 Highlight so if there are no comments. Then we can start with this week's session. So. Before we even start with this week's session, I just want to highlight some things. I've sent an email through my UNISA, so you should have received the notes for today's session that we're going to go through. You on that email, there should be two links for those who haven't joined the MS team for this group for the tutorial. Your lecture also shared the code for the sessions that he will be having on MS teams to answer any technical issue questions that you might have on MS team. So you need to also join the lectures MS team as well. And I think all the other tweeters as well will be on that group. And it will be everyone who is registered for STA 1610. You are more than welcome to participate in that group as well. It's just mainly for administrative issues. And if you also want to talk to your lecture. Like I also mentioned previously that every first week of the month your lecturers have an MS team or a Zoom session where they discuss issues that you might have and they give you some hints as well. So you are more than welcome to participate. So it means in STA 1610 we go an extra mile. So including also the lecture is not only us as tweeters, but we also support your lecture also supports you. And you need to also use that because it's going to benefit you in the long run. Anyway, I also want to highlight one last thing that for all the tutorials I have pasted them in a way that it allows you to have an understanding of the content and then also have an understanding of the activities themselves. And that is why on Saturday we do the activities. And I have pasted in a way that it will also help you to understand and be ready to write or do your assignments. So your assignment one is due on the 24th, but I encourage you to start doing your assignments now. And you have three chances to do your assignments. The sooner you do your assignment, the better. Remember those, you're not going to get your score immediately. You will only get your scores after the closing date. So now since the closing date has been extended to the 24th, you only going to know what is the percentage that you received after the 24th. But remember also, those who scored less than 50%, your lecture will still send you emails to give you another try to submit your assignment as well again. Only those who received less than 50%. So if you did well, he's not going to send you again, but you have three chances to submit. So you can do it the first time and feel like, ah, it didn't go well, let me retry, let me do it again. Take it as a practice assignment. Take it as you trying to understand the content as well. Don't look at it as an assessment or something like that, but use it as a practice in order for you to learn some contents because three times and your highest will be recorded. So I will take all the three chances that I have and use that. So please use all the opportunities that you are given. OK, so we can discuss this later on as well. So let's start with this week's session. Welcome to your session number five. Today we're going to be doing basic probability. We're going to learn by the end of the session, you should know what are the key concepts in the basic probability. What are the basic probability rules that we apply in cases of applying the general addition rule, independence, multiplication rule, mutual exclusives, and so forth. And then lastly we'll end the session by looking at conditional probabilities as well. It is going to be a very long session. We will do some activities and exercises together, remember we're going to use the chat group as well, the chat function to put our answers on there and interact. Remember once a person have posted in the chat, you can like that answer and let's see how many people have answered the same way. OK, so we still continue in that manner and we will still do activities together. So let's start with basic probability concepts. So in general, probabilities are things that we use on a mostly like, it's like we use them on a daily basis as well. And I want to find out from you because I think once you understand what probability is about, then I think it makes it easier. Do you know what probability is? What is a probability? And that is a question that I'm posting to you. Do you know what a probability is? Can I take a guess there? Yes, you can. Probability is the possibility that something could happen. OK, yes. That is a probability, yes. And can you give an example of a probability? Or how we use probability? I'm thinking. Whether it's going to rain or not. Aha, there we go. So we constantly use that. And these days, like for example, we like to look at the weather forecast. And on the weather forecast, they tell us about the likelihood of the rain. Like they say it's going to be 60%. There is a 60% chance. Those are probabilities. So you have been exposed to probabilities. So now we're going to learn how to calculate those probabilities. How do they come up and say that is a probability of 60%. You're just going to learn those basic things about those probabilities today. So a probability is a study of chances. It tells you, it gives you a chance that a certain event will OK or not OK. And if an event will happen, that probability will have the probability of a 1. If an event will not happen, it will have a probability of 0. So it means probabilities are between 0 and 1. So it means most of the probabilities, either we represent them as decimals and they can never be more than 1 or they can never be less than 0. So they will always be between 0 and 1. So if you're doing any calculations during your assessment or your activities or your exam and you calculate the probability and the question was asking you about the probability and you get 2,3. You must know that you did something wrong with your calculation. Your probability should be only between 0 and 1. But sometimes probabilities, because they are in decimal, they are in a relative frequency format, we can multiply that by 100 and get it into a percentage. So in terms of a percentage, 0 will be 0%. 1 will be 100%. So it means a probability can be only between 0% and 100%. If you get anything above that, it's not a probability. So what are some of the terms? So we said a certain event, an event that we know that is going to happen. Like we know that the sun is going to come up. The sun will come up and we know that the sun will go down. That isn't a certain event. So that probability will be 100% all the time because we are certain about that probability. An impossible event is an event that would not happen. If I can think of an event that can never happen. earthquake. Pardon? earthquake. earthquake. But an earthquake can happen. They happen here in Cape Town as well. We do get earthquakes. So the chance might not be 0%. It's possible that an earthquake can happen. So I'm trying to think of the sun being green. The sun being green. Yes, that's an impossible event. So we always know that the sun will never be green unless something, I don't know. But that is definitely a zero. The sun cannot be green. Yeah, so an impossible event will have a zero probability. So when we talk about probability and we spoke about the sun coming up, the sun going down, and the