 We are going to start now, welcome to your first session. I can't believe that it goes so quick. Today's session, we're going to do study unit 4, which is basic probabilities. It's a very long session and demanding session because there is too much information in Chapter 4. But we will manage to get more exercises and need to be able to do a lot of activities so that you are familiar and are able to answer some of the questions when it comes to basic probabilities. Like any other sessions that we're going to have, all sessions are interactive. If I am moving too fast or you don't understand anything, please don't wait until the end of the session. Stop me and ask your question. Sometimes when I'm presenting, I don't see the chat because my screen is full screen in front of me. So I will advise you to just unmute yourself and stop me and then I will assist you so that we don't leave you behind. Like I said, it's too much information today. Okay, so the agenda we're going to follow today is we're going to have 30 minutes to do the basic concepts and look at a little bit of other rules that you need to understand in order for you to calculate basic probabilities. Then we go into the next additional 30 minutes from half past half to one. We're going to look at the basic rule, but only this one we're going to only look at the additional rule or the addition rule. Then we'll take a 10-minute break because this is a two-hour session and by the time we go to a break, I will stop the video so that we can record the second part of the session. And then when we come back from the break, we will do the basic probabilities where we look at conditional probabilities. And then at the later stage, within the last 10 minutes, if we still have time, we will do a lot of exercises as well. Okay, so by the end of today, you should be able to know and understand the basic concepts of basic probabilities. You should be able to understand how to calculate the basic probabilities and how to use or apply the rules of basic probabilities. You should be able to answer questions when it relates to conditional probabilities or a probability event that another event has happened with the key concepts. First question actually is, who can tell me what is a probability? Without me even explaining it. Anyone who wants to take a step? Wouldn't it be an outcome? An outcome? Who else? I think it's the likelihood of anything happening. Likelyhood. The likelihood of anything happening. Lightlihood as well of something happening. Okay, anybody on a day-to-day basis, where do you hear more about probabilities or the likelihood of things happening? What examples can you give? Weather? Weather. The likelihood of rain. The likelihood of rain. A new infection of COVID-19? New infections of COVID-19? Just a new infection, but that is not the probability. So you meaning like predicting the new, okay, new infections? All right. Where else? Elections? Elections. Where else? Where else would you use probabilities? Anybody who bet lotto? Tatama chance? Tatama millions. That's a chance of winning lotto and things like that. So that is what you're going to learn. Those concepts that they use in doing so many of other things, you're going to learn them today. And you're going to learn how to apply some of these rules to get the likelihood of something happening or not. For example, some of the terminologies that we use in statistics is first to describe what is the probability? The probability is the likelihood or the chance that an uncertain event will happen or a care. The probability that the sun will go out is always going to be 100% because it's going to be in the morning, the sun will always come out. The probability that the sun will set down, it's obvious. The sun always comes out and the sun always goes down. So the probability of that happening is always equals to one. The probability of a person flying without having a mechanical help or anything just due as a person flying are zero because a person cannot fly, you can only jump. So that probability will be equivalent to zero. A probability of tossing a coin, a coin has two sides to it, a head or a tail. And the probability of that coin landing on a head is a 50-50 chance. And those are the things that you're going to learn today. Like I explained, a certain event is an event that we are sure that that event will happen and that probability will always be equals to one. An impossible event is the likelihood or the chance that that event will not happen. And that probability will always be equals to zero. I'm talking about the probability of events. What are these events? I'm saying the sun will come out, toss the coin. Those are what we call events, things happening. When things happen, there is an outcome that comes out of there. When the sun rise up, it becomes there. So those are the outcomes that comes out. Problem event. We need to also understand that there are different types of events. For example, there is a simple event. The sun will come up. That's a simple event. It's an event that describes one thing happening at that time, at one time, plus. Not two things, only one is a simple event. The probability of a day in January is only one event that is happening. But we also have what we call a joint event. Therefore, it means there are two things happening at the same time. The probability that the sun will come out and it will be rainy. Two things, sun come up and it is rainy. The rain and the sun are two different things. Those we call those events that are happening at that particular time. We call them the joint event. When we have a joint or an event, there is also what we call a complement event. And a complement event is an event that complements the other one. So it's an event that form part of all the event, but it's not the same as the initial one. So let's say, for example, how do I put it