 Hi, I'm Zor. Welcome to a new Zor education. We continue the course of advanced mathematics for teenagers, and we are talking about theory of probabilities. I have spent some time in the previous lecture explaining certain more mathematical approach to theory of probabilities based on the set theory and the measure theory. Everything, whatever I'm talking right now about is presented as a course on unizor.com, and I do recommend you to read the notes for this lecture, which is basically a set of problems before you are listening to this lecture. Try to solve these problems yourselves. There are answers, so you can check yourselves and only then listen to this lecture. Very briefly, let me just repeat the results of the previous lecture about the definition of the probabilities through measure theory based on the set theory. We are talking about random experiment which results in certain elementary events, and this random experiment has certain number of elementary events, etc. Now, in case when probabilities are equally distributed among elementary events, if there are n events, then every one of them has a chance, the same chance as any other, and the probability is assigned as 1 over n. The measure theory actually tells you that you can consider this set of elementary events as just a set of certain elements, and every element is allocated a certain measure, which is equal to in this case 1 over n. And this measure is additive, which means that any subset of this set has a measure equal to a sum of measures of its elements. So, in this case, I would like to introduce certain number of problems which are actually nothing but a very trivial repetition of the same idea. Just to exercise how you can approach the probabilistic problem through the viewpoint of set theory and measure theory. So, here is the condition which I have. This is basically the condition of all the problems in this lecture. We have eight people in the room. A, B, C, D, E, F, G and H. That's their names. Now, the person A is a male and 19 years old. Person B is male and 18 years old. C is female, 18 years old. B is male, 20 years old. E is male and 24 years old. F is female, 28 years old. G is male, 33 years old. And H is female of 16 years old. These are people which are in the room and my random experiment is just pick any particular person from the room. Okay, that's an experiment. So, what's the results of this experiment? Well, the results can be either A or B or C or G, etc. So, there are eight different results. And since I'm talking about a random picking of the person from the room, I should really allocate the probability measure of one-eighths to each and every one of them. So, here are the problems which I would like to consider. The problem number one. What are the set and its elements representing the results of this random experiment? What are the elementary events? And what is the measure of the probability of the entire sample space? Well, basically, I have just already told you. We are talking about a set theory and a measure theory. So, from the set theory standpoint, I can say that the set which represents this experiment is basically a set of eight names or letters or whatever you want to see. That's my omega. That's my sample space. It's a set which contains elements from A to H. Each element is representing an elementary event. So, my set represents a random experiment and each element of this set represents a particular elementary event which is a result of this experiment. Now, what's the probability measure which is allocated to an entire set? Well, that's basically the entire probability of basically having something, which means, by definition, actually allocate the measure of one to an entire set. So, in this particular case, probability is equal to one. So, that's the first problem. I mean, these are the problems we are dealing with. They are very simple problems and only the purpose of them is to illustrate the ideas of using set theory and measure theory to study the probabilities, the theory of probabilities. Basically, again, the theory of probabilities is nothing more but a measure and set theories combined together and applied whatever the mathematics about sets we know. Alright. Next problem. What is the measure of probability allocated to each elementary event? Okay, we already talked about this. The measure of probability of A is equal to one-eighths. This is eight. And one-one-eighths. That's what it is. And this is exactly the same as the measure of probability of any other event, B, C, D, E, etc., up to H. So, each of these elementary events are allocated a measure of one-eighths. And any combination of events is basically having a measure of some of the elementary events which are included into this subset. Let's say if I have a subset which contains only two elements, G and H, well, the probability of this event, which is basically a measure of a subset, is equal to some of the measures of these components, which is one-eighths plus one-eighths, which is two-eighths. Alright. Next problem. Okay. What is the subset that represents the event randomly selected person is a female? And what is the measure of its probabilities? Alright. Females. Okay. What are the females we are talking about? This is a female, which is C. This is a female, which is F. And this is a female, which is H. So, a subset, which contains only these three names, subset of an entire set called omega, which contains all the names from A to H. So, this is the subset. It represents basically the subset of all the females, or it represents an event, a random event of picking a female out of the set of eight people. So, if I have a set of these eight people, event that randomly picked person is a female is represented by this subset. And the probability of this is equal to