 Hi, I'm Zor. Welcome to Unizor education. We continue talking about theory of probabilities. This lecture is part of the Unizor.com website dedicated to advanced mathematics for teenagers. Now, let me very very briefly remind what we were talking about in the previous lecture. The previous lecture and this one are dedicated to a little bit more formal, more rigorous definition of the concept of a probability. So the previous lecture was basically about probability of certain events and elementary events in a particular case when we have our sample space which is basically a set of all elementary events. Arranged in such a way that every elementary event has exactly the same chances of occurrence as any other elementary event. So it's equal chances. Now for that particular case we have modeled our sample space as basically a set of of certain elements and the elements are elementary events. So they are elements. Now, and equal chances basically it meant that every elementary event has the same probability of occurrence as any other and the probability was basically introduced as a measure which is equal to 1 over n for every elementary event where n is the total number of events. We are talking about finite sets of the elementary event. So we have a finite set. We have elements. Each element is assigned a measure of one nth and every subset is basically a model of an event in the lingo of theory of probabilities. So whatever we call event in the theory of probabilities on the set theory language is basically a subset of this set. It contains a certain number of elements. Any subset contains a certain number of elements in the finite set, right? So, and the probability of this subset or the probability of this event is a measure of the subset which is m over n where m is number of elements of elements which are included into this subset. So basically graphically, this is my set. These are my elementary events. Each one has weight or measure of one nth and any event is basically a combination of certain elements and the measure of this as basically an area when you're measuring on the plane is basically a sum of the measures of each element which is inside of this subset. In particular, a subset which contains no elements, which is an empty subset, has a measure of zero. So the probability of occurring of no event among whatever the elementary events we have, like we are rolling the dice and what's the probability of having non, not one, not two, not three, not four, not five, and not six. Well, the probability is zero if it's a normal dice. And the probability of the entire set as a full subset is equal to one because we're summarizing all the elements in it. So what's the probability of having anything, one or two or three or four or five or six when you are rolling the dice? Well, the probability is one because with a frequency of one, something does occur. And then every subset, which is in between the empty and the full subset, let's say the probability of having only even number on top. That's two, four, and six even numbers, right? So that's basically a combination of three elementary events and to have the measure of this subset or the probability of the corresponding event. You just have to add them up. All right. What's the difference between this when the distribution among the different elements is even and it's all equal to one over n and a little bit more complicated case? What if the distribution of the measures is not one nth of every element of this set? What if it's different? Well, can it be different? And what happens in this case? Well, the answer is number one. Well, it's our theory. We can do anything we want with this theory. We can obviously assign elements different weight rather than one over n, right? Does it really present any problem? Quite frankly, from the theory standpoint, there is no problem. I mean, we still have certain measure allocated to every element and we can still summarize the measures which are inside. Like, for instance, in this graphical representation, what if the measure of each elementary event is actually the ratio of the error of this particular figure, whatever, divided by the area of an entire drawing, whatever I have here. Well, then obviously each one of them has a different measure. Some of them is equal to one, right? Because you are adding together all the areas and divided by the same area. So it's basically exactly the same properties. The only difference is the distribution of the measures is not equal. So my elementary events are not equal chance. They're not occurring with the same frequency if we are rolling the dice again and again. So, yes, answer is this is possible. And the other answer is that we can actually have certain practical situation when this is the case. Well, to tell you the truth, probably most of the practical situations ideally are not where the chances are completely equal, right? So it's only in some very, very artificial conditions. For instance, if you are rolling the dice, are you absolutely sure that the dice is absolutely symmetrical? Of course not. Besides, there were cases when the dice was loaded, actually. They put some lead or whatever else inside the dice. So one of the sides was actually falling on the bottom because it was heavier more frequently than the others. So let's just consider this situation. For instance, you have loaded dice and your statistical results show that one particular side is on the top like twice as often as all other sides. So we have one, two, three, four, five, six on the top and let's say this one is occurring twice as frequently as any others. What is the distribution of probabilities in this case? Well, let's just solve the problem. What do we know about the probabilities? Well, we know that if this probability is x and this is x, this is 2x, this is x, x and x and we know that the total probability of everything together should be equal to 1, right? So we have the equation. So 7x equals to 1, x equals to 1 7. So the probability of this is 1 7, this is 1 7. The probability of 3 is 2 7s and then 1 7, 1 7 and 1 7. Sum is equal to 1, 2, 3, 4, 5. 5 7s and 2 7s is 1 as it's supposed to be. So basically we have a model, we have the probabilities and and now using these probabilities, we can calculate the probability of let's say an event even number on top. Now even number on top is 2, 4 and 6 and you sum these three probabilities and you will have 3 7s. So that's even. And the odd number on top should be the rest of it, right? So the odd should be 4 7s. So the sum is equal to 1. But odd is 1, 3 and 5. So it's 1 7, 2 7 and 1 7. So that's 4 7s, right? So the purpose of this particular lecture is to introduce you to the fact that the distribution of probabilities among elementary events, the finite number of elementary events might not be necessarily equal. So it's not like a symmetrical distribution. The distributions in practical life are usually not symmetrical, but however either the knowledge about the particular random experiment or statistics which you have accumulated before can actually show you which elementary events, the outcomes of our random experiment are occurring more often or less often and based on that we can assign different probabilities to these events. So basically that's all I wanted to talk about today. So we have introduced the concept of distribution of probabilities. So if you have a sample space which is basically the set of all different outcomes of your random experiments, in certain ideal cases you can count on every elementary event to be the same in its chances to occur as any other element. But in some non-ideal situation, like loaded dice for instance, or in some other cases, the situation might be different and the elementary events can have different probabilities. Let's say a group of students is passing certain exams. Well, they are not equal in their knowledge, which makes the probability of, let's say, any particular student getting any particular score on the exam. It's actually a random experiment, but if it's a good student, then most likely the score will be higher than if it's the bad student. So basically all I'm saying is that the distribution of probabilities is not exactly ideal, like in case of an ideal dice rolling when everything has one-sixth probability. And we actually have to take it into account when we are dealing with certain practical situations. Now, as far as the problems which will be discussed, many problems will be discussed in the theory of probabilities, most likely these problems will be about ideal cases, which means that it's extremely important to basically realize what is exactly the sample space, what are exactly elementary events, and considering the situation is ideal, just the number of elementary events brings you to the probability of each one of them. So if the number is n, then the probability of each elementary event is 1 over n. So you know the distribution of probabilities. It's evenly distributed among n elementary events. Now, to understand what's the probability of any event, all you have to do is to find out if this is your elementary events, then you have to realize what is the event you're dealing with, what kind of elementary events it consists of. And then just add the probability of this and this, whatever constitutes this particular event. And that's how you can understand, that's how you can find out the probability of any particular event. And again, everything goes to a set theory with a set being a model for a sample space with elements of this set being model of elementary events. And any subset of that set being a model of any event. With the measure which we allocate to every elementary event and consequently using the additive property of the measure to every subset, we can actually model the concept of a probability. So probability on the sample space is basically a measure on the set. And again, we're talking about finite sets right now. The more complicated case of infinite sets and infinite number of elementary events and different probabilities, etc. These are all topics which I will not touch in this course. This is more for college level students. That's it for today. I do recommend you to read the notes for this lecture on Unisor.com. That would probably help you to better understand the concept of the probability and how it's related to more rigorous approach based on the measure and sets and subsets, etc. That's it. Thanks very much and good luck.