 Hi, I'm Zor. Welcome to Unizor Education. Continue talking about theory of probabilities and in particular it's a little bit more formal description using the set theory and the theory of measure. I would like to remind just very, very briefly the analogy which we have established before. What we have done, we have made the comparison between the sample space of the random experiment, which is basically all the results, all the outcomes of this experiment, all the elementary events basically, with certain set, with every elementary event being an element of this set. Now, every random event, which is basically a combination of certain elementary events, in this case would be a subset of this set and we are considering a situation when all the elementary events are, they have equal chance to occur, which means that the concept of a probability can be actually interpreted as a measure, an additive measure, established on every element. So if we have n elements, then each element has a measure of 1 over n and it's additive in the sense that whenever we have any subset which contains certain number of elements, the measure of that subset is basically a sum of these elements and if there are m elements which are in the subset, then the probability or a measure allocated to this subset is m over n. So the probability is an additive measure and this is basically our model. We are using this language actually to translate something which is a little bit intuitive, like probability and events, random events, etc. We translated it into a little bit more formal language of the mathematics, which is using concepts of sets, elements of the sets, subsets and the additive measure. This is methodization, if you wish, of more intuitive concepts of the theory of probabilities. Now, I will actually move it further because I would like to introduce the continuation of this modeling, of this equivalency and right now I'm going to address the logical operations on the events. For instance, we would like to know not just event A or event B, but something like what's the probability of event A or B, something like this. So we have a logical operations. Now, the operations I'm going to consider are OR and NOT. So what's the probability of NOT having something or what's the probability of having something and something else? So these are three major topics which I would like to address right now. All right, so let's start with an operation OR. So we know this equivalency, all this modeling with the set theory and let's consider a concrete example. Now, the one which I have here is the following. Let's consider the game of craps and we would like to know the probability of having 7 or 11 on two dice. So again, let's go to the probabilistic side of it. Now, what are the elementary events? What's our sample space? Well, it's basically all the different results of rolling two dice, right? So if we would like to get 7, then the dice number 1 and this is dice number 2. What we can have is 1, 6, 2, 5, 3, 4, 4, 3, 5, 2 and 6, 1. So we have 6 different pairs, right? Now, if we would like to know what are the pairs which result in 11, then it's actually, I put it here, 5, 6 or 6, 5, right? So this is my result for 11 and we have two pairs. So altogether, we count all the different pairs of two dice which result in something which we would like to consider as our good positive event, 7 or 11. Well, there are 6 and 2, there are 8 pairs out of how many? Well, if each dice has 6 different results, then two dice have 6 by 6 which is 36 and that's what makes our probability to be equal to 8 over 36. 6 plus 2 is 8 over 36 which is 2 9s. Now, I would like to address exactly the same problem from the set theory standpoint. So first of all, what I would like to do is I would like to model my sample space, right? So what is my sample space in mathematical viewpoint? Well, obviously the easiest way to represent this sample space is as a set of all pairs where the first and the second are integer numbers from 1 to 6. So it's like 1 6 or 2 2 or 3 5 etc. etc. So there are obviously 36 pairs. That represents my set which is a model of a sampling space of my random experiment. All right now, what we are interested right now is to get 7 or 11. So let's consider these two events separately. So I'm considering this positive result of rolling a couple of dice when the sum of 2 is equal to 7 or 11 as two individual events 7, 11 and conditional between them is or. So what's the a subset of this set which represents 7? This subset contains 1 6 pair, 2 5 pair, 3 4 pair, 4 3 pair, 5 2 pair and 6 1 pair. So this is a subset of this set. Okay now the sum equals to 11 is another subset 6 5 and 5 6. These are two subsets of my set. Now obviously the operation of union which is an operation of the set theory we know how it operates. In these two cases what is the union? Well the union is obviously all these and all these all together eight different pairs out of 36 which makes the probability equals to 836 because each pair has a measure of 1 over 36 right? Each pair has a measure of 1 over 36. So eight pairs, eight different pairs have the probability of 836. So it's exactly the same probability obviously it's the same. What I would like to point out is our modeling our mapping if you wish of logical operation on the events the purely probabilistic kind of a standpoint and mathematical operation of union on two subsets of certain set. So all we have to do is properly model our random experiment with its sample space into this set of pairs with numbers integer numbers for 1 to 6 each. Then our event 7 on the top on the sum of two dice is mapped into this subset of this set. Now event that the two dice result in 11 if you sum them up it maps into this subset of this set and so the whole operation the whole result of the operation of operation or getting on the top of two dice 7 or 11 is mapped into a union of each individual subset. So the or logical or between events corresponds to union between subsets which represent these events. Well this is a little bit easier case because these are all completely different pairs. What if these two models these two subsets have something common set have some common elements. Here is another example where everything works as well. So consider you have one dice and you are interested in divisible by two or three result. So same thing if you have one dice its model in the set theory is obviously a set of numbers from 1 to 6 integer numbers obviously right. So