probability of an earthquake happening, or the sun turning green. When we talk about those, we're talking about events that are happening because the probabilities are calculated from events. So an event needs to occur before you can calculate a probability. Does it say recording? I don't see that. Oh, you are recording? Yes. So in order for us to calculate the probability, then there should be an event happening. So there are different type of events that can happen. And they can be simple event. So within an event, there are possible outcomes which are sun coming up or sun going down. So those are the outcome. In a simple event, there is only one event that is happening at a time. Sun coming up, that is an event. That is one event of the sun rising. A day in January from all the days in 2014, it's also one event because we're only looking for that one day in January. We can also have what we call a joint event. And a joint event is an event that describes two or more characteristics happening. So for example, an event that has two or more characteristics happening, so it means representing two events happening at the same time. Like taking an umbrella and it is raining, something like that. Oh, gosh. Taking an umbrella and it's raining. Or being accepted at a university and you are a male. So those are two events. So being accepted and being a male are two different events that can happen. Can you come up with another example? Maybe that makes sense to you. Two events happening at the same time. Working while studying. Working while studying. That's another joint event. Anyone else? Texting while driving. Texting while driving. That's another event. Those are two other possible events that can happen. Sometimes, because events happen in a population and that because when an event happened, there are possible outcomes from that event. And those possible outcomes, if we add them together, they create what we call a sample space. So sometimes one event from, let's say, we're looking at only one simple event. One event has an outcome, one outcome, and has another outcome. So let's say, for example, we have a coin. A coin has two. If I toss a coin, I'm creating an event. And I know that that coin can either land on a head or can land on a tail. So those are the two events that I have. So sometimes when we talk about events, we also talk about complementary events because a coin has two sides. One event will be a complement of, yes, one outcome will be a complement of the other. So an event can be a complement of the other event as well. So what is a complement of an event? A complement of an event is another, from a same sample space, the other event that is not happening will be a complement of the event that is happening. So for example, when I toss a coin, it will be an outcome. When I toss a coin, when it lands on a head, a tail will be a complement of a head. And that's what a complement event is. Tossing a coin, falling on a head, and it didn't fall on a tail. So therefore it means tail is a complement of a head. If I look at 2014 as my sample space, as my population, and I need to select a, create an event where I select a day, and I only select a day in January. All other events, all other days that are not in January are going to be complementary event of January. I hope that makes sense to you. And then we're going to learn how we calculate a simple event, how we calculate a joint event, and how we calculate a complement event. When we work with probability, do we have a set of events? Yes? Are you asking a question? Please make sure that you always mute if you do not have a question. Okay. So, like I already explained about the population which is called the sample space. So a sample space is a collection of all possible events that are happening. For example, if I have a die and a die has different sides or outcomes, all the faces of a die creates a sample space for that die that I have. Because when I create that event of rolling a die, it can either land on a one or land on a four or land on a six. For that die, I will have one, two, three, four, five, six as my sample space. And when I create an event, one outcome of that event will create an event. And I will get one outcome. Same applies when I have 52 cuts or a cut, deck of cuts. They all create a sample space. And when I take one cut out of the deck, I am creating an event which when I draw that one cut, I might draw an ace of heart or an ace of a diamond. I'm creating an outcome called ace of diamond from that event taken from the sample space. We can also represent events in different ways, like we can organize them and visualize them in a VIN diagram. So we've already seen what the VIN diagram looks like is the one that we used in the previous, when we were explaining the event, a VIN diagram. Let's say the block outside represent the sample space. So it means everything that will be inside the block will represent what we call event. So here we have event green and event yellow. Event green represent all days in 2014 that falls on a Wednesday. And event yellow represents all days that are in January. Now, we know that a calendar, let's say 2014 has January up until December. All those other days are inside that white area. So then it means a simple event Wednesday and a simple event January, they create a joint event called January and Wednesday. And that's where you will represent your joint event. So a simple event Wednesday and a simple event January, where they meet, they create a joint event for days that falls in January and also are on Wednesday. And those will fall in this category. And that's how you can visualize events. Other ways of visualizing events, we can use what we call a decision tree. And a decision tree says we start with the sample space. Let's say our sample space is a coin. So depending on how many times we want to toss this coin. So let's say we toss a coin. The first time we toss a coin, it lands on a head. And then we take that coin and toss it again. It can either land on a head or it can land on a tail. Let's say it lands on a tail. So therefore that event that was created should be a head and a tail, which is part of the outcome for that process or for that event that we created. If it landed on a tail, when we toss it for the first time and it lands on a tail, it can either land the next time we toss again, it can either land on a tail or it lands on a head. And if it lands again on a tail, it would have created an outcome for that whole event, a tail and a tail. And that's how you visualize events. And if you're interested in doing stats and doing machine learning and doing this data science or becoming a data scientist, one of the most used visualization visualization is and doing probabilities as well and selecting the likelihood of something happening and following it up. They use what we call a decision tree matrix or network or they use a neural network, which also form a type of a probability visualization method or graphic that you can use to follow the probability of something happening. The likelihood of an outcome being a yes or no or a fail or a pass or being at the risk or not being at the risk. So they use the decision tree to make those decisions. It's very interesting. Okay, so in this module, I would like us to like