 Hi, I'm Zor. Welcome to Unizor Education. We're still trying to understand what the probability is all about. This is the third lecture in this series. This lecture as well as all others can be found on Unizor.com with notes and that's what I recommend you to basically go to and read the notes before or after the lecture. That would be very, very useful for you. So the previous two lectures were about two different approaches to probability. One is frequency-based. So if you are repeating the same experiment under the same conditions again and again and observe certain condition, whether it occurred, certain event actually, whether it occurred or it didn't occur, and then use the frequency, number of occurrence divided by the total number of experiments. If this number tends to some number, that can number actually be called the probability. Another approach is based on the knowledge where if we know everything about how a particular experiment like flipping the coin actually is conducted, including all the forces involved, moments and air resistance and gravitation fields, etc., etc. If we know everything, then we can actually say that this is not a random experiment. It's a deterministic experiment. We can definitely predict the result of it, right? So if we don't know what the result of the experiment is, that means we don't know all these factors. So basically the probability is the measurement of how much we don't know about certain event. All right, now this is the lecture about why do we actually go through all these extremes and try to understand what the probability is all about. Well, let's just think about it. Mathematics is an abstract kind of a subject. But before being an abstract, before mathematicians converted it into an abstract subject, it had actually certain practical usefulness. Now, what's the usefulness of probability? Where did we start from? What's the point where people realize that the probability and learning the probability, studying the probability, is really a needed activity in our observation of the world? Well, there is a very, very simple explanation. Well, people needed the probability and studied the probability basically to predict something. You see, the only purpose of studying the probability is to say that, well, you know what we think that something and something might or might not happen with both certain probability and the quantitative characteristic of this probability signifies the likelihood of this particular event to happen or not to happen. So the prediction, predictability of everything which is related to the future which we would like to know in advance, that's all about the probability. And that's the purpose. Now, let me go one by one to certain points which I would like to actually make. Okay, the first point is we know that the probability of something which is related to, well, let's take the dice rolling. We would like to talk about the probability of having, let's say, three on top. Right? So dice and we would like to have a number three on top. Now, if we don't know anything about how this dice is made, what forces are involved, what's the gravitation forces, air resistance and stuff like this, then we can just say that any number can be on top with basically equal chances. And since there are only six numbers on the dice, that means that every one of them has the probability of one six. Okay, so the probability of having a number of dice of three is equal to one six. This is the symbol which I will use in this particular case. It means that dice falls with a three on top. And the P stands for probability. All right, now, what does it mean actually? Does it mean that from every series of six throwing, we will have exactly one event when the three is on top and others will be different? Absolutely not. That's absolutely not the case. How about if we will increase the number of experiments? Let's say 600. Does it mean that it will be 100 times when the three will be on top, the one six of the total number of experiments? Absolutely not. It can be 99 or 117 or anything like that. However, what the theory of probability says, and this is the frequency approach, that if number of experiments will tend to infinity, then the frequency will actually be closer and closer to one six. That's what it says. So, what I can actually say that the frequency of dice of three will tend to one six as number of experiments tends to infinity. That's what basically theory of probability says. So, if you have 600 experiments, yes, one six of this is 100. It means that it probably will be relatively close to 100, but not necessarily exactly. If you will take 6,000 experiments, it will be close to 1,000. But exactly 1,000? No, not necessarily, again. But this closeness in relative terms as the number of experiments goes to infinity will actually be closer and closer. The number of real frequency, real number of occurrences of number three on top divided by the number of experiments will be closer and closer to one six. So, it will be a sequence of values which are getting closer and closer to one six. That's what theory of probability says. Nothing more than that. It does not predict exact numbers at all. Now, similarly, let's take a look at the coin. Coin has two sides, heads and tails. The probability of each is one-half if we don't know anything about the coin. So, the probability of coin falling on the heads is equal to one-half as well as probability of falling on the tails. Now, same thing here. Does it mean that from every, let's say, 100 experiments, it will be 50 tails and 50 heads? Absolutely not. But, however, the number of heads divided by the total number of experiments will be closer and closer to one-half as we increase the number of experiments. That's what theory of probability says. Now, let's consider some other events, not elementary events like in this particular case. What can we say about the coin throwing? If you would like to study the event that either tails or heads will actually be the result of the experiment. Yeah, but, hey, there are no other results, right? So, the total probability is