 Hi, I'm Zor. Welcome to Unizor education. We continue talking about probabilities today. This is about conditional probability and this lecture is part of the advanced math for teenagers course presented on Unizor.com and that's where I recommend you to to go to listen to this lecture. It has notes and notes are very very useful. So you might actually choose to read the notes before you listen to the lecture and then listen to the lecture and then read the notes again. Anyway, so let's talk about conditional probability. I have this plan which I'm going to use and here is my first statement. If you have certain random experiment and you have absolutely no knowledge about results of this experiment, you did not conduct this experiment in the past. So you don't know the frequencies of occurring different different elementary events. You don't have any knowledge about the particular features of this experiment. So you cannot really predict with any certainty the one result over another. So you must assume that if you are really assigned the task to set the probabilities of different elementary events, which are the outcomes of this experiment, you basically have absolutely no choice but to assign them exactly the same probabilities. That's the start, right? So if you have a random experiment with n different elementary events, which are the results of this experiment and you have no reason to believe that one is is tending to occur more often than another, then you obviously have to assign the probability of every elementary event equals to 1 over n. All right. Now that obviously happens with flipping the coins, rolling the dice, shuffling the deck of cards, etc. All right, fine. So now we have actually spent some time and decided to put it on a little bit more formal, more mathematical standpoint. And our approach was as follows. The set of elementary events, which we called sample space, we have basically made an equivalency between the sample space and a set. Well, this is a set of elementary events. Set can have many different types of elements. So why not elementary event to be an element of this set? So that's one thing. The second thing is the probability of this elementary event, which happens to be 1 over n, if there are n elements in the set, n elements, is basically a measure. A measure introduced on the elements of this particular set. In as much as we can measure the area or the angle or we can measure the speed or anything, whatever we can measure. So the measure is a numerical weight, if you wish, of each elementary event. So basically we have associated a number, number 1 over n with every element. We also consider every random event, which is basically a combination of elementary events. Well, what is a combination of different elements in the set is a subset, right? So any random event is treated as a subset of our set. Our set or is basically an equivalent of the sample space, which is a set of all the elementary events. And what's important is that the measure should be additive. Which means, if you have certain random event, which contains certain elements, then the measure of this random event, the subset is equal to a sum of all the elements, which comprise this particular subset. And if you have two subsets, if they do not have common elements, then you just add the measure of each of them to get the combined subset, which we have introduced as an operation or, for instance, or in some other cases you can manipulate in a similar fashion with the measure. So measure is additive. It means it can be added up in exactly the same fashion as you are adding up areas of different figures. If you put them together, if they don't intersect, then the measure of the result is the sum of the measures of the components. All right. So I will use this measure theory equivalent or formalization of the theory of probabilities in the course of explanation of what exactly the conditional probability actually is. Maybe not in this lecture, but definitely in some other lectures dedicated to conditional probabilities. All right. Next. What's next is the following. You have assigned initially the probability of each of the n elements of each elementary event. The probability would be equal to 1 over n, because you don't have any knowledge about what's going on. But let's consider you do. Let's consider that, for instance, you can conduct this experiment time after time after time and you accumulate the statistics and you see that one elementary event occurs more often than others for whatever reason. Well, that's the reason for you to reassign the probabilities slightly differently. So instead of getting one nth for each elementary event, you might redistribute this probability and put a little bit more into the elementary event, which occurs more frequently. Now, there might be even some cases when certain elementary events don't occur at all. Well, then that's the reason to assign the probability of zero to these elementary events. Let me just give you an example. Let's say you are rolling the dice and you have decided that, well, you don't know how the dice will fall. So you have decided to use the first 10 digits as the result of this experiment. Yeah, but yeah, you can tell me that there are only six sides on the dice and there are numbers from one to six. Why did I include this? Well, number one, because I have the right to include anything I want into my set of elementary events. But now if I am a reasonable person, I should assign the probability of one six to this, one six to this, one six to this and zero, zero, zero and zero to all these guys, which basically never occur. So it's not wrong to include certain elements in your original set of elementary events as long as you know what's the probability of these. And if they are impossible, well, I mean, there are certain cases, maybe rolling of the dice is not such a good example because it's obvious that seven, eight, nine, ten cannot actually occur. But in some other cases that might not be such an obvious case. So you are free to basically assign the probabilities to any events as long as these probabilities correspond to either your experience, how you observe in the past, or your knowledge about the internal structure of your experiment. So your knowledge about the internal structure of the dice is that it has only six sides. So it cannot actually have seven, eight, nine and ten. So if somebody wants to include these elementary events as the results, that's fine. They are free to do it as long as they assign the probability of zero. So