 Hi, I'm Zor. Welcome to Unizor Education. This lecture is part of the advanced mathematics course for teenagers presented on Unizor.com. All lectures are there, have notes, there are some past exams, problem solving, etc. And for registered students, there is basically the way to organize the whole process as an educational process which contains most of the topics presented in the regular course of mathematics. And I do suggest actually to register which would enable you to get enrolled in the course and take exams, have your scores examined, etc. Everything is free on the site and it definitely makes sense to do it on a serious basis like an educational process rather than pick this or that particular lecture. As a course, it does make this kind of a sequential study of one topic after another. Alright, after this deviation from my main purpose of this particular lecture, let me start. So the theme of this lecture is the main task of the theory of probability. Well, it might be a little presumptuous to basically state that the theory of probabilities has this main task. However, as a general representation of the ideas behind the theory of probabilities, well, it does make some sense in as much as you can say that let's say the purpose of arithmetic is counting. The most important purpose of algebra is probably solving equations. I mean, there are some other, obviously, applications of algebra, but this is probably one of the most important. And in this particular, from this particular viewpoint, we can really state what is actually the purpose of the theory of probabilities. And here is how I would like to present it. First of all, the theory of probabilities is studying random experiments. That's the foundation. We are talking about random experiments. Random experiments have certain elementary events, like individual results, individual outcomes, which in many cases, at least in this course, are presented as events which have equal chances. Like, for instance, if you are rolling the dice, then 1, 2, 3, 4, 5, 6 on the top of the dice have equal chances to occur. If you are dealing a deck of 52 cards among four players, then every distribution of 13, 13, 13 and 13 cards to these players seems to be having equal chances and can be qualified as an elementary event. So, we have a set of elementary events, like the minimum and the smallest event, which we can talk about, which is the result of the experiment. And there are some bigger events which we can build from these smaller events. Like, for instance, if you would like to know, let's say, something about rolling two dice and have six on both of those. I mean, this is a more complicated event. So, the random events, which are also the result of the random experiments, can be constructed from the elementary events. Okay. Now, let's talk about the probabilities. Now, when we are talking about elementary events, we usually assign certain probability to each one of them. Well, it can be actually built on many different foundations. Well, first of all, for instance, we can just roll the dice million times and see that the six comes more or less one-sixth of the times, and three also comes about one-sixth number of the times. So, it can prompt us that the probability, which is basically a concept which can be derived from the frequency of occurrence, can be associated with each number one, two, three, four, five, six, as one-sixth of the total number of rolling the dice. And similar considerations can be applied to other elementary events in other random experiments. Another source which we can derive our probabilities is certain knowledge. For instance, if we are knowledgeable in combinatorics, we can actually calculate what should be the frequency of any particular distribution of 52 cards among four players. What's the frequency of occurrence of every particular distribution as the number of deals would actually go to infinity? So, the frequency, observed frequency, our experience, and then our knowledge, both are input, so to speak, into our view on the theory of probabilities. And this input allows us to basically understand what are the probabilities, which is some kind of a limit number for the frequencies of each elementary event. All right, so far that's not the main task of the theory of probabilities, although it is part of it. So, the input part into this particular problem. So, what's next? Well, next is from different elementary events, which are basically our sample space, so to speak. We can construct more complicated events. As I was saying, for instance, we can talk about two dice rolling and have, let's say, whatever numbers on the top. Or we can have, let's say, a coin which we are flipping, let's say, two times. And we want to know what's the probability of having two heads on two flips of the coin. So, these are a little bit more complicated events. And now we have to understand how these complicated or more complicated events are constructed from our elementary events. So, the input information which we have, either experience and frequencies or knowledge, are contributing to our understanding of what the probabilities of elementary events are. Now we have to build upon it the next layer. Let's talk about more complicated events. If we know how they're built from the elementary events, we can calculate their probability. Okay, so that's our next problem. And why do we have to do it? Well, primarily, we are interested in more complicated events. And I was saying before that the probability allows you prediction. Now, since if we know that the probability of something is equal to something, we can actually say that the frequency of occurrence of that something would be, well, such and such. So, we expect it to occur a certain number of times on average if we are providing a certain number of experiments. So, that's what's very important. The important is the output, the result of all these manipulations which we are making. We had input to have the probabilities for elementary events. We have the way how we construct the events. And now we have