equal to one-half plus one-half. I'm looking to both results. There are no other results. So, the probability of this is equal to one. So, when probability is equal to one, only in this particular case we can say that the particular event really will happen. So, that's when deterministic process actually takes place. So, what does it mean that the coin actually falls tails or heads? It means it's not really standing on the edge, right, which probably is impossible. So, the probability of having either tail or head is actually one. That's when the random experiment actually is converted into deterministic one. Let's consider another event, again also related to the coin throwing. What's the probability of not having either tails nor heads? Well, but, again, there are no other variations, well, except on the edge, which is impossible, right? So, the probability of this event is equal to zero. Now, why the probability of this particular event is equal to zero? Well, you know that the probability of any event is constructed by adding up elementary events from which actually it's built, right? So, what elementary events are in the event that it neither has nor tails? Well, none of the elementary events. And that's why the probability is zero. We don't have anything to add up. And conversely, if you would like to know the probability of either heads or tails, then we have to add both probabilities, one half and one half, and that would be one. So, the probability of one means the event will actually happen and our experiment is no longer random. It produces consistent result and the result is happening of this particular event. And the probability of zero is also a deterministic process when the particular event is never actually happening at all. So, we can predict the result. Predictability is certain when something is zero or one. In all other cases, it's chance, right? Okay, next. Next is an interesting question. Which actually kind of counterintuitive. If you throw the dice ten times in a row, is it possible that you get three on top ten times in a row? Well, the answer is yes, it is possible. However, how can you calculate the probability of this? This is actually an interesting question and it's very much related to combinatorics, which I was telling you a few times before that combinatorics is actually prerequisite to theory of probabilities as we are studying it here. So, let's talk about the dice, which we are throwing ten times in a row and we are actually interested in certain number of times out of these ten to have three on the top, right? So, this is basically a task. Consider ten times we throw the dice and we need to be interested in K times three on top. Okay. First of all, whenever you are talking about some kind of a probabilistic problem like this, it's very important to realize what are your elementary events. Well, look, if we are throwing a dice ten times, then our result of the experiment is ten numbers, which are on the top. Each number is from one to six and we have a sequence of ten numbers. For instance, one, two, one, six, four, three, one, two, what is it? Four, four, three, five. For instance, this is ten numbers and these numbers are result of the tenth throwing. Any other sequence of ten numbers can be the result of ten throwing. What we are interested in is how many of these different results contain exactly K times number three. In this case, we have two times. We need K times. So, every result of throwing the ten times the dice is the result of the tenth throwing. It's basically equivalent to any other result because the throwings are completely independent. We don't really put any kind of dependency on the result of the previous, let's say, throwing onto the next one. We just randomly pick up the dice and throw it, which means that any number can have the same chances to appear on every time we're throwing the dice, which means that on the first place, we can have one or two or three or five or six and on the second place, we can have exactly the same and on the tenths place, we also can have one or two or three or four or five or six, which means that the total number of combinations of ten numbers, each one of them being from one to six, is six times six times six times six, so six to the tenths degree. That's total number of elementary events. So, these strings of numbers, ten numbers, each one of them from one to six, each such a string represents an elementary event. And all the elementary events are completely equivalent, completely symmetrical. They have exactly the same chances, which means that since total number of events is this, then the probability of each event, like this one, equals to one over six to the tenths degree. Right? Okay, fine. We have determined our elementary events. This is always the very first thing which you have to think about when presented with a theory of probability problem. Okay. Now, we have to basically find out how many of these elementary events construct an event which states that there are k number threes among these ten. Well, again, this is a combinatorics problem. First of all, we have to pick which k places three should actually be. And that can be done in number of combinations from ten by k. Okay? Now, after we have chosen one particular set of k places, we place three there. So, there is no choice actually. It's three and three and three and three, whatever. But the other ten minus k places, they can have actually any number on any of those places except number three, right? Because we have reserved k spaces for number three. And the other ten minus k spaces are supposed to be anything but three. And there are only five candidates. So, five candidates for ten minus k places, give us five to the ten minus k number of combinations. And we have to multiply them because for each of these, there are all of these. So, that's number of different combinations of ten numbers with exactly k numbers three among them, on different places. Now, since this is a number of elementary events which satisfy our event which we are talking about, the number three occurs k times. And the probability