by 52 factorial. That's the probability of each particular distribution. Now, since we can always find out how to construct any event from the elementary events, we can always add the probability of each elementary event from which our event occurs, and that would be the probability of the event. So if we are looking about even number on the top of the dice, even number is two, four or six, right? So it's three different elementary events out of how many? Out of six, right? So two has the probability of one-six, four has the probability of one-six, and six has the probability of one-six. So if I need two or four or six on the top, then I'm adding these probabilities, and I will get three-six or one-half. So by introducing symmetrical elementary events and understanding how from these elementary events any other event which we are interested in can be constructed leads to the probability of that particular event through the calculations. What's next? Okay, now I would like actually to give you two examples of how this mechanism is working. So example number one. I have two dice, and I'm throwing two dice together. I'm interested in the event at least one dice has greater than four on top. That's an event, right? If I'm throwing two dice, I'm interested in a combination when at least one of them has a greater than four number. How can I calculate this probability based on whatever I know by now? Well, as I was saying before, we have to understand what are our elementary events from which we can construct our event, and the elementary events we are talking about must be, well, in some way symmetrical so we can very easily determine the probability of each one of them. So what's the elementary event when I'm throwing two dice? Well, it's a combination of two numbers, where each of them from one to six. How many combinations are? Well, six for this and six for this, obviously 36 different combinations. Now, is any one particular combination has some kind of greater rights than another? Let's say combination three-five. Now, is the combination three-five is supposed to occur more frequently than the combination two-one? No, answer is no. All these combinations, I have chosen specifically these combinations in such a way that they are really symmetrical. So any two numbers, and if the dice are made of solids without any kind of extra weights on any side, then the probability would be the same for every elementary events, and I have 36 different elementary events, so each one of them has the probability of one 36, right? So the probability of, let's say, one four is equal to one 36. The probability of two two, one 36. Or any other combination of two numbers. Okay, that's good. So we know our experiment, we are throwing the dice unlimited number of times again and again and again. We know that the frequency of occurring of any two numbers is approaching one 36 as the number of experiments goes to infinity. That's a reasonable assumption, which basically is derived from the symmetrical of our experiment, right? So now let's go back to our event. At least one of these two dice has greater than four on top. Well, we can actually calculate how many elementary events are supposed to construct this one. Well, let's just think about it. At least one of them greater than four. So I have actually two different ways to calculate it. The way number one, now what does it mean that one of them, so the first one should be either five or six, right? Now the second one can be one, two, three, four, five, six. And this one, one, two, three, four, five, six. So these are the combinations and these are combinations. Or maybe it's the second dice, it has five or six, right? So we have first dice can be anything and the second should be five or one, two, three, four, five, six and six. Now these are six combinations and six combinations and six and six, it's 24. But very important, we count it twice. The combination which contains both, five or six in the first and the second dice because since both dice are within five or six range, we counted it once for this group and once for this group. And there are two combinations each, so it's two and two, so it's four. Four different combinations, five, five, five, six, six, five and six, six. We count it twice. So we have to subtract four and we will get 20. So 20 different combinations of numbers, one and two, where at least one of them is greater than four. We're counted this way. We can count it another way. Now what does it mean that we have at least one of them greater than four? Well, it means the same thing as if we subtract from the total number of combinations, which is 36. We subtract those combinations which do not have five or six, which means these combinations are only from numbers from one to four. So the first dice is having one of the four numbers from one to four and the second one, so it's four times four, it's 16, and if we will subtract this, we will get the same number 20. So 20 is the number of elementary events which comprise our event which we are talking about. At least one dice has a greater than four number. Well, what does it mean? Well, very simply. We know that there are 20 different elementary events which comprise together one event which we are talking about, right? All these elementary events are the same in a way that they have exactly the same probability of 136, right? Any pair of two numbers has a probability of 136. Twenty pairs of these satisfy the condition of the event we are looking for which means that the combination we are looking for has the probability of 136 which is, what, five-ninths? Maybe one-half, by the way. It doesn't matter. So this is how we calculate. So first we think about what our set of elementary events and it's preferable to have these elementary events as symmetrical in some way and not intersecting to each other. Like in this particular case, the number of elementary event is any pair of numbers. Two numbers. Each one from one to six. And they're obviously not intersecting. The combination two and four is definitely different from four and four or two and six or there's no seven. So any other combination. So we have the number of elementary events. We calculate how many of them are and if they are symmetrical and hopefully they are, the probability of each of them is one over the number of elementary events just to have the complete full event to be equal to one. The probability of full event to be equal to one. And then from these elementary events we construct our event and think how many elementary are falling into our event. And I actually draw you a picture. Let me just draw it again. For instance, you would like to measure some area because the area has some shape, difficult shape. How can you measure this area? Well, you take a measurement unit. Let's say one square with a side equal to whatever, one meter. And then you put all these squares here and you cover the area with these squares and you just count how many fall into your area. So you know the measurement of this square and you multiply it by the number of squares which fit to this area. Well, hopefully they fit exactly as in our case. Alright, so that's how we deal with probability from the frequency standpoint. Repeatable random experiments, elementary events which are supposed to be symmetrical and that's why these elementary events can have the probability which we can actually calculate. And then from these elementary events we construct any event we want to and count how many elementary events belong to this particular event we are interested in and that's how we calculate this probability. Now, I would like to complete this lecture with a very interesting example which is actually a wrong example. Sometimes people are saying, well, the probability of the next earthquake is like 90% or the probability of the flood in New York which is similar to the Sandy hurricane flood is whatever, a number of... They put some kind of numerical characteristic on this. But think about it, from the frequency standpoint and I'm talking only about frequency. There are other aspects, there is a knowledge aspect that would be the next lecture, but from the frequency standpoint this has absolutely no sense because there is no such thing as an experiment like a flood in New York which you can repeat again and again with the same conditions or earthquake or anything like that. So all these are actually completely irrelevant and all these statements they are more emotional or ideological even and they basically represent certain level of risk which people perceive subjectively to this or that event but the numerical characteristic which they put on it usually has very, very little foundation behind it. So that was my end of this story and hopefully you will go back to Unizor.com and read this lecture again. It's only reading, there are no problems to solve but I do suggest you to read because maybe as you read it and writing I can explain certain things maybe better than orally. So that's it, thank you very much and good luck.