and love to use what we call a contingency table. Easy to use, easy to follow, like other than a VIN diagram or a decision tree. And you will see that most of the example that I do, I represent them in a contingency table because it's the easiest method to use to visualize the data. So a contingency table is just your summary table that shows your categories on the rows and other category on the columns as well. We call this a cross-tabulation as well. In the exam or in the assignment, if they didn't calculate the total, you can add the total at the end because they are very important. We're going to use them. Later on, you will learn that the total calculates what we call the simple event. All these totals, they calculate the simple event and whatever it's inside the table, it's where you find the joint event. For example, this is an event that a person selected is a main regardless of whether the person is promoted or not promoted, there are 960 of them. That is a simple event. But if a person is promoted and is a male, then that creates a joint event and there will be 288. You also need to calculate the grand total because you will need that most of the time when you are given the events. Yeah, you also need to remember this. Events are like whole numbers. So if you have a whole number, you must know that you're working with events. If you are given decimals you are working with, if you are given decimals or percentages, it means you are working with probabilities. So at the moment, these are events. You will need to calculate the grand total because for probabilities, especially the simple event and the joint event, you are always going to calculate the probability by dividing by the grand total, which is your sample space, which is your N. So later on when we do the calculations, you will learn what I'm, or you will understand what I'm referring to when I say you need to calculate the totals, you need to calculate the grand totals. So let's learn how to calculate those probabilities that we have been talking about. So a simple probability is calculated from a simple event. And the formula to calculate a simple probability is the number satisfying that event divided by how many there are, which is the total sample space, which is your N, which is also referred to as the grand total in a contingency table. Let's get an example. Like I said, we're always going to use a contingency table. So let's say we have this table that shows promotions data by gender. And we want to calculate what is the probability of being promoted? Because we only interested in a simple event of being promoted. So being promoted, regardless of whether the person is a male or a female, there are 324 people who were promoted out of 1,200. So looking at the formula it says, the formula says number satisfying the event divided by the grand total. So this is our grand total. This is the number satisfying this event because it satisfy the single event of being promoted. So it means our X will be 324 divided by our N, which is 1,200. Gives us 324 divided by 1,200. Gives us 0,27. I want you to calculate, since I have been talking a lot, now is your chance, now is your chance to calculate the probability of main. Calculate the probability that the person selected will be a main. Number satisfying the event divided by the total, the grand total. Irregardless of the person being promoted or not being promoted, we just want to know what is the probability of being a main? And I see Fiso has already posted. If you agree with his answer, just like it. Okay, the two Fisso's have posted. I'm going to get my... Yes. Can I just ask, where are the posting the answers? On the chat. On which chat? On MS team. There is a chat function. You don't know where to find the chat function. It is a... I will check quickly. If you join, did you join with your laptop or with your phone? I've joined with the laptop at Valya, definitely. Okay, so if you joined with it, it should be at the bottom. Sorry, it should be way, way, way. So... It's next to the hand. Do you see the hand? Yes, next to the hand. Okay, I got it. You got it. Thank you. Okay. Okay, let's see how many people agree. Garavo and Fisso, they say 0.8. I'm going to keep you get my card later while we do that, you are busy. Remember, you can like the other person's answer if you agree. Okay, you said the probability of men, regardless of being promoted or not. Yes. Okay, then I take back my 0.3. It's 0.8. Okay, so, okay, since we are all, we are already taking back some answers, how do we calculate that? So remember, probability of men, it will be equals to number satisfying that probability divided by n. The number satisfying the event, regardless of whether they've been promoted or not, they are 960. Divide by, we always divide by the grand total. So divide by 1,200, and that will be 0.8. And that's how you find the probabilities. Any question before we move on? No question, so everybody's on the same side. Okay, let's look at joint probabilities. Joint probabilities, remember, we get them from the joint event. And joint events are two events happening at the same time. Joint event formula, we use the probability of A and B will be given by the number satisfying that event, the number satisfying the joint event divided by the grand total. You can see that it's almost the same as what we just did with the simple event. This is number satisfying the joint event. To calculate the probability of men and being promoted from the same table, it says number satisfying the event divided by the grand total. Where do I find the joint event? Men and being promoted, that will be my joint event. Therefore, it means I'm going to use this my X and this is my N. So it will be 288 divided by 1,200. And that gives me 0.24. With that, your exercise calculate the probability of being a man and not being promoted. Men and not promoted. Sorry, ma'am. Will you please, I think what you're saying about 288 over 1.8. I didn't copy you well then. Joint probability comes from joint events. The formula says for a probability of A and B, which means the joint event A and B, it's given by number satisfying that event divided by the grand total. From our table, men and being promoted is a joint event. That means this is our number satisfying that joint event. Divide by the grand total. That is our grand total. Then we're going to just substitute our X, which is 288 divided by 1,200. And that gives us the joint probability of a joint event, men and promoted is 0.24. Your exercise, calculate the probability of men and not promoted. I see. In this class, do I only have three people? It's Fiso and Carabo. And Carabo. You must like the answers there, so that I can see that many people are doing the exercise. You're not just watching us on the screen, you are also taking part. That's the only way I can see. If we were in class in a normal face-to-face class, then I would be walking around and seeing what you guys are doing. So we are in an online environment. For me to know that you are working is by you participating on the chat. Please. Yes. I think my problem, my chat channel is still frozen, so I'm not sure what happened. Remember I reported some weeks ago and I still have that problem. What is frozen? The chat channel, I can't type. Next time when you join my UNISA, you need to join with your UNISA, my life email. Oh, not the link that is under this location. Oh, okay. No, it's fine. It means you're joining with your private email address. UNISA restricted private use, you will be joining as a guest instead of... As a student. Yes. So you need to join with your student email. Oh. You need to participate. For all communications with UNISA as well, when you're sending emails to your lecturers or to UNIS, if you use your private WEC email address, it will not go answered. UNISA does not even accept them. Okay, no, thanks. Not at all. Do that next time. Yes. Thanks. Okay, so probability is divided by N, number satisfying, joint probability, the joint event main and not promoted is? 672 divided by 1200. 