 Hi, I'm Zor, welcome to Unizor Education. This lecture is a continuation of the probability course, which is part of the advanced mathematics for teenagers on Unizor.com. This is a free site, so everybody is welcome, obviously, to study mathematics using this site. It's an advanced course, which means not everybody might actually like it, but its primary emphasis is solving problems, proving theorems, etc. Just to develop your creativity, your analytical thinking, and logic. So this is the second lecture related to problems and theory of probabilities, as it is defined through measure theory and the set theory, which makes actually theory of probabilities a real mathematical subject. In the previous lecture, I was exemplifying all these problems using certain set of people, which are, let's say, they are in a room, for instance, and they are males or females, these are m or f, these are their names, actually, and this is the age of these people. So I was using this set of people to demonstrate that the probabilities of certain events can be actually expressed as a measure of certain sets and subsets. Now I will try to do the opposite. So let's say I have this type of arrangement and I will present sets and subsets and then the problem is how to derive what is a probabilistic sense of certain subsets of these elements. All right, so the elements are all the people which are collected in that room. So basically my total sample space is eight letters, their names. And I also know that the probability is allocated evenly among all of them because the random experiment I'm talking about is basically just selecting any person from this room and the result is either or these particular members of this group, right? So my question now is if my random experiment results in all these elements of this set as results, then certain events which I am actually is interested in, if I'm interested in, for instance, the event of picking a male out of this group or a female or a person older than certain age, etc. So these events I'm talking about, I will try to represent certain subsets of this particular set and then the question is what exactly an event is associated with this. So my first question is describe in words, which means the describing probabilistic terms, an event X consisting of the following elementary events, C, F and H. So I have a subset X, which is C, F and H. Now first of all, this is a subset, which means in our model it should represent certain event. This is the event of choosing the person C and F, C or F or H, right? So it's either this person or this person or this person. So my question is what's the reasonable way to describe this particular subset of people? Well let's just think about it. C is a female of 18 years old, F is female of 28 years old and H is a female of 16 years old. They are all females and there are no other females. So basically I can say that this particular subset of my sample space is representing an event of picking a female out of this group of eight people. That's it, that's problem number one. Problem number two, event A, B, C, D, H. By the way the probability of this event is obviously three eighths because there are three elements and each one of them is allocated to one eighth's probability. So the next is A, B, D, right, A, B, C, D, H, A, B, C, D, H, A, B, C, D, H. So A, B, C, D, H. Let's think about A, male of 19, B, male of 18, C, female of 18, well actually it's males and females so probably the sex doesn't really, the gender doesn't really have any meaning for this particular subset. So it's 18, D is 20 and H is 16. So probably the H is a good representation. So I have 19, 18, 18, 20 and 16. So everything is 20 and below. So what I can say is that this is a subset which represents the event of picking a person who is of 20 years of age or younger, 20 or younger. So that's what actually the common quality of this particular event. Next, the next is C, just one, just one element C. Okay, well this is easy. Basically if I want only a subset which contains only one element then basically I should specify that the person is supposed to be female of 18 years old. There are no other females of 18 years old. There is a male 18 but not female. So my event is what's the probability of picking up a female of 18 years old out of this group of people and the probability is obviously is equal to one eighths. Next, EFG, okay, EFG, EFG. Let's note it and ABDEG, ABDEG, ABDEG. And I'm interested in X and well I can use the intersection or X and Y, whatever you prefer. So what kind of event is this? All right, let's just think about it. First of all, what is EFG? EFG are three oldest persons, 24 and older. What is ABDEG, ABDEG. These are all males so that's what's common about them. So what I'm interested is, first of all the person is supposed to be male and no, this is supposed to be older, 24 or older so this is 24 and older and this is male. So the event I'm interested in is picking up the person who is 24 years or older and male. Well male is spelled differently, this is male, okay. So a male of 24 years or older, that's an event I'm interested in. So if I'm interested in probability, for instance, the question would be what's the probability of the event of picking up a male of 24 years or older. Now how to calculate this probability? Well very easy, first you have to really construct what is an intersection between these two guys and the intersection is E and G are common, right. And the probability therefore would be equal to two eighths which is one quarter. Next, basically the condition is exactly the same, X is EFG which is older, 24 and older and Y is all males which is ABDEG. And the task I'm actually asking you to perform is express the same thing or X and Y but in slightly different terms using a negation knot in front of the H condition. So instead of saying that this particular male, this is a male, should be of 24 years or older, I can say it should be no less than 24 years old. So the condition which I can, the event which I can actually specify as the one this particular task is describing, this particular model is describing is the event of picking up a male of no younger, of not younger than 24 years old, not younger than 24 years old. It would be equivalent to 24 or older basically, right. So I'm supposed to pick a male of no younger