to use all this information, all this knowledge, and derive with an output, which is the probability of events which we are interested in. And this is needed for predictions of the future. Again, why do we need the probability at all just to predict the future? Not precisely. We cannot say that exactly this particular event will or will not happen. But if we know the probability, we can say that, well, with a certain degree, frequency, etc., approximation, we can evaluate how many times out of how many experiments we can expect this particular event to occur. So, that's what our output is. So, let me just summarize it. What do we have? We have the input into our theory, which is experience. We observe the frequencies and knowledge. We also have the way how we construct different events. So, from which elementary events, which we know the probabilities of, our more complicated event, event we are really interested in is constructed. So, as a result of this, we come up with an output which is a probability of different events. So, that's actually something which we have examined the whole process. And we did actually a few examples before. So, maybe for just illustration purposes, let me just give a very simple example. Just as an example, let's take the flipping the coin. And we are interested in two coins, actually, which are independently flipped. And the event we are interested in is having two heads on the top. Now, how can it be related to whatever the knowledge about the coins we have? Well, we can observe that if we will flip the coin million times, then something about half of the time will be tail and half of the time will be the heads. Not necessarily exactly half, but more or less. And if we will make instead of a million, we will have a billion flips of the coin, then the relative frequency will be even closer to one half. And in general, the more we do, the closer the frequency occurs. So, it makes us to believe that the probability of having heads and tails is equal to one half. Alternatively, we can actually examine how the coin is manufactured. And we see that there is no grounds for putting more probability on tails rather than heads, because the manufacturing process is relatively symmetrical. And the tiny details are not really significant. So, we can say that from our knowledge about how the coin is manufactured, we can state that the probability of flipping the tails or the head should be exactly the same and equal to one half. So, that's our input. Now, let's talk about our event we are interested in. We have two coins and we independently flip them. So, what's our elementary events we are interested in? Well, there are four elementary events, right? We can have heads-heads, we can have heads-tail, tail-heads, and tail-tail. So, there are four different elementary events of the experiment with two coins. And again, there is absolutely no grounds for putting more probabilities on one than another. And in this case, if we are interested in two heads, this is just one of the four different elementary events. Actually, our event is elementary event. It's constructed, we have constructed this event from one and only one elementary event. And we know that the probability of all of these should be equal to the same number and some of the probabilities should be one. So, it's one quarter for each of them. So, the probability of head-head is supposed to be one quarter. All right. In some way, whenever we have constructed our event from elementary events, and when we know that the probability of each elementary event is such and such, whatever it is, our task of obtaining the probability of the event we have constructed is very similar to, let's say, you're talking about weighing an assembly. And the assembly contains certain details. You know the weight of each detail. Now, how do you weigh the whole assembly? Well, you just summarize. I mean, if you cannot weigh it physically on some kind of weights, whatever, then you just summarize the weights of each component. And that's how you get the result, the weight of an entire assembly. So, that's how the probability of the event is constructed based on, or calculated rather, based on the way how it's constructed, how this event is constructed from elementary events. Or similarly, and this example I will probably use more extensively in this course of probabilities. If you have to know the area of this particular figure, well, probably you have to understand that it's constructed from a square and a triangle. You know how to, you know the area of the square if you have these sizes. And you know how to calculate the area of triangle if you have sizes. So basically, you can just add the area of this particular square and the area of a triangle. And that's how you get the area of more complex figure. That is how the probability of any more complicated event is calculated based on the probabilities of the elementary events it's constructed from. So, this is a very important example which I will probably use all the time. Now, a couple of examples from the theory of probabilities, kind of a set of problems. Let's say you are shuffling the standard deck of cards and you are interested in the event queen of spades is on top. After you shuffle the deck, you are interested in the event that queen of spades is on top. Actually, there are more than one approaches to this particular problem. Let me just exemplify two of them. Approach number one. What is the result, the elementary event, so to speak, of each shuffling of the card? Well, it's the sequence of the cards in the deck after we have shuffled it. So we know that the number of permutations of 52 cards in the standard deck is 52 factorial. So the 52 factorial of different sequences of the cards in the deck is the result of our shuffling and this is our sample space. Now, this sample space contains 52 factorial of different elementary events. There is absolutely no grounds to consider one particular sequence more probable than another because shuffling is supposed to be really random. So let's just assume that we have some really good shuffling mechanism and it provides you... Now, this definition of the good shuffling mechanism is actually the equal chances for any kind of a sequence. All right, now, out of all these 52 factorial different sequences, how many different sequences construct our event when the queen of spades is on top? Well, obviously, if queen of spades is on top, so the top card is fixed and the other 51 cards can be in any sequence whatsoever. So we have 51 now different sequences, 51 cards put in any sequence we can and the number of these sequences is obviously 51 factorial. So if we have 52 factorial of the elementary events, the probability of each of them is obviously one over 52 factorial and 51 factorial of these events are those when the queen of spades is on top. So I have to add 51 factorial times one over 52... 