1200, and the answer is? 0.56. 0.56. And that is joint event. We also have what we call marginal probabilities, but you do not have to worry too much about marginal probabilities because marginal probabilities are your simple events. If you look at the table, let's go back to the simple probability. When we calculated the simple probability, we used the total value because we said regardless of whether they are men or women in this instance, we're going to just use the total. With marginal probabilities, we don't look at the total, but we look at the in-betweens, those that are breaking down the promoted. We look at the men and women. We add them together to calculate the probability of the simple event, which is what we call marginal probabilities. So how do we do that? So marginal probabilities are the same as simple probabilities. And with marginal probabilities, we just add all the joint probabilities together to create a simple event. Let's calculate probability of being promoted. We calculated it previously and we used the total. So now we're not going to use that theory of its simple probability. Let's say we were not given the total, but we are told that for the people who were promoted, 288 of them are men, 36 of them are female. What is the probability that the person would have been promoted out of the 1,200 people that were there? So to calculate that, then it means we go into... Oh, sorry. To calculate that, then it means we go into calculate the probability of... No, no, no, no. Men, no, men and promoted, men and promoted, plus the probability of women and promoted. Now, as you can see, I'm writing end as an end. So I need to also mention this before we move on. The probability of A and B is the same as the probability of A and B. One and the same thing. For A joint probability, we can use an intersect or we can use an end. So they mean one and the same thing. So now we're calculating the probability of promoted by using the joint event men and women, plus the joint event women and promoted. And for that, we're going to find that for the probability of men and promoted, there are 288 divided by the grand total of 1,200. Remember, simple events, oh, sorry, events, we always going to divide by the grand total, plus 36 divided by 1,200. And that gives us, since they've got the same denominator, we just add the top part. So it's 288 plus 26 equals 324. 324 divided by 1,200. Because our common denominator is 1,200. And since they've got the same common denominator, we just add the top part. And this will be the same as what we have calculated previously. And it should give us 0,27 as our marginal probability. And that's what it gives us. Your exercise, calculate the probability of a women using marginal probabilities, probability of women. It's a bit confusing, regardless of being promoted or not. No, without looking at the total, calculate the probability of a women using marginal probabilities. So it means you're going to look at the probability of a joint probability of women promoted, women not promoted. 0.2. It will be 0.2, because yeah, we're going to look for the probability of women and promoted plus the probability of women and not promoted. And that will give us 36 divided by 1,200 plus, not promoted, there are 204 divided by 1,200, which is 36 plus 204, which gives us 240 divided by 1,200, which is equal to 0.2. And that is probability of women using marginal probabilities. Sometimes there are events, joint events cannot happen at the same time. There are events that this events cannot happen at the same time. A coin cannot, a normal coin, let's say, a normal coin cannot have two tails or two heads. Always have a head and a tail. If we have two events, that cannot happen at the same time. For example, a day happening in January and a day in February, those two days, they can never happen at the same time, because one day is in January and the other one is in February. So they can never happen at the same time. So it means mutually exclusive events are events that cannot happen simultaneously. If we choose, like I already did this example, if we choose 2014 calendar and we select A as a day in January and B a day in February, those two cannot happen at the same time. Therefore, event A and event B are mutually exclusive. And since they are joint probabilities, for event A, sorry, since they are joint event, the probability of event A and B happening will be equals to zero. So it means for joint events that cannot happen simultaneously, their probability will be equals to zero. And you need to always constantly remember this. The probability of event A and B, if they are mutually exclusive, it means that probability will be equals to zero. You also have collectively exhaustive event. It means events that include everything. They include every element of the sample space, let's put it this way. So with collectively exhaustive events, if you have events, so at least one of the event must happen from the sample space. And it needs to cover the entire sample space. Let's also continue in this manner and look at the calendar for 2014. If A, represent a day that falls on a weekday, B represents a day that falls on a weekend. So it means a weekday, Monday until Friday, a weekend, Saturday, Sunday. C is a day in January and D is a day in spring. Event A, B, C and D, all of them are what we call collectively exhaustive events because they include a day happening in a weekend, in a weekday, January, spring. So even those that are not in January and also fall on a spring, there are those in 2014 that would fall on a weekday or on a weekend. So A, B, C and D, all of them, they are collectively exhaustive. But A and B are mutually exclusive. Or we can also say, but they are not mutually exclusive to one another in terms of how we look at them individually. Because in January, they can be a day on a weekday and they can be a day in a weekend. January also falls in spring. When does spring start and end? Okay, spring day can either be on a weekday or a weekend and so forth. Event A and B are collectively exhaustive as well on their own because in 2014, a day, whether it's January or whether it's in spring or whether it's in February or March, it can be a day on a weekday or a day on a weekend. They are, both of them can be collectively exhaustive of 2014 days. And they are also mutually exclusive because they cannot happen at the same time. So a day in a weekday cannot be a day on a weekend. Then we say those A and B are mutually exclusive. Any question? If there are no questions, then we