in this manner? When I toss a coin, I have two outcomes that can happen. There is an outcome of a head and an outcome of a tail. When I talk about a complement of an event, so when I toss a coin and it lands on a head, that is an event that is happening, that a coin is tossed and it lands on a head. That is an event with an outcome of a head. When I toss another, I toss the same coin and it lands on a tail that is another event with an outcome of a tail. But they all come from one sample space. So it means a complement of a head will be the tail because they all come from a coin. A coin has two events or two outcomes, a head and a tail. So a head will be a complement of a tail and a tail will be a complement of a head. And that is what we're talking about when we talk about a complement. I introduced something and I said a sample space. What do I talk about? A sample space, like population that we did the last time is the same concept. A sample space is a collection of all possible events that are happening. For example, I like using a coin because it only has two options. It can go either one. A sample event is a collection of all outcomes or all, oh, sorry, a sample space is a collection of all events which has an outcome. For example, if I roll a die and I know that a die has six sides, all the sides of the die from part of what you call a sample space. The same applies if I have a deck of cards, all 52 cards that are inside that box form part of a sample space. But the card can have, I can have an event where I draw a card and the card is an ace which is black or is red, things like that. When we talk about events as well, we are able to visualize them. And visualizing the event, we can use different form. Some people prefer using the VIN diagram which gives you the entire sample space. And from the sample space, let's say we use all days in 2014. So we take the calendar for 2014 and we want to say that calendar for 2014 is our sample space. If we only interested in the days in that falls on a Wednesday, that will be our event Wednesday. We might also want to know those events or those days that falls only in January. That will be event January. We know that there are days that are in January and they are also on a Wednesday so that form part of our joint event. Am I talking about? A day on a Wednesday, it's a simple event. A day in January, it's a simple event but a day can be in January and can also fall on a Wednesday and that form part of what we call a joint event. That's one way of visualizing the event. The other method of visualizing the event, we can use what we call a decision tree or a tree diagram, which also starts with the sample space. Let's say this one, we are using a coin. With toss a coin, it can land on a head or land on a tail. If it landed on a tail or if it landed on a head, when we toss the coin again, it can land on the head or it can land on the tail. And if it land on the head, we can toss again, it can land on the head or it land on the tail and those are events. The same for that coin, if it landed on a tail, if I toss again the second time the coin, it can either land on a head or a tail and those are how you will define your event and you should be able to calculate the probabilities as such. And at the end, you will have all the outcomes that happens when you toss that simple fair coin. Other methodology or the other visualization that you can use, which I always prefer to use is the contingency table. A contingency table, it's easy to use because contingency table gives you your event. For example, this is for gender and also for whether the person has been promoted, the promotion status at work. Inside the table, it will give you your joint event because this is the joint event of men who were promoted as police officers in the past two years and it will also give you the simple event which are those in the totals. Now, in the exam or in your assignment sometimes they might give you a contingency table but do not give you the total. You have to calculate the total because they will help you to calculate your simple event. So this is a simple event, it got less of whether the person who was promoted or the police officer that was promoted, whether it was a man or it was a women, there were 324 and that is the contingency table. You will see that I like using it in all the exercises that we're going to be doing, most of them you will see that I apply the contingency table to visualize some of the probabilities. Okay, those are the basic concept that you need to know. Now, how do we calculate the probability from the events? So a simple probability can calculate it from a simple event which means it will be the probability of a simple event happening. Let's say I have the same table that we introduced earlier of the policemen, women, police officers, whether men or women who were promoted in the past two years. If I have this table and I want to calculate the probability of promoted, I know that a simple event because promoted is just a simple event. It's just those who are promoted irregardless of whether they are men or women. The formula I've to calculate, the simple probability is, the probability of an event A is given by the number satisfying that simple event divided by the grand total or what we call the sample space. If you are given the contingency table and you are not given the total, you have to calculate the total so that you can calculate the grand total which it is what we call the sample space. It's just your 324 plus your 876 and even if you add from the bottom, your 960 plus 240 will also give you 1200 and that is your sample space which I refer to as the grand total. Which is the number satisfied if I need to calculate the simple event probability, the number satisfying the simple event divided by the grand total. And you will notice that every time we calculate