 Good afternoon everyone and welcome to another session of your academic literacy, so especially the numeracy literacy, where we're looking at the basic skills for the basic skills, looking at statistical skills. Please make sure every week you join the session, you complete the register, the link will always be shared in the chat. I will keep on reminding everyone during the course of the session as well and then I will also repost the link as the session progresses and then also at the end of the session. If you have any technical issues or you do not see any recordings or you are unable to get to the schedule, please send an email to CTNTAT at unisa.ac.za and use your My Life email address when you communicate with Unisa. If you have any questions relating to your module, where you touch on the content of your module, you are not understanding how certain things you need to be answering certain questions, you can send an email to me at unisa.ac.za and copy CTNTAT at unisa.ac.za. Those are the two important emails that you should always have. Like I said, welcome to your session. The basic statistics for human sciences, where we look at the statistical skills of your PSYCH3704. For the day session, we're going to concentrate mainly on basic probabilities. The plan, not for August but for September, is as follows. On the 6th of September, there are your dates from the 6th of September up until the 27th and the topics that we're going to be covering each week. So please familiarize yourself with this. The session plans and the notes are always uploaded onto the platform, My Unisa platform. You can access it via the link. I think if you are already on the WhatsApp group, you already have the link to access the notes and the recordings and the session plans. They're all in the same, almost the same place. Before we start with today's session, are there any questions, comments or queries that you have, anything that you want me to address before we start with today's session? Remember, there is a thing called unmute. I know that in the beginning, we put up a slide that says mute yourself. So when I ask a question, I expect you to unmute and talk to me. Gabby, we will share the WhatsApp link closer to the end of the session or as we progress. Please talk to me. Let me hear you because I'm not sure if you are hearing me unmute yourself and talk to me. Do you have any questions, comments? All good. No questions from me. Thank you very much. Oh, yeah. Thank you. At least now I know that I'm not alone. Yes, I'm here. Thank you. Hello. So let's hope we keep this momentum of talking to one another and not let it be a one way. Yeah, let's make the session fun. Okay, so since there are no questions or comments or queries, just give me a second. Okay, so let's get you to understand and learn more about probabilities. For this session, you just need your calculator and your formulas. If you don't have a calculator, you need to go buy yourself a calculator or go borrow a calculator from someone because most of your module, because it's research, it's statistics, it's quantitative. You are required to do some little bit of calculations, but I will show you how to do some of these calculations. They are easy, straightforward, but you need to know the formulas. It's very, very important to always remember the formulas and know them. When you were writing face to face, when UNISA was writing face to face, they would give students the formula sheet that makes it easier for them to know which formula to use. Since you are writing online, I'm going to assume that they allow you to bring your own formulas to the exam and you can use that. So you need to make sure that you have a formula sheet that you create for yourself for every little thing that you do so that when you write either your assignment or your exam, you have something close by that you can always refer to. Okay, so by the end of the session today, we're going to learn the key concepts of basic probabilities. Remember, this is also not Psych 3704, it's not tutorial, so I'm going to touch basically at the high level statistical concepts. So it might not relate 100% exactly to your content that you have in your module, because this is not about Psych 3704, it's about statistical literacy. So but I will try by all means to find examples that matches up to what your module refers to. And I will also try to look for certain concepts and refer to them the way your module refers to them so that then we have a synergy or I don't create any confusion. But basically, like I said in my first session, I'm not a Psych 3, I'm not a psychological research person, I don't have a background in terms of psychological research. I've got background in terms of pure statistics. So I will try by all means to bring everything to the spike research level in terms of explaining some of the concepts. So bear with me if I explain something and it still doesn't make sense for you, let me know so that I can find another example to make it easier for you to understand. So we're going to look at the basic concepts of basic probabilities, we're going to look at the probability rules, including the general addition rule, independent multiplication rule, mutually exclusive rules and conditional probabilities as well. What I'm not touching on is the Bayer's rule because Bayer's and conditional probabilities almost look exactly the same but with the Bayer's rule it means you look at the conditional probabilities prior and use the prior knowledge to create your probabilities. But I'm not going to touch on that. Okay, so looking at basic probability concepts. So we normally do use probabilities on our daily basis or on a frequent, but you sometimes are not aware that you are using the concept of basic probabilities. For example, who of you in the morning watches the weather and look at whether it's going to be raining or it's going to be hot? Most of us we do, right? And when you look at the weather they always give you some percentage in terms of whether it's going to rain or not then they might say that there is a 80% chance of rain or 60% chance of rain and that percentage that they tell us helps us to make a decision based on that. So if they give you the likelihood of something happening that is part of probability. So you are using probabilities. So hopefully by me making this example of probabilities you are now feeling at ease in terms of probabilities. You always use things like the chance that I'm going to make it tomorrow is this. I might not make it tomorrow. You are already making probabilities that you are calculating your chances. If any one of you who plays lotto you are playing a game of chances because lotto is about the chance of winning. It's about the likelihood of