can move on to the next part. Hi, listen. Yes. Maybe I'm slow, but this collectively exhaustive event, the one that you just explained, maybe I need to revisit it later on maybe tomorrow, but yeah, it's getting me confused. I had in previous ones. Yeah. All what you need to remember with collectively exhaustive event is, what is my sample space? Are all events that are listed, are they collectively exhaustive in terms of my sample space? Do they include every other outcome of my sample space? Do they cover the entire set of my sample space? Let's say, for example, I had, I can remove, let's say we don't have weekend. Let's say it's A, C and D only. They will not be collectively exhaustive because then a day in 2014 needs to also include a day in a weekend, unless B was replaced by other, what do you call this, the seasons, other seasons that is not, that are not spring. Then you would have covered every, every day of 2014. But if I remove weekend and I don't replace it with any other event that might also replace some of, that might fill the gap that is missing to complete the entire sample space, then A, C and D will not be a collectively exhaustive event. It can only be collectively exhaustive event together if they cover all entire set of a sample space. We didn't even need to have C and D to get a collectively exhaustive event because weekday and weekend are collectively exhaustive events within that entire sample space of 2014 days because it covers all the days of the week. Okay, now let's summarize what we just learned. So in terms of the contingents, sorry, do you have a question? Yeah, I just wanted to know or what type of questions can we get from this chapter? Calculations and theory. And theory will be one of those things that we just mentioned now. Talking about collectively exhaustive event, talking about what type of events are mutually exclusive, talking about what are the simple events, what are the joint event, what are compliment events. So you will get questions that can ask you to calculate something, calculate the probability of something happening. Or you can get a question asking you about compliment event or a compliment event or a joint event or a simple event or a collectively exhaustive event or mutually exclusive event. On Saturday, when we do the activities, I will bring as many questions as well in terms of theory and calculation so that you can see the type of questions that you might get in the exam as well or in your assignments. Okay, so now moving forward, let's recap. In terms of the contingency table, because I said I like this table and I think you can also, when you get a question and you are given a contingency table, you can answer your questions as quickly as possible and easily if you understand what is happening. Remember A and B are events where there are outcome A1 and A2 or B1 and B2. When an event A1 happen and B1 happen, we call that event a joint event. And that is there, joint event A1 and B1. And we can calculate the probability of that joint event by using the value there. When an event B1 happen, it got lots of weather, A1 or A2 happened. And that is what we call a simple event. And we can calculate it, can calculate the probability of that simple event by using the total. If you are not given the total, you will be given the simple event you just add them to calculate the total. So you will add A1 and A2 joint events and calculate the total. And you use the total to calculate the simple event. When you add A1 event, A1, B1 joint events there and you add it to the joint event B1, A2, you are creating a marginal event where if you add the probability of the marginal event, you're creating a simple event. The sum of all joint probabilities within this table should be equals to one. The sum of all the total should be equals to one. The sum of this total should be equals to one because the sum of all probabilities should always be equals to one because probabilities cannot be more than one. Those are our joint events. That's where we calculate marginal or simple probabilities. Recapping so far, what we have learned? We have learned that a probability is a numerical measure of the likelihood that an event will happen and we learned that the probability of an event must be between zero and one and we know that zero is an impossible event and one is a certain event. We also learned just now that the sum of all probabilities should always be equals to one. Oh, the probability, not the sum actually, probabilities lies between zero and one and now we learned that the sum of all probabilities should be equals to one and if A, B, C are mutually exclusive. So if event A and event B and event C are mutually exclusive and collectively exhaustive they should, the sum of those values or the event should be equals to one. You need to learn this from now on until you go write your exam because every week from now on, the same concept that we just dealt with now, the sum of all probabilities is equals to one. Probabilities are between zero and one. Even the next discrete probabilities that we're going to be doing, you need to remember this. When we do hypothesis test and you need to remember this. When everywhere from now on until you go write the exam you always need to remember this. That probabilities lies between zero and one. The sum of all probabilities are equals to one. And if the sum of all probabilities are equals to one, remember in the beginning we introduced what we call compliment events. And we said compliment event is an other event that are not included in the one that you are looking at. Remember that. So a compliment event, now I can introduce it since we've dealt with so many other events now. A compliment event is defined to be an event consisting of all sample points that are not in another one. So if I'm looking at A, so the compliment event will be an event that is not included in A. Therefore, if I know that is not included in A so then it means the probability of A plus the probability of not A should be equals to one. The probability of A plus the probability of not A should be equals to one. That is compliment event. How do we write the compliment? We can write it as not A or we can write it in this form. So sometimes we write the compliment event as P, A or with a C or we write P, A with a copy. They both represent a compliment event. So that is a compliment event. Or we can write it as in full the way we're saying it because we say it's not in A so it's not A and the sum of it will be equals to one. So if I want to calculate the compliment event, so I will say one minus probability of A because I will just bring probability of A onto the other side. And then... Orcologists, can I disturb you? Yes. Okay, I had an impression that mutually exclusive and collectively exhaustive probability, when you're explaining according to the weekdays and weekends and then January and spring, an impression that that event was impossible. So now I'm looking at this example here that you have here. I'm thinking it's, isn't it supposed to be denoted as being zero? According to this example. Nope. Remember last bullet, event A and B are collectively exhaustive, but they are also not mutually exclusive. A, B, C and D on their own, they are not mutually exclusive because an event in A can have, or