probabilities, we always going, especially when we are giving events, you are always going to use the grand total. So how do we then calculate the probability of promoted? The number satisfying the event promoted is 324 and we'll have to divide it by 1200. So the probability of promoted will be given by 324 divide by 1200 which gives us 0,27. Remember the probability, the either between zero and one. So if you calculate the probability, let's say I'm calculating this and I get the answer as 1,27. It means I am doing something wrong. Always your probability should only be between zero and one. So since you know how to calculate the probability of promoted, here is your exercise. Calculate the probability of main. I'm going to give you five minutes or not even five minutes, two minutes. Calculate the probability of main. Are you done calculating the probability of main? Let's see if you got the answer right. The probability of main will be given by the number satisfying the simple event main regardless of whether they were promoted or not promoted. Therefore, they are 960 main. So it will be 960 divide by 1,200 which gives us 0,80. Agree, and that's how you calculate the simple probability. How do we then calculate the joint probability? A joint probability refers to the probability that two events are happening or more than two events are happening at the same time. The formula that you will be given in the exam as well, the probability of A and B, which is the joint event of A and B is given by the event satisfying the joint event divide by the sample space. If I need to calculate the probability of main and promoted, remember using our same example of table, the joint event are those probabilities that we see inside the table. So the joint event of main and promoted will be 288 because those will be the main who were promoted. To calculate it, you say 288 divide by 1,200 that gives you 0,24. You have two minutes to calculate the probability of main and not promoted. Okay, did you get the answer? Let's see if you have the correct answer. The probability of main and not promoted will be given by the joint event of main and not promoted, which is 672 divide by 1,200, which then gives us 0,560, the joint event. The probability there are also what we call the marginal event, since we're using the contingency table. Also need to understand that marginal probabilities are just the same as your simple probabilities, but because we are using the joint probabilities or the joint events, let's go back to the table. Marginal probabilities, we're going to be calculating them by using the total tabs because it's the same. When I look at a contingency table, I can see that 324 is made up of 288 and 36, which is made up of a joint event of main promoted and a joint event of women promoted, makes up that simple event, which we can also call the marginal probabilities if we are using the joint events. So how do we then compute this marginal probability? Computing a marginal probability because if we have a contingency table with more than three columns, or we have more than three rows, then we want to calculate the simple event for either the column or the rows, you're always going to add the joint probabilities to calculate that marginal probability, which is the same as a simple event. So for example, if I want to calculate the probability of promoted and I'm given, let's say I'm not given the total, but I'm given the probability of the event main and promoted equals 288, and main and women and promoted is 36. Therefore I can just say the marginal probability of promoted will be 288 divided by 1,200 and 36 divided by 1,200 and that will give us 0,27, which is the same if you take 324 divided by 1,200, which was the simple event that we calculated. The probability of promoted if we use the simple event formula, which says the number satisfying the simple event will be 324 divided by 1,200, which also gives us 0,27. So whether you use marginal probabilities or you use the simple event, depending on what in your assignment question or your exam question they have given, you need to know which one you're going to be using if they have given you the totals, you can calculate using the simple event. If they've given you the joint probabilities, you can calculate it using the marginal probabilities, means one and the same thing. Calculate the probability of women, not using the simple event, but using the marginal probability. I am going to do it with you now. So to calculate the probability of women, therefore this column of women, it means I'm going to use the joint event of women promoted and the joint event of women not promoted. So it will give me 36 divided by 1,200 plus 204 divided by 1,200. You never did math and get confused why I just give you the answer. Let me show you how to calculate this manually as well. Because it's a fraction and we're using an addition slide, we can always say the common denominator is 1,200. And when it's an addition, what we do, we add what is at the top, 204. Therefore, that gives us 240 divided by 1,200. As you can see, it gives you the same as the total that we have there, which then gives us 240 divided by 1,200, which gives us 0,2. And those are the marginal probabilities. I know that the first part of the session is over. We should be in the next part of the session. So just continue from here because we're going into the rooms. Now, since we are also learning about the basic concepts or the key concepts within the probabilities, there are things that you also need to know and remember when you deal with probabilities. Probabilities can also be mutually, or events can also be mutually exclusive. And when they are mutually exclusive, you should be able to also calculate all their probabilities. What do we mean by a mutually exclusive event? A mutually exclusive event is an event that cannot happen at