than 24 years old. So that's using a slightly different word description of the same event. So event can be described as either picking a male of 24 years old or older or picking a male of not younger than 24 years old. Next, X and not Y. Okay, well again the same two subsets and now I'm interested in X and not Y or if you wish you can do it this way. This is a knot or sometimes it's a vertical bar on the top. All right, so X and not Y, which means it should be 24 or older and not male. Well, so you can say that it's supposed to be of 24 years old or older and not male or you can say it's 24 years old or older and female. Not male means in this particular case a female. Or you can say it's a female of not younger than 24 years old. And by the way, the result of this is the following. What is not Y in this particular case? If I am negating this event, it means I'm staging the opposite, which is what? C, so not Y is C, D, E, F and H, right, that's what not Y is. Which are C, F, H are females, C, F and H. And now I'm supposed to intersect it with E, F, G and the only common is F. So the result of this particular operation is F, which is female of 28 years old. So that's the only result which can be in this particular experiment. So if I'm interested in the person greater than or equal to 24 years old and not male, the only candidate is one person, which is this one, and therefore the probability of this event is one-eighths. Okay, now I'm not going to wipe out anymore because I'm using exactly the same X and Y, two events. But in this case it's not X and Y. Okay, I'm interested in not X and Y. Now not X means less than 24, so it's younger than 24, and Y means male. So I'm interested in a younger than 24 years old male. So that's the event I'm interested in. And the probability of this event is, well, if it's younger than 24, that's the opposite to this one, right? E, F, G and the opposite is, so not X is A, B, C, G, H, right? And I'm supposed to intersect this and this, right? So the common elements are A, B, D, and that's it. So this is not X and Y. And the probability of this event obviously is three-eighths. So not X means younger than 24, which means these guys and this one. But I'm interested only in males, which means this one is out and this one is out. So I have male 19, male 18, and male 20. A, B, and D, right? That's exactly what I have here. So the formal description of this kind, or this if you wish, which is exactly the same thing, not X and Y, not X intersect Y. So this is the formal description using the more mathematical approach. And in words means males younger than 24 years old is the event I'm interested in when I'm selecting one particular person from the group. Okay, what's next? Okay, these are different. C, F, H, C, F, H. That's my X, C, F, and H. That happened to be all the females. Okay, that's my first event. Second event is A, B, D, E, G. A, B, D, E, G. A, B, D, E, G. Now, what is this event? Let me see. A, B, D, E, and G. These are all males. Okay, so these are all females. And these are all males. Okay, so what am I interested in? X or Y. Okay, so I'm interested in X or Y, or union. Now, what's the union of these two subsets? Well, if these are all females, these are all males, I'm interested in an event which I can say, okay, I'm selecting the person and I'm interested in this person to be either male or female. Now, obviously, that encompasses all my different elementary events, which means I can basically say that, okay, I'm interested in selecting a person out of a group of person and I'm selecting one person. So, obviously, this encompasses the whole sample space. So, this is actually the same as sample space omega and the probability of this, obviously, is equal to one. All right, and the last example I have, A, B, D, E, G, so that's X. A, B, D, E, G, that's X. So, that's my X event and what's my Y event? C, F, C and F. So, both are females of different ages. All right, so what am I interested in here now? X or Y. Okay, X or Y, union. Well, let's think about it. So, what I'm interested in? I'm interested in picking up either a male or a female of 18 and 28 years old. Now, these are two females. I have another one, 16. So, I can say that I'm interested in either a male or a female older than 16 years old. That's how it can be worded. I mean, it can be worded different. I can say actually older than 17 years old and it would be exactly the same thing. Or older or equal to 18 years old. So, there are many different variations in this case, but basically it means that I'm interested in the person who is either a male, regardless of the age, or if it's a female, it should be older than certain level. In this case, I can choose older than 16, for instance. That was my last problem. The purpose of this exercise was, again, to bring together these two things. The probability theory, which considers sample space and elementary events and random events, which are combinations of elementary events. Consider this in relationship with the set theory and measure theory, where the set is actually the sample space we are talking about, but more formalized as a set, which contains elements. Each of them represent the elementary event, which can be a result of my random experiment. And there is a measure, which is allocated, in this case, evenly among the elementary events, which I'm modeling as a measure allocated evenly among elements of this set. And the measure is additive in that respect, that whenever I have any kind of an event, which is, in the theory of probability, represented as a combination of elementary events, but in the set theory is represented as a subset of my sample space. And the measure of any subset is equal to some of the measures of the components. With obvious extension of logical combination of different events using the words like or and not, with corresponding set theory operations of union or intersection or negation. Well, that's it. Thanks very much. I do recommend you to go to Unisor.com and review this lecture again just by yourself. It's a very useful exercise. And obviously you're always welcome to register as a student. The site is free, so at your disposal you will have basically an ability to enroll into certain topics and take exams, which I believe is very, very important for studying. That's it. Thank you very much and good luck.