in January, for C, it can also include a weekday event. So it will not be mutually exclusive in this instance. All of them combined, not mutually exclusive, but A and B are mutually exclusive. And that is why I asked, when does spring starts and end? Spring is in September, around September, October. I don't know. I think it ends in November, right? Yeah, it's somewhere there. Yeah. So it means C and D are also mutually exclusive because a day in January cannot be in spring. So C and D, C and D are mutually exclusive, but they are not collectively exhaustive on their own because what about the other days that are not in January and what about the other days that are not in spring? So C and D on their own will not be collectively exhaustive, but they will be mutually exclusive. So you need to remember that. So in terms of, and if they are, oh, sorry, let's go back there. So if I come here, the sum of all this should give me one. So if I add all these values, the probabilities, if I calculate the probability of this, the probability of that, the probability of this, it should, regardless, the sum of all of them should be equals to one. And that is what we're referring to there. Remember, if they are, oh, sorry, they will not be equals to one because they are not mutually exclusive, all of them together. Only if I add weekday end, weekday, get it right. So because they are not mutually exclusive, the sum of all of them will not be equals to one, but the sum of the probability of weekday and a weekend, only for this event will be mutually, will be equals to one. So the sum of only A and B will be equals to one because they are collectively exhaustive and mutually exclusive. They include all the days in 2014, in the sample space. Regardless of whether it's January, February, or spring, they will be covered. It will include all the events. Okay, yeah, that makes sense. That is the compliment event. Your question. The probability that it will rain today is zero comma seven. What will be the probability that it will not rain? This is a compliment event. Remember, probability of A plus the probability of a compliment of an event, A, should be equals to one. If A is rain today, then what is the probability of not rain? Okay, zero and rain. A would be one. So this would be zero point seven plus the probability of not rain, which will give us, we take zero point seven on the other side. The probability of not rain will be equals to one minus zero comma seven. Therefore, our probability or a compliment event will be equals to zero comma three. That's right, Selah Lady. And whoever liked your answer. Any question? Easy stuff, right? We're not gonna finish everything probability today. So we will continue on Saturday with the rest of the probabilities and then do exercises, activities on Saturday. More activities. Now let's move on and look at the rules. The addition rule states that the probability of A or B is given by the probability of A plus B minus the probability of A and B. What does that mean? Probability of A or either B happening is given by the simple probability of A happening plus the simple probability, the simple event of B, probability of the simple event B happening minus the joint probability of both of those events happening. Also, probability of A or B, we can also write this as probability of A union B. A or B or A union B mean one and the same thing. So if you look at the past exam papers or some examples somewhere or in your assignment, they put the A, U, B, you must know that they mean one and the same thing. So the probability of A or B is given by probability of A plus the probability of B minus the probability of A and B. In the exam, you will get this formula. It's one of the formulas for the probabilities that you will get. You will get the probability of a simple event. You will get the, or anyway, you're writing online. Okay, my bet. But let's say they go back into the venues, you will get the formulas. It's one of those formulas that will be included in the exam. Okay, if event A and B are mutually exclusive, then the probability of A and B is equals to zero. And in that case, therefore it means this is equals to zero. Then the formula will just be the probability of A or B is given by probability of A plus B. When do I use one or the other? Only if A and B event A and B are mutually exclusive. You only use the second one. For mutually exclusive event, the probability of A or B is given by the probability of A plus B. Otherwise, if they ask you, calculate the probability of A or B, in regards of what's happening, you use the normal formula. So this is the formula we use. The probability of A or B is equals to the probability of A plus B minus the probability of A or B. And only if they say event A and B are mutually exclusive, or you calculated the probability of A and B and found that those two events probability is equals to zero, then if they ask you the follow up question and they ask you to calculate the probability of A or B, you must use probability of A plus probability of B. So how do we do that? Let's look at calculating the probability of men or promoted. We know the formula is the probability of A or B is given by the probability of A plus the probability of B minus or it's missing the probability of A and B. And remember this can also be written as the probability of A union B is equals to the probability of A plus the probability of B minus the probability of A intersect B. It can be written like that. I mean one and the same thing. So let's look at our table that we have been using all along. Calculating the probability of A or B, we know that the probability of A or we need to calculate the probability of men being promoted. So we need to go and say the probability of men or promoted, we need to write the formula down. Let me not write it in pen because I think I have it here. We can write it as probability of men or promoted will be given by the probability of men plus the probability of promoted minus the joint probability, which I also left the P there, the probability of men and promoted. Probability of men, regardless of whether they've been promoted, there are 960 of them. Probability of promoted, regardless of whether they are men or women, there are 324. Probability of joint event of men and promoted is 288. Substitute into the formula, 960 divided by 1200 plus 324 minus 288 divided by 1200 gives us 0,83. Your exercise. Your exercise is to calculate the probability of women not promoted. Probability of women or not promoted will be given by the probability of women plus the probability of not promoted minus the probability of the joint event of women and not promoted. Women, regardless of whether they've been promoted, there are 240, 240 over 1200 plus not promoted regardless of whether they are men or women, there are 876 all of them minus the probability of women and not promoted, there are 204 over 1200 always divided by the grand total. Do you have an answer? My answer is 1.1. Probabilities are always between zero and one. And if you get that answer, it means there is something wrong, right? Mine is 0.76. It's 0.76 because 240 divided by 1200 is 0.202 plus 876 divided by 1200 it's 0.73 or is it 0. Yes, 0.73 minus 204 divided by 1200 it's 0.17 0.2 plus 0.73 minus 0.17 is equals to 0.76 