that's another point. So the probabilities are changing. That's what's most important. As long as you know something about your experiment, your probabilities might deviate from the original equal distribution among the elementary events to something else. That's a very important observation, actually. Now, I actually would like to use my equivalent with a set theory to put it in some kind of a more picturesque format. How can you actually graphically represent what I just said? Well, for instance, you can do it this way. Let's say you have certain area divided into certain parts. Now, each small area represents an individual outcome of your experiment. So in this case I have whatever, 12 different areas. So I can assign the probability of one-twelfths to each of these little squares. Now, there are 12 squares, so the total probability of all this is equal to one. Now, each square represents an elementary event. Now, the combination of the elementary events is basically some kind of random event, right? So I can say that the combination of this, this, this, this, this and this, it represents the event. Now, some of these numbers, which are inside, that's one-twelfths, how many? Seven times, represents basically the probability of the event itself. Because as we know, the probability is an additive measure. So it's exactly like the area. So the area of this figure is equal to some of areas of little squares inside, right? So basically this type of representation of the probability is very useful because it actually implies this additive feature of probability. As long as you know how to draw some picture which represents the results of your experiment and you assign the probability of each individual elementary event, then by combining elements into some kind of figures, you can always come up with a picture representation, the graphical representation of the event you're interested in. And the probability of this event is very easily calculated basically exactly like the area of something. Now, if for whatever reason the probabilities are changed, so instead of one-twelfths to each one of those, I have, let's say, 0, 0, I have 0, 0, 0 and 0 and the rest I have one-eighths, one-eighths. Now, I have exactly the same event as before, but now the probability is 0, 0, 0 and 3 times 8, so it's 3-eighths instead of, as it was before, 7-12th, right? So the graphical representation is very convenient because it really reflects how every event has the probability equal to the sum of the elementary event's probabilities which comprise this bigger event. So I'm going to use this graphical representation in the future. So I did not really touch the conditional probability yet, but I'm still kind of trying to introduce you to certain concepts which would lead me to introduction of the conditional probability, all right. So we have actually touched a particular case when from the initial distribution of probabilities among elementary events, we can switch to something new based on the knowledge or frequency or whatever else which we have. And let me just give you an example of certain case when something becomes impossible, okay. Let's consider we have a special dice. A special dice, when it's rolled, it shows only even numbers, 2, 4 and 6. Now, the odd numbers are not actually shown. And therefore, my distribution of probabilities instead of 1-6 to each should actually be switched to 0, 1-3rd, 0, 1-3rd, 0, and 1-3rd. So I have three possible outcomes, 2, 4 and 6, and three impossible, 0, 3 and 5. That's why I have assigned the probabilities of impossible events to 0. Now, it doesn't really matter whether I can make such a dice or not. This is a purely theoretical consideration. So that actually is an example of how exactly the red distribution of probabilities occur based on some knowledge. So what's my knowledge? And here is the link to the conditional probability. What is my condition? Condition is, or my prior knowledge, if you wish, about the experiment, is that this is a special dice and it falls only to show even numbers on the top. So under this condition, the probabilities have different values, as you see. So this is basically an example. Now, another example, a little bit more practical, because you can say that, well, it's probably impossible to have this dice, et cetera. But here is a practical example. Let's say you are playing Blackjack and you're not the only player. There is a player before you, so you're number two, let's say. So dealer first gives the card to the number one and then gives the card to the number two. Now, if you just don't know anything at all, what's the probability of getting a king? Let's say you're playing with one standard deck of cards or 52 cards. So you have 52 cards, and as far as the kings go, there are four different kings and four different suits. So if you don't know anything about anything at all, you would consider that the probability to get the king would be for 50 seconds, because there are four kings out of 52 cards, so the probability to get the king would be this. But what if you observe that the person number one, who actually got the card before you, got the king? Well, under this condition, there are only three remaining kings, right? So this particular condition that the first player got the king makes your probability instead of four, 350 seconds, right? So again, the prior knowledge or certain condition which you might not actually be aware of in the beginning, but now if you are aware of these conditions, that changes the probability of the event which you are interested in. I mean, obviously the elementary events are changed, everything is changed. I mean, as long as you know that the first guy got particular card, that changes completely the probability of everything else for you and for all other players if they are there. But again, if you don't know anything about what exactly that first person received, you have absolutely no information about getting anything and you've probably decided that your probability is still 450 seconds. Alright, so that's just examples of how knowledge about something or statistical experience or whatever else, which basically represents certain condition your experiment is actually conducted. How that changes the probabilities of elementary events and any random event. So these are just examples. Alright, next we will do a little bit more precise one particular example and I will use the graphical representation which I have. Let's say you are playing with two dice and I will represent graphically the results in this way. 