51 factorial times one over 52 factorial probability of each elementary event. That's how my big event, the queen of spades is on top, is constructed. So each one elementary event has this probability and I have 51 factorial of elementary events which are combined together to construct my event I'm interested in. So that's the result. And obviously, since factorial is the product of all the numbers from one to this one, so this is from one to 52, which means from one to 51 and then 52 and this is from one to 51. So 51 factorial can be reduced and we will have this result. That's the probability. At the same time we can consider the whole situation slightly differently. Let's assume that our elementary event is some particular card is on top and we don't care about the rest. Well, I mean it's a different approach. Now how many different elementary events we have in this case? Well, since we have 52 cards, 52 cards can be on top so it's 52 elementary events. Is there any grounds for having one particular probability of one element over another on the top of one card over another? Is the probability after the shuffling of the, let's say, queen of spades is greater or less than, let's say, ace of cards? No. Again, if our shuffling is really random, we really have to have exactly the same probability for any of those events and we have 52 different events so each one should have one over 52 probability and we have exactly the same answer for one particular event, our elementary event. Now we call this elementary event in this particular approach which is queen of spades on the top. So that's one problem which I have. Now another problem which I would like to address and present it in a similar fashion is, let's assume you have a box with n different pairs of socks. Now pairs are different but within the pair obviously two socks are identical. So now you pull from the box two socks. What's the probability of them to belong to the same pair? Well, let's again think about what is our input. So the random experiment is we are pulling the socks two socks. What's our sample space? Sample space is the set of all elementary events. Well that's all the different combinations of two socks out of whatever the number of socks we have. Now we have two n socks, right, n pairs, so we have two n socks. Now and we are pulling out two socks out of two n. How many different ways to pull two socks out of two n socks? Well it's number of combinations from two n by two. Now this is the number of different elementary events which constitute our sample space. And again, since there is no grounds for putting one particular combination of two socks, more probable is more probable than another. So all of them should have exactly the same probability and that probability is equal to one over number of combinations from two n by two. Great. Now question is our event, our random event we are interested in is we are pulling two socks which belong to the same pair. So how many different elementary events are comprising our event? Well we have n different real pair of socks, right? So whenever we pull any one of these n pairs that's the good event, right? So that's the elementary event which must be included into our event of pulling a good pair of socks, right? So we have n different elementary events which constitute our event we are interested in. So we have to multiply this by n and that's the answer n over number of combinations from two n by two. So we have examined what's our input, what's our sample space, what's our probabilities of elementary events. We have understood how our event is constructed from the elementary events and then we just summarized all the elementary events which are part of our random event. We are interested in to get the probability. So all these examples I was just making for one and only one purpose. This is actually the main task of the CREF probabilities to identify your input which means your sample space your elementary events and assign the probabilities to these elementary events. Understand how our event we are really interested in is constructed and then apply just plain arithmetic adding up all the probabilities of the elementary events which comprise the event we are interested in to get the output and the output is the result of this task the answer actually to this particular problem is the answer to a question what is the probability of certain random event which I am interested in because most likely I'm not really interested in individual elementary events that's kind of a simpler job that's the input the output is to evaluate the probability of certain event we are really interested in. Well that is the conclusion which I would like to make I suggest you to read all this material in the notes for this particular lecture on unisor.com and again I encourage you to register on the website on the website as a student have some supervisor get enrolled in some classes which will allow you to take exam and you know evaluate yourself as much as you can for self-evaluation basically purposes. It's always good to solve the problems yourself and the exams actually allow you to do this not every lecture has exams associated with it but I'm working on it so thanks very much and good luck