the same time. Let's say, for example, we choose a day from the calendar in 2014 and we define those days as, A represent a day in January and B will represent a day that happens in February. There will never be a day that is in January and can also be in February. Therefore, A and B are going to be mutually exclusive because they can never happen at the same time. And that's what we define. And if we calculate the probability of this, therefore it means the probability because mutually exclusive events will always happen when there are joint probabilities. So therefore the probability of A and B for the joint probability will be equals to zero. The other thing I need to also make you, if we write the probability of A and B, we can also write it as such. The probability of A intersect the probability of B. They mean one and the same thing. You will see sometimes maybe in the past exam paper they use the intercept, sometimes they will use the end. So you must know that the two refer to the same thing. So the probability of A and B and the probability of A intersect B is the same thing which will be equals to zero. Those are mutually exclusive events. Then they can also be a collectively exhaustive event. So if your events are collectively exhaustive event, it means one of the events must, okay. For example, it will also include that a set of events will cover the entire sample space. So if we're looking at the calendar, all possible scenarios of the calendars need to be met so that everything in the sample space needs to be accounted for. What do I mean? Still going to use the calendar. If I choose a day in a weekday and a day on a weekend, already the week is complete. A day in a weekend and a day in a weekday. And I choose a day in January and a day in spring. If you notice, does not include February much, but those days are included in a weekday. And those days are included in a weekend for other months. And I'm not including other seasons. I'm only including spring, but I know that there will be those days that are in summer, in autumn, in winter that are included in a weekend. And they are also included in a weekday. Therefore, event A, B, C and D are collectively exhaustive because they cover all days of the calendar. But they are not mutually exclusive because a day in a weekend can be a day in January and it can also be a day in spring in a way. Event A and B are collectively exhaustive for the calendar and also for the week. And they are also mutually exclusive because a day cannot be on a weekday and also can be on a weekend. And that's how you will define a collectively exhaustive event and mutually exclusive events. In the next show, what we just discussed just now was to say, if I look at the contingency table, they will give me the event A and the event B. And with event A and B, I will get the joint event for all event B that happens for the outcome of one and all the event B that happened with the outcome of two. And I will also be given, which we call them all those, we call them the joint events. And you are also given, or you can calculate your simple or marginal probability or marginal probabilities from your events, from your simple events. What have we learned so far as well? We've learned that the probability is a numerical measure that shows us the likelihood or the chance of something happening. And what we have learned as well is that probability of any event must be between zero and one. It can never be less than one. So, sorry, less than zero and it can never be more than one, inclusively. And we know that if it's zero is an impossible event, if it's one is a certain event that it will happen, if it's 50%, it's a 50-50 chance. What we know as well is the sum of all probabilities should be equals to one. So, if I add all the probabilities, they should all make up one. A compliment event, we also learned that a compliment event is an event consisting of all sample space points that are not in another event. So, if I need to find the compliment of A, it means I need to find all other events that are part of the sample space, but they do not form part of A. So, the probability of A, I can find the compliment of it by finding one minus the compliment of that probability. What do I mean? Remember, if we know from the probability in our sample space, if we know that the probability of A plus the probability of a compliment of A, which is denoted by C, so the compliment can be the compliment they can use in A copy or sometimes they can use by at the top. They all mean the same thing, it's a compliment event. So, we know that the sum of all probabilities should be equals to one. If I need to find the compliment of A or the probability of A, it's just to take the compliment of A and move it to the side so that I can find the probability of A. Your exercise, then we move into the additional rule. The probability that it will rain today is 0.7. What will be the probability that it will not rain? Therefore, what they are asking you is find the compliment event of rain, which is not rain. Remember that, the probability of rain will be given by one minus the probability of not rain. Not rain. Or you can say as well, the probability of not rain, it's the one minus the probability of rain. They mean one and the same thing. Did you find it? The probability of rain is one minus the probability of rain, which is 0,70. Therefore, the compliment of rain is just 0,30. That is a compliment. So the question was asking you to find the compliment of rain, which is not rain. Any question? Then we take this 15 minutes to do the additional rule. In the absence of questions, we move on. We're going to learn how to apply the addition rule. What do we mean by addition rule? Addition rule? In the exam, they will not ask you apply the addition rule or what is the addition rule or something like that. But they will ask you, what