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6. So I have two dice and the result of my experiment would be in these squares and the square has row and column. Row means the first dice result and the column means the second dice result. So if I throw a dice and I have a combination 3, 4, so it would be 3, 4, that's the square which represents the result. So the results of my experiment are in this 6 by 6 matrix, 36 different squares and if I don't know anything about how the whole experiment actually is conducted, then I can say that 1, 36 is the probability associated with each particular result. So each square has an area of 1, 36. I have 36 squares so all together we have 1 as it's supposed to be and now if I would like to know let's say what's the probability of having let's say even number on a second dice. So the second one which is the column should be even. Alright, so it's and I don't care about the results of the first one, right? So this square and this square and this and this and this and this represent the results of the my experiment if the second dice is 2. Now these are, the result is 4. The first one is irrelevant and this is the result is 6. So all these squares which are marked with a point represent an event. It's a combination of elementary events. So each one of these is elementary event when the first dice falls whatever it wants and the second one is even, right? So that's what I have. Now if I would like to know the probability of the second dice to be even, I just have to add together the measures of each of the pointed square and there are 6, 6, 6, 18 pointed squares. So the probability is a sum of 18 by 136, so it's 1836 which is 1 over 2. So the probability of 1 half. If I throw two dice, I have the second dice even. Alright, fine. That's unconditional. That's when I don't know anything about the dice. Now let's consider a slightly different case. What if my dice are special and they are so special that their sum, so the dice number 1 plus dice number 2 always comes to 6. Now, by the way, it can be actually practical. I mean you can say that this is impossible to have such two dice that will always fall in the way that the sum is equal to 6. But I can make a different experiment. I can just completely disregard all the events or all the experiments when the result is not 6. So I'm throwing the dice, a pair of dice, time after time, if the sum is equal to 6, I consider this to be my experiment and they don't count anything else, right? So basically my experiment is only this. So I'm on the condition that two dice always come up, always sum up with the sum of 6. Now, what does it mean? Well, it means that I have to completely redistribute my probabilities. Instead of 1.36 to each one of them, I have to redistribute it differently because there are only how many? 1.5 gives me 6 in sum, right? 2.4, 3.3, 4.2 and 5.1. So only these results of two dice give me a sum of 6. So only these 5. So what does it mean? It means that I have to assign the probability of 0 to anything else and the probability of 5 of them, right? So 1.5 to these combinations. So 1.5 should have the probability of 1.5, 2.4, 3.3, 4.1, 4.2 and 5.1. These are only possible choices which I have. Everything else is impossible, which means everything else is 0. So all elementary events which are not marked with 1.5 actually have the probability of 0. This condition actually completely changes the distribution of probabilities among events. Without this condition that the sum is equal to 6, each of these squares have the probability of 1.36. Now, under this condition the probabilities are changed and these guys have 1.5 and everything else has 0. So what is the probability of having even number as the second dice results? Well, let's just see. I have this one, 4.2 and I have this one, 2.4. That's the second one is even. All other cases are either impossible, I mean this, this, this, this, this, this, this, this, this and all these are impossible. They do have zero probability, right? So if I will start adding, this has 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. If I will start adding all elementary events which comprise my, the result which I'm interested in, that the second dice is even. I will have to really add all these which are in blue but these are zeros, right? So only these are important. So what's important is this one and this one, right? Out of these which are my result which I'm interested in, only these two have non-zero probability. So I have only 2 out of 5 and my conditional probability of having the second dice as an even number, my conditional probability under the condition that the sum of two dice is 6 is equal to two-fifths, not one-half as used as was before. So as you see, my condition which I have imposed on my experiment has changed the all elementary events, now all elementary events are either one-fifths or zero and as a result it changed the probability of the event I'm interested in. So the second dice shows the even number. So that was my illustrative example of how condition changes the probabilities associated with the results of experiment, alright? So let me just summarize this lecture. If you don't know anything about the random experiments which you are conducting then you are forced basically to assign even probabilities to all the outcomes and if you have n outcomes you have 1 over n as a probability of each one of them. But if you have certain knowledge, like for instance knowledge that these are special dice which sum always to 6, that knowledge completely changes the probabilities of elementary events and as a result it changes the probabilities of your events which you are interested in. So the conditional probability is this changed probability based on the knowledge about certain conditions which you have. And graphically it might be represented as a table or in next lecture when I will be going into more details it will be actually represented in the language of the set theory which I was trying to just touch once before. Well that's basically it for this introductory into what conditional probability actually is. Conditional probability is the changed probability of the events based on certain knowledge, certain conditions which you impose on your random experiment. And this is an example of how to apply this particular conditional probability, how to calculate it. Well that's it. I do suggest you to read again this lecture on Unisor.com in the notes for this particular lecture. And that's it. Thank you very much and good luck.