So remember, the other thing is all of them have the common denominator. You can just say 240 plus 876 minus 204 divided by 1.12 So you will say 240 plus 876 minus 204 equals 900 and so you will get 912 divided by 1200 when you divide that by 1200 you get the same answer 0.76 Oh yeah, I see. I added instead of minusing the 204. Yes. That's my mistake. Okay. So we left with 30 minutes. In this next 30 minutes we are going to do this exercise. And this is one of the complex exercise. It looks easy. It looks straight forward. It looks like you can answer the question because the question is asking, where is my question? Oh, sorry. The question is asking in eventually. Huh. I'm gonna get to the question just now. So the question actually was asking, find the probability of B given A because we haven't done that as yet. But I just wanted to show you that from this complex question you need to know all these steps. And remember, this is one question in the exam. So if a question comes in that format for example, so we're going to use this one as an example. So with this question, we're going to answer all three of this question because in the exam or assignment, probability of B, they will give you the answer there. You just confirm whether that is true or false because you're looking for the correct answer in your steps because you're writing a multiple choice question. So all the options will be one of the calculations that you need to be making. So in state of us following that, I just want you to practice with the actual question. So let's go. So given that statement, if probability of A or B or the probability of A complement is 0.5 and the probability of A and B is equals to 0.1, what is the probability of B? You are given the simple event, you are given a complement event B, you are given the joint event A and B. Find the probability of B. Already, Lungile answered the question. So because we are given the probability of a complement, so this will be one minus the probability of B complement, which then it's one minus 0.5, which is the same as 0.5. That is as easy as that. The next option that I have given you the probability of A or B. So you need to calculate the probability of A or B. Finding the probability of A or B, you need to know, remember the formula because the formula is the probability of A plus the probability of B minus the joint probability of A and B. You just substitute the values onto the formula and calculate the probability of A or B. So Lalei said 0.7 and Fiso says 0.8. Let's see who agrees with either one of them. Like, like, like. Let's see. And Lungile says 0.8. And Lungile says 0. So you can like the answer. Let's see which one you guys agree with. 0.8. 0.8. 0.8. 0.8. Okay, so probability of A is given, which is 0.4. Probability of B, we just calculated it, it's 0.5. Minus the probability of A and B is 0.1. 0.4 plus 0.5 is 0.9. Minus 0.1 is 0.8. How did you even get to 0.87 if we're only working with two decimals? With one decimal? Remember, when you are given decimal numbers, you are given probabilities, right? When you are given events, they will give you as whole numbers. Okay, so the next option they would have asked you, if A is a mutually exclusive event, find the probability of A or B. What will be the equation for A or B? For mutually exclusive event, it will be the probability of A plus the probability of B. 0.9. And that will be 0.4 plus 0.5, which is equals to 0.9. And that's how easy it will be, because all these will be options that you just need to verify based on that statement that they would have given you. Okay. Hi, Ms. Liz, can I ask something? Yes, you may. Sorry, I just joined in from the other class. In terms of the probability of A and B, is it always gonna be given or you can just add up the probability of A and B once you've calculated it? Because I think that's what you did. Okay, so for example, when you see probability of A and B, you must not get confused between the joint probability A and B. It doesn't mean adding A and B together to get A and B, right? This is a joint event. It has its own values. This is the probability of a simple event. In this simple event, there is A and B as well. In this simple event, B, there is A and B. So you cannot add A and B and say it is equals to the joint event. So a joint event is a probability on its own. A simple event is a probability on its own. In terms of event given, for example, A and B are general letters that we use to represent certain things, to represent gender, male or female, to represent whether we use A and B as a general formula. So when you say A and B will always be given, hey, it's very difficult to say because you might not be given A and B, you might be given B only, and you might be given the joint event A and B and ask to calculate what A will be. So you must just read the questions carefully because you will be given some sort of information to enable you to do certain calculations. For example, let's say this question that we're asking calculate the probability of A or B. They have given us the complement event A which is the probability of B given, sorry, the complement of B, then didn't give us the probability of B. So we need to first calculate the probability of B in order for us to find the probability of A or B. So that will give you to some extent some values, but you need to derive some of the values in order to get to the answer. Okay, so, oh, we still have a lot of time, 20 more minutes. Not sure if I need to confuse you even further with the conditional probabilities, but since this is a recording, you can always come back to it and watch the recording. We will recap before we start on Saturday. So let's continue and look at the conditional probability. What conditional probabilities are event happening given that another event has already occurred? If we want to find the probability of A given B has already occurred, that probability will be given by the joint event of A and B divided by B. So we're always going to divide by the given event. Or we can find the probability of B given A, which will be the joint event of A and B divided by the given event, which is the probability of A. And we know that probability of A and B are joint event and the probability of A or the probability of B are your simple event or your marginal event. So if the question asks you to find the probability of an event given that another event has already occurred, then you need to use conditional probabilities. Let's say they give us the statement of the people hired, 27% of them have been promoted, 80% are main and 24% are both male and are promoted. Ask us to calculate the probability of a given event. Sometimes it's not easy when you are reading the information straight away like this, unless if you understand your probabilities. So the easy way for me is to visualize this in a contingency table. Then I draw my contingency table. I have my main, my women promoted, not promoted. Then I start populating. The first statement said those people who were hired, 27% of them are promoted. So I'll put them regardless of whether they are male or a main or a women, promoted. They will go into the total of promoted because that is my simple event, promoted. So that's where my probability will be. Remember this is in a percentage or in a