obviously three-eighths. Because there are three elements and each one of them has a measure assigned, probabilistic measure assigned to each of them is one-eighths. Next. What is the subset that represents the event randomly selected person is a male? And what is its measure of probability? Okay, so now we are talking about male zone. So, let's just do it again. So, what's the probability, what's the subset of our set, which represents all the males? Obviously, it's A, B, D, E and G. And what's the probability of this subset? It's five-eighths. Because there are five elements, each one of them has one-eighths measure allocated to it. And the measure, probabilistic measure is additive, which means that the subset has a measure equal to the sum of its components. And each component is one-eighths. Okay. Fine. Next. What is the subset that represents the event randomly selected person is older than 20 years old? Older than 20 years old. This is not good, this is not good, this is not good, this is exactly 20, but I'm looking for older than 20, which means it's this, this and this. So, E, F and G. Only these three people qualify for being older. So, how can I express as a subset this particular event? Well, the event contains E, F and G elementary events. So, my subset, which represents an event that the randomly picked person is older than 20 years old, is a combination of these three elements of my sample space. And the probability of this is equal to one-third, sorry, three-eighths. Three-eighths. Okay. Next. The subset that represents the event randomly selected person is a female and older than 20 years old. Female and older than 20 years old. Okay. This is an event which is older than 20 years old, right? So, this is event older than 20. Now, what is a subset which represents an event that I randomly picked the female? Well, it's C, F and H. These are females. So, this subset of my total sample space, omega, is representing an event that I'm picking the person older than 20 years old. This subset represents the event that I'm picking a female. Now, I'm talking about end condition between these two. So, it should be both female and older than 20 years old. Well, in the set theory language, it's called intersection. So, basically I have to intersect these two subsets. Now, what's the intersection? Intersection is common elements. There is only one common element which is F. So, F is the only female which is older than 20 years old. All other females which is this one is younger and this one is younger. So, my event which I'm interested in consists of only one element of this set and the probability of this is equal to 1-8. Okay, next. What is the subset that represents an event a randomly selected person is a person? And what's the measure of probability? All right, well, it's an interesting question. What is the event that the person is a person? Well, that's basically the... Any result of my experiment fits this description. Because whenever I'm picking a person, I'm picking the person, right? So, how to represent it in this more formal language from the standpoint of the set theory? Okay, which elementary events correspond to whatever my experiment actually is supposed to produce the person? Well, each and every one of them. If I'm picking A, it's a person, B, it's a person, etc. So, basically a subset of my sample space which consists of all elements is basically a representation of this particular event. So, what is my event again that randomly picked person is a person? Well, that's the representation of this event in the set theory. And the probability measure is obviously equals to 1. Because we combine together all eight elements, each one of them has the measure of 1-8. So, the result is basically an entire probability measure allocated to an entire space, which is 1. Alright, what is the subset that represents an event randomly selected person is younger than 10 years old? Alright, now we are talking about younger than 10 years old. And there is no younger than 10 years old, which means that in this particular case, a subset which I am interested in is empty. Well, you know from the set theory that there are empty subsets. This is an example of a full subset which is basically the same as an entire set. And this is an example of the subset which is empty, which contains no elements at all. And the probability of this is equal to zero. Why? Well, because there are no elements. Each element is assigned a probability of 1-8, a measure of 1-8. But there are no elements here, so zero. So, I just presented these two extreme cases when the probability is equal to 1 and the probability is equal to zero. Well, just to introduce you to a little bit more extreme cases. Yes, there are events which have the probability of zero. I don't want to make any examples, but this is a good example. What is the probability of the person which I am picking is less than 10 years old. And then there are events which are definitely happening. So, if I am interested in this particular case to pick a person and I am picking a person, then I always get a person. So, the probability of that event is 1, that's the full event. So, there is nothing wrong with these extreme cases. They do not represent anything like unusual. I mean, what's the probability that it's raining if it's raining? Well, the probability is 1. And what's the probability of the rain when you have a sunny weather in the window? Zero. Okay. Now, the opposite to the previous one. The previous one was younger than 10 years old. Now, I would like to introduce certain logical operations and or and not. So, in this case, I am talking about not. So, what's the probability and what's the subset which represents the event and what's the probability of this event? And the event is I would like to not younger than 10 