of each such event is one to the sixth to the tenth degree. Then, obviously, if I will have so many things and each of them has this particular probability, then this is the probability of number three to occur exactly certain number of times, k times. In this case. Now, in the notes to this particular lecture, I put exact numbers for all k from zero to ten. Now, if I'm not mistaken for k equals to zero, it's something like zero point sixteen. For k equals one, it's zero point three, two. For k equals two it's zero point nine. For k equals three, zero point, I think it's fifteen. For k equals four, it's zero point zero five. And then, down, down, down. So, the probability to have ten, for instance, for k equals ten, to have ten numbers three is very, very low. Actually, we can calculate it exactly. It's supposed to be three and three and three and three and three and three. So, that's actually one of the elementary events. So, that's one to the sixth to the tenth degree, which is a very low number. So, basically, the probability is concentrated more within these four. So, the number of numbers, the count of number three being on top among ten throwing is somewhere between zero and three, with a maximum between, like, one and two, which is basically around one-sixth, right? One-sixth of ten is what? One point six or something like this. So, it's somewhere between these two. And that's why these two have the highest probability. So, statistics actually will follow the combinatorics in this particular case. And that's really the base for frequency-based concept of the probability. So, we can say with certainty that, you know, what was a very high probability, the number of trees among ten throwing will be somewhere between zero and three. It's very unlikely that it will be, like, four or greater. I mean, it's possible, but it's unlikely. So, if we will have a series of ten throwing, and then another ten throwing, and then another ten throwing, then the three, number three will be between zero and three times, is significantly greater than all other variations. And that's what it means that the probability of this particular event is whatever it is if you will summarize these numbers. What is, it's something like, it's like zero point nine altogether. So, something like 90 percent probability that among ten throwing you will have no more than three times number three occurred. Okay. So, that's something which I wanted to talk about evaluating the probability using the frequency. And that's where the combinatorics actually plays very important role. By the way, if you will put it on a graph, you will have something like this, zero ten. So, it will be something like this, with zero, one, two, and then three, and then down. Okay. Okay. What's next? All right. What's very important about approaching the probability with the frequency of occurrence is to have exactly the same experiment repeated again and again. If you would like to evaluate the probability using the frequency of occurrence of particular event, you better make sure that the experiment is exactly the same, that the conditions of experiment are exactly the same. And then, only then, you can say that, okay, the probability is something which can be approximated with the frequency of occurrence as number of events is increasing to infinity. Now, infinity is obviously very far away and we will never reach it, which means that we will be closer and closer to the probability, but well, if not necessarily, we will be, you know, exactly where it is. Certain knowledge, knowledge of combinatorics, etc., might actually help you to calculate it and then you can verify it using the real experiments. So, this is how we can calculate, actually, the exact probability. And we can say that, okay, if we will conduct the experiments of throwing ten times the dice, we will have more or less these results. Okay. Now, a perfect example of this approach, which is not really working because the conditions of the experiments are changing is stock market prediction. Now, there are two different approaches to stock market predictions. One is technical and another is based on fundamentals. So, the technical is, okay, let's just investigate how the stock behaves in the past. And based on this, we calculate certain statistics, certain trends, certain whatever, and that's how we predict what will be in the future. Well, as you know, nothing works perfectly. There are people who are, during a certain period of time, are winning using this strategy and then they lose it. So, it's not really that clear. Now, why is it happening? Well, it's happening because the market conditions are changing. They accumulate the statistics based on certain statistics. They have certain time frame. And then, the time change, time is changing, certain economical conditions are changing, currency tribulations, political upheavals, wars, or God knows what else, they all change the market. They change the behavior of the people participating in this market. And that's why the strategy is no longer working. And then those people who are involved in this technical analysis, they are starting from scratch, basically. Okay, forget whatever it was before. Now, we have completely different conditions, new experiments, and let's just calculate it again and again. And here I must actually add that there are basically two approaches to this technical stuff, technical analysis of the market. You can, for instance, you can check the market day after day after day the closing numbers. Or you can change it, you can check it like minute by minute. What's important is that if you are using the minute by minute statistics and let's say forecast it for the next minute. So you watch it for 10 minutes and then forecast for the 11th minute. Then you watch it again, you use the experience of 10 minutes and forecast for the next 11th one. Now, in this particular case, the chances that your conditions are changed very low because one minute is not such a big time. But from day to day, especially overnight, changes might actually be much more substantial. So it's more probable, if