we have six different elements in this set they represent our sample space of the experiment. Now the logical operation two or three well let's do it separately. What is the model of the event divisible by two? Well that's two four and six subset of our set of six elements it's three elements. Now what is the model of the event divisible by three? Well it's three and six. Now divisible by two or three well let's get again back to the probability kind of case. What are the elementary events? Well out of the elementary events which we have divisible by two or three we have to cross out this and cross out this. Everything else is divisible either by two or by three right. So one two three four. So we have four different elements we have four different elementary events which satisfy this condition and so the probability should be four six right which is two third. That's how it should be from the probabilistic standpoint. Now let's address it from the measure theory. Now we know that probability of this guy is equal to one half right three out of six so it's three six. Probability of this is one third. We cannot add them up as we did in the previous case right. In the previous case we just added up two numbers number of combinations pairs which resulted in seven and then combinations which resulted in 11. They were all different so we just add it. We cannot add this because there is a common element six. However again if we will apply a correct operation of union back to the union between these two sets. So what's the union of two four six and three six. Well the union is all elements which belong to either this or this two three four and six. This is a correct union between these two. Six is supposed to be only ones in the result and that's why we have a correct probability of four out of six because the number of elements in the union of these two is four. Although the sum of elements three and two is five actually but that's completely irrelevant in this particular case. Again the operation or logical operation or on events is supposed to be mapped into the union between the subsets which represent these events from the theory from the set theory standpoint. Alright now let's move further. We have a next condition is end. I mean probably by now you understand that everything is mapped quite well into the operations on set theory. So the end operation should be mapped to intersection. Right now let's give an example. Let's consider you would like to again roll the dice and you are interested in the event that it's divisible by two and three. So the previous example was or now I'm talking about end. So what's divisible by two and three. Well out of all these elementary events divisible by two and three is only six. Right so we're supposed to get one six. Well let's just consider from the set theory standpoint. Divisible by two. Again we are using the same set of six numbers as our set. Numbers from one to six. Every number is an element obviously and we are talking about subsets. Divisible by two is a subset two four six. Divisible by three is a subset three six. And what's the intersection between these two? Well the common element is only one which is six. So the intersection is equal to one and only one element which is six exactly the same as in this case. And the last operation I wanted to consider is operation not. And obviously you understand that the operation not should be mapped to a complement which is a set difference or minus with a subset. Now what's the example? Example for instance not divisible by three. Okay so what's not divisible by three if you are rolling the dice? Again these are divisible by three. So that's what's left. One two four and five. Well let's consider the mapping of this onto the set of six numbers. And obviously I'm talking about the event divisible by three and divisible by three is a subset three and six. And what's the complement of this? Well the complement is a total set without three and six. So what's left? Exactly the same thing is here. So my not operation on the random events is modeled as a complement to a subset which is a model of the event which we are negating. All right so my main point actually of this entire lecture is that all our intuitive understanding of the theory of probabilities can very well be represented using the language of theory of the set theory and the measured theory. And there are a couple of requirements obviously. So the measure is supposed to be additive. The measure of every element of the set and we are talking about only finite case as you remember. The measure of each element in the previous lectures we were introducing the equal chances or not equal chances. So equal measure of the elements or not equal doesn't really matter as long as every subset of the set is measured as a sum of measures of all the elements which comprise this particular subset. And another requirement is that the measure of the entire set is supposed to be equal to one because that's kind of a probabilistic approach we have to model the frequency right. So the frequency of an entire results of all the different results from the random experiment the frequency of occurring is basically one obviously because it's always something which occurs. So these are you know few requirements but basically the whole probability theory is just a part of mathematics if you approach it from the set theory and measure theory standpoint with a couple of requirements to make this theory of probability a particular case of set theory and measure theory. So both these concepts the concept of a set and the concept of an additive measure applied to this theory basically gives you a complete picture of the theory of probability. And that's why the theory of probability is a mathematical theory it's not just some kind of you know half intuitive half precise half analogical and half eulogical and have some guessing and predictions and stuff like this all these concepts are much less mathematical than the set theory and the measure theory. So that was my point to show you that this is actually a precise and mathematical subject if approached from this particular way. I recommend you to read the concepts which I'm talking about today on theunisor.com there are notes to this particular lecture so if you read it again it might actually better provide you better understanding of this mathematical concept mathematical approach to theory of probabilities. Well that's it for today thank you very much and good luck!