years old. So, if I am picking the person from this set and I am talking about the event, I am picking the person who is not younger than 10 years old. Well, all of them are not younger than 10 years old. They are older, right? So, again, the result of this particular event is any person. So, we are talking about A, B, C, D, E, F, G, H. That's my event and the probability of this event obviously is equal to 1. So, what's the probability of picking a person who is not younger than 10 years old? 1. Because any person would qualify, which means the entire sample space is good. Any elementary event corresponds to my event I am interested in. Okay. What is the subset that represents the event? Randomly selected person is not younger than 25. Not younger than 25. So, it's not younger than 25. And not a male. Okay, let's talk about this. So, you have this end condition, which means we have to have this particular subset and this particular subset and make an intersection out of them, right? Alright, let's do it. Not younger than 25. Not younger than 25. It's 28 and 33 only, right? So, we have F and G. That's the subset, F and G. Now, not male. Male, male. This is not male, it's a female, right? So, it's C. Male, male. Not male. F. Male. F. So, these are not younger than 25. These are not male and I have to do an intersection between these two. So, what's the intersection? F is common, right? So, I have only one element in common. So, that's the only subset which contains only one element of this set, which corresponds to this particular condition. So, what's the probability of this? Well, obviously, it's equal to one-eighths. So, if my event is, I'm picking the random person from this set of eight people and I'm interested in the event that this person is not younger than 25 and not male. Well, the only subset which I can get is if I'm picking only one person out of this. Everything else does not correspond. And the probability is one-eighths. Next. What is the subset that represents event? Randomly selected person is not younger than 25 or is a male? Okay. Not younger than 25 and is a male. Okay. We have to do exactly the same thing. We have to first find out the subset which represents this event. Not younger than 25. And we already did this. It's F and G. These are both not younger than 25. 28 and 33, right? Now, males are A, B, D, E, A, B, D, E and G. And we have to do an intersection between them. I see only G. Oh, I'm sorry. The problem was OR, not ENT. OR. Now, what does it mean that we are talking about OR? Well, you remember that OR is represented by a union of two subsets. Now, union is any element which belongs to either this or that. So, what do we have? A is good because it belongs to this one. B is good because it belongs to this one. D is good because it belongs to this one. E belongs to this one. F belongs to this one. G belongs to both of them. But we count it only once. So, this is the set of elements which are either not younger than 25 or a male. And we have probability of this equals to 6 eighths or 3 quarters. Okay. Next. What is the subset that represents event? Randomly selected person is not a male or not older than 20 years. Okay. It's something similar. Not a male or not older than 20 years old. Okay. Not a male. Okay. Male, male, not a male. F, male, not a male. H. So, that's my subset which represents an event that this is not a male which I have chosen. Now, not older than 20 years old. Not older than 20. So, these three are not good. 24, 28 and 33. They are older than 20, right? Everything else is good. A, B, C, D and H. A, B, C, D and H. That's another subset. And I have to do the OR which is union. Which means I have to pick up the elements which belong to either this one or that one. Right? So, which ones? Well, A because it belongs to this one. B belongs to this one. C belongs to both. So, I have to count it. D belongs to this one. There is no E. F belongs to this one. There is no G and H. And the probability of this is equal to 6, 8, which is equal to... Okay. Next. And actually the last one. What's the subset which represents the event randomly selected person is not a male. Not a male. And... So, we are talking about intersection. Not older than 20 years old. So, it's basically... The conditions are the same but they are connected with and rather than OR. The previous problem was OR. Now it's AND. So, what do we have? Well, same thing again. Not a male are C, F and H. Not older than 20 years old. I have to exclude these. So, it's A, B, C, D and H. A, B, C, D and H. But now I have to connect it with an intersection because I need the condition AND between them. Right? So, I have only common elements. Now the C is common and H is common. Only two elements are common. F is here but not there. And C, H are in both. And the probability of this is equal to 2, 8, which is one quarter. Yeah. That's it. Why did I spend so much time basically talking about these absolutely trivial problems? My purpose was not to exercise in theory, in a set theory in unions and in intersections, etc. But to demonstrate that the theory of probabilities is basically nothing but a set theory and measure theory, additive measure, which is assigned to the elements of the set with the total measure of an entire set equal to one. Because this is just basically a condition of the theory of probabilities. Because we always have to talk about a relative part of the subset relative to an entire set. So, that's why it's convenient to have an entire set as one, as a unit. All right. That's it. I do recommend you to go through the same problems again. Just go to Unizor.com and just solve all these problems. There are answers. So, it's very easy to check yourself and see if you really understand this conceptual equivalence of talking about events and probabilities.