you wish. I have used the same word, probability. It's more probable to predict the market correctly if you're using a frequent checking of the market with a very small time interval between the checkings and you project it onto a very small interval. Like you're checking a minute by minute and then projected for the minute. The chances you are correct are greater than if you are checking, let's say, day after day after day and projected for the day. Or months after months after months and projected for another month. Because the conditions are much more changeable during a longer period of time. So now, I think we have exhausted the frequency-based probability and how it's related to predictions. Let's talk about the knowledge-based. So as you know from the previous lecture, knowledge is very important for calculating the probabilities. And the more you know, the more uneven the elementary events might actually be. So if you know that your dice is loaded and one particular side is much heavier than others, then you can say that the probability of that side to be on the bottom is greater than the other side. So that actually brings whatever is on the opposite side on top as a much more probable result of the experiment. So it's not one-six, one-six, one-six, etc. It would be one-half, let's say, for one and something less, much less than one-six for others. Now, if you know like everything about everything involved in a particular experiment, you basically act like God. So you know all the forces, all the momentum, all the resistances, everything, then you can basically tell that, okay, the result of experiment will be such and such and you will be exactly right. That's very, very important. So knowledge is very, very important. And the probability of not being equal among elementary events is basically a manifestation of the fact that you have certain knowledge about this process. So let's just consider a couple of examples. So back to stock market, you have people who are using the fundamentals so-called. So they investigate the state of economics. They investigate what this company has invented during the last time. What are patents which the company has, what's its market share, etc. I mean, all the conditions which are basically involved in making a judgment about whether the company is good or bad to invest money in. Now, those people, if they are capable to basically have all these factors into account, they will be relatively precise with their evaluation of how the company will behave in the future. And they will be able to give a more or less exact recommendation to buy or not to buy that particular stock. So that's the probability which is kind of skewed from being completely neutral towards something where you know exactly what you want to do. And let me add one more example. Example related to atom bomb. In 1930s, people realized that based on the famous equation of Einstein's theory of relativity, if you split uranium, there might be something like a chain reaction and a lot of energy would be basically, it would be a big explosion with a huge amount of energy because the parts which are splitting the nucleus of uranium have less energy based on this equation than the energy of the uranium. So the amount of energy which is supposed to be freed during this reaction might be actually very, very big and can be the basis for the atom bomb. So the question is how much uranium you need to start this chain reaction because there are certain neutrons which are being freed during this particular process and they are breaking other atoms of uranium, et cetera. So that's how the chain reaction. So if you don't have too much of uranium, then the number of neutrons produced might not be sufficient but if you accumulate sufficient amount. So the question was basically what is the amount of uranium which is necessary to produce the bomb so it will basically explode. If you have two halves of this necessary amount and put them together, it will blow up. Well, obviously at the time, it was 1930s, people didn't have a lot of knowledge about how all these reactions are supposed to happen. And there were actually two, well, I don't know how many people, there were certain scientists who conducted certain purely theoretical calculations. So they used whatever the theory, whatever the knowledge they had to predict how much exactly we need, how much uranium we need to start the bomb. Now, obviously there are so many factors in this particular equation like, for instance, the purity of the uranium, temperature, pressure, I don't know, many, many factors. So obviously not every factor they were actually included into the calculations which means they did not have an absolute knowledge about the fact. They had partial knowledge. But even that partial knowledge allowed them to estimate the approximately amount of uranium should be from this to this. And exactly when they conducted actual experiments, obviously, and it appears to be that they were almost right. So I would like to say that instead of just saying, okay, we don't know, we just have to conduct experiments and be on the frequency-based, they decided to be on the knowledge-based of the probability that they have narrowed down using that knowledge the number of elementary events and the probabilities which would be happening in this particular case. So that's how knowledge is contributing to the quality of the predictions. So that's exactly my point. The quality of predictions would be great if you have the probability of one as a result of this. If you don't, at least have 0.9 or 0.8. Because if you don't have any knowledge, then every elementary event is equally probable and that actually doesn't produce any reasonable results for prediction, for predictability. So predictability is basically shifting off the equal distribution of probabilities among elementary events. So that's all about predictions. I do recommend you to read the material of this lecture on Unizord.com. It's a little differently maybe presented there, but basically the same material. And well, that's it. Thank you very much and good luck.