decimal form. It means this are already probability calculated. So I don't have to worry about dividing by the sample space and X divided by that. Already these are probabilities. The second one says 80% of them are main regardless of whether they have been promoted. I will put them under the total of main because that's where my simple event main is at. The last one says it's a joint event of both main and promoted, the 24% which is 0.24 which I will put it on a joint event of main and promoted. What is left? I can just populate the whole table and I can calculate all the other probabilities at a go. Because let's say we assume that this is a statement they give you and they have five statements that you need to check which one is correct. So it means you have to go through each one of them to calculate and find out which one is correct. So in order for you to do simple, easy, quick thing to do, populate the table. We know that the sum of all the probabilities is equals to one. So therefore it means my sample space will be equals to one. Then easy, I can populate already that because this is a total. It means it's main and promoted plus main promoted will give me my total. I can calculate this because it's one minus 0.27 will give me my 0.73. I can calculate this value here because one minus 0.8 will give me 0.2 and I can do the same with women promoted. 0.27 minus 0.24 will give me 0.3. And I can also find my promoted, not promoted, not promoted, my not promoted. 0.80 minus 0.24 gives me 0.56. And the last one, I can just use 0.2 minus 0.3 or I can say 0.73 minus 0.57. It will just give me 0.17. Now I have my joint probabilities and I also have my simple probability. So I can answer any question with regards to conditional probability or any probability that they ask me. If the question was, find the probability that main given that they have been promoted. So it means joint probability, main and promoted divided by promoted. So main and promoted is 0.24. Given that we use, we divide by the given which is promoted. So regardless of whether they are main or women promoted is 0.27. Then just substitute and calculate the probability of main given that they were promoted. I want you to calculate the probability of women given that not promoted. And that will be given by the probability of women and not promoted divided by, we always divide by the not promoted by the given. So it will be divided by the not promoted. The probability of women and not promoted is 0.17. 0.17 divide by the probability of, sorry, I forgot to put there, probability of not promoted, regardless of whether they are male or female is 0.75. So seven threes divide by 0.73. And therefore my probability of given will be 0.17 divide by 0.73, which equals 0.23. And you can go on and calculate the probability of being promoted given that it's a main, or probability of being promoted given that it's a women. And you follow the same concept. Any question? And then we're wrapping up. If there are no questions, so to wrap up with conditional probability as well, we have what we call a multiplication rule. And a multiplication rule for two event A and B, we can find it by using the probability or the conditional probability. And what it means with this, it says for a joint probability, the joint probability of event A and B can be given by the conditional probability of A given B times the probability of B. And if and only if A and B are independent, therefore the probability of a given event will be equals to the probability of A for independent event. The probability of A given B is the same as the probability of A. Then if that is the case for independent event, then the probability of A given B will be replaced by A, then it's the multiplication. So it becomes the probability of a joint event A and B will be given by the probability of A times the probability of B. And this is only if A and B are independent. Otherwise, the joint event of A and B is given by the probability of A given B times the probability of B. When do I use the conditional probability? We use the conditional probability, sorry, we use the multiplication rule. If I have the probability of A given B, they given me that, but they didn't give me, if they didn't give me the probability of A and B, and they also gave me the probability of B. If they give me the red, but not give me the black, then I need to calculate the multiplication rule because then this, if I need to make A and B the subject of the formula, so this will be, I will be left with the probability of A and B on the side, and I need to get rid of B, probability of B, then I must multiply the side by the probability of B and it means I must multiply the side by the probability of B, B and B cancels out, then I'll be left with the probability of B and this will be the probability of B times the probability of A given B, which is the same as what we had. If I rewrite it, the probability of A and B will be equals to the probability of A given B times the probability of B, which is multiplication rule. If and only, if A and B are independent, then the probability of A and B will be given by the probability of A multiplied by the probability of B and that's what we just explained in that nice slide. And that's all what I explained in this slide. Must also remember that the probability of A given B is the same as the probability of B and the probability of B given A will be equals to the probability of B. In regardless of what happens to B, whether A is known or not known, it does not have any bearing on B, what happens to B. So that is why the probability of a conditional probability of B given A, if events are independent, therefore, it will be the same as the probability of that event. On Saturday, we will start with this exercise after we recap and also discuss the independence probability, which I just explained now, but we'll just explain it again and then continue with the exercise and do a whole lot of other exercises. With that, it ends today's session. Just to recap on what you have learned, you have learned the key concepts of probabilities. We've learned the rules of the probability. We looked at addition rule and we looked at the multiplication rule and we also looked at the conditional probability. On Saturday, we will continue and look at the independence rule and then we would have been done with chapter four and you should be able to start practicing and getting ready to do your assignment two, which actually is due on the 30th. So if you're still waiting to do your assignment one age, you're going to be under pressure because you won't have enough time to do chapter four, chapter five, for your assignments two. Within six days, if you're still struggling with the simple easy stuff of chapter one, two and three, when it comes to the probability, you will need more time and that is why you need to start now practicing, not doing your assignment, practicing, so that by the time you go look at your assignment, you know what is happening. Okay, with that, end today's session. Any comments, any input, any query, you can ask. Thank you so much, Ma'am, that's informative. Okay. It's interesting what you did and that's why we are still overwhelmed a bit, but we'll go through it again and then we'll see. Thanks a lot, it's very informative.