is the probability of an event A happening or event B happening or either event A and B happening? A question will come up like that. And in your mind, you need to know that now I need to apply the addition rule. So which state? The probability of A or B, not both of them, but either one of them happening, it's given by the probability of event A plus the probability of event B happening minus the joint probability of both of them. Why do we subtract the joint event? Because in the probability of A, if you remember the contingency table that we had, remember this is where you calculate the probability of A. If this is A and this is A complement. Here will be your joint event of A and B. If this is B here, that will be your joint event of A and B. And if I calculate the probability of B here, in the probability of B, because I'm using those two joint probabilities, it includes the probability of A and B. And it also in the probability of A, it also includes the probability of B. So in order for us not to double count, we have to subtract the probability of A and B. So in the exam, you will get this formula. The probability of A or B is across the probability of A plus the probability of B minus the probability of A and B. The reason why I'm mentioning, you will get this in the exam, that is in case you go for a venue-based exam, let's say they open up for a venue-based. Those who are writing online, you will have your formulas because you have all your study material with you. You can then calculate this probability as well. But before we calculate that, if A and B are mutually exclusive, remember, if the event A and B are mutually exclusive, we know that the probability will be equals to zero. Therefore, this will be equals to zero. And the probability of A and B will be the probability of A plus the probability of B. If and only if they tell you that A and B are mutually exclusive, then you apply this formula. If they don't, that is the probability formula that you're going to use to calculate the probability of A or B. So how do we then calculate it? Let's say we want to calculate the probability of main or promoted. To calculate the probability of either a main is promoted, then using this contingency table, we know that the joint probability is here. We know that we're going to use the probability of main by using that value, that event. How do we know that we're going to be using the probability of promoted? So we know the formula. It says the probability of A or B is equals to the probability of A plus the probability of B minus the probability of A and B. If we convert this to use main or promoted, it will say the probability of main or promoted will be given by 960, which is the probability of main divided by 1200 plus the probability of promoted, which is 324 divided by 1200, because these are events minus the joint probabilities of 288 divided by 1200. And since it's addition and subtraction, the common denominator is 1200. So we can say 960 plus 324 minus 288 divided by 1200, it will give us 0,83. And remember, the reason why we subtracting 288 is because we do not want to double count. Your exercise, calculate the probability of main or not promoted. Are you done? Anyone who wants to calculate with me? Anybody? Nobody? Okay, women or not promoted will be given by the probability of probability of women, probability of women, plus the probability of not promoted minus the probability of women and not promoted. That's the formula we're going to use. To get the probability of women, because we're using event, it will be 240 regardless of whether they were promoted or not. So that will be 240 divided by 1200 plus the probability of not promoted, which is the draw, which are 876. So there will be 876 divided by 1200 minus the probability of women and not promoted. Others, 672, sorry, we're looking for women, not promoted, not men, not men, but women. My razor, so we're looking for those ones because we're looking for the women, which is minus 204 divided by 1200, which gives us 240 plus 876 minus 204, which is 912 divided by 1200, which gives us 0.76. We're going to go into break, and when we come back from break, then we will do the conditional probability, but before we do the conditional probability, because now when you go to break, I'm going to give you this exercise to continue doing. We'll meet again after 10 minutes. I think I said at 10 past one. So let's say in the exam, they give you a question and they say, if the probability of A is 0.4, the probability of a complement of B is 0.5, and the probability of A and B is 0.1, which of the following statement is incorrect and they give you a whole lot of statement that you need to answer. You need to be able to know what you need to do with those statements that they have given you. For example, if you look at this, they have given you a complement of B, so therefore you should be able to calculate the probability of A, and this is a joint event, all these are simple events. So in this exercise, I am going to give you three key statements that you're going to calculate and you're going to prove them right or wrong. Number one, you need to find the probability of B, which means find the complement of the B complement, so which is the probability of B. That number two, you need to find the probability of A or B. And if you forgot the formula for the probability of A or B, remember it is the probability of A plus the probability of B, minus the probability of A and B. Let's use an N sign. And the last statement that you need to validate B are mutually exclusive, find the probability of B. And those are the three statements that you need to use during the break. And we will come back, actually let's come back at five plus one because it is five to one. We'll come back at five to one. If you have any question during the break, you can ask, I will be online.