you winning the lotto and so on. So you are using probabilities so feeling so we're not going to be introducing things that you're not familiar with. So a probability is a chance that an uncertain event will occur and that probability will always have a value of between zero and one. So what do we mean by zero and one? It means you cannot get a negative value when you are calculating probabilities. You cannot get a value more than one because always your probabilities are represented in decimals. Unless if they ask you for a percentage or a proportion probabilities are always in decimal format like 0.5, 0.3, 0.2, 0.1 and 1 if I'm calculating in percentage form so it will be 0% to 100% so it will always lie between 0% and 100% or it will be between 0 and 1. So when you are calculating any question they ask you about probabilities and you get a value like 2. You know that there is something that you have done wrong. You haven't calculated your probability correctly. So a certain event that is the event that will happen definitely will happen for example a sun will come up in the morning. Definitely always 100% the sun always comes up in the morning. That is a certain event and that event has a probability of one. An uncertain event will be a probability that you are not sure. An uncertain event can either be you are not sure close to 60% not sure or 20% not sure. That is what an uncertain event an impossible event would be an event that cannot happen like it will never, never ever. It will never happen. An example of an event that can never happen. Which example can I use that can never happen. There is nothing that comes to my mind right now. A horse cannot become a dog. A dog horse. A horse that is a dog. That is an event that can never happen. A dog is a dog and a horse is a horse. That is an event that can never happen. An event is something that it's occurring, it's happening. That is what we call an event. When we assess the probabilities sometimes like I spoke about the sun coming up. When we talk about the sun coming up we're talking about an event. An event is when the sun is coming up. There are three ways that we assess probabilities. Other than knowing that we know that we have an impossible event and a possible event, an uncertain event. There are three approaches for us to assess the probability of an event happening. The next one is called a priority. A priority refers to the chance or the likelihood of an event happening when there is a finite amount of outcomes. Each and every outcome has the likelihood of occurring. That is the normal probability that we use. When you know that an event will or might happen. In that event you are sure of you are able to count the number of outcomes that can come out of that event. And each and every outcome from that event has the likelihood of happening. That is what we call a priority. And usually to calculate the probability of an event happening or an occurrence of an event. We use the number satisfying that event divided by the total. And the total year might be a sample space. For example, we know if I have a dice. If I call it in, I'm from that side of Harman's Grand. We don't call things like we don't speak normal English. We speak English. So we say dice, even though it's one, we say it's a dice. So a dice, right? A dice has six sides. If I roll a die, I'm creating an event. So if I need to calculate the probability of a die landing on a head or a die landing on a one, right? So I know that the total outcomes of that die, it's either one, two, three, four, five, there are six outcomes. So that is my T, that is my total number of outcomes that I have. If I need to calculate the probability of that dies rolling and landing on one, there is only one of the signs, right? So it's one divided by the total, which is six. And that is what the priority probability will be calculated using. So this is the formula that we will use every time when we calculate the probability of an event happening. But we do have another approach that we can use to assess the probability. And this is what we call an empirical probability, which refers to the likelihood of an event occurring based on the historical data. So now here we based this probability on what happens previously. Also it's the same because we need to know what probability or what outcome or occurrence that we need to calculate this probability for. And we also need to know what was the total number of outcomes that existed before, right? So for example, if we collected data about rain for the previous 10 years, that is historical data. We can use that data to calculate the probability of rain or of a day that the rain will rain and it rains, hey, we can calculate that if we have historical data around the rain patterns. And that is what empirical data is about. So also it will be the outcome satisfying the event divided by the total outcomes that occurred. Then these two events or the ways of assessing the probabilities, we normally use them usually because you will see in your data when, or not even in your data. So when they ask you questions, they might ask you questions around the priority or they might ask you questions around your historical data, which is empirical probabilities. Now, the last one, which is a subjective probability, this one is based on the researcher's opinion. So as a researcher, I can state what probability I want to work with or what kind of event because then this will be based on my previous experiences. So subjective probabilities is based on the combination of your individual past experience, personal opinion, analysis of a particular situation. So as a researcher, you will base the probability on things that you have experienced before or you have observed before or it is of your personal opinion. For example, we normally use this as people in this, as long as you have been living in South Africa, at some point you have made a probability which is subjective. For example, I can just give you an example of the Minister of the MEC of Health in Limpopo. Based on her own experience, she made the subjective probability calculated in her head probably to determine that the X amount of the chances are the majority of people that are getting healthcare in Limpopo are from Zimbabwe. Because of certain events that happened previously or certain knowledge that she had. And those are the things that we do as researchers. We always, based on our previous experiences, we might make some probability, create some probability or calculate some probabilities. It's still fine because it's subjective. It's subjective to that particular period that you are calculating those probabilities for as well. But those are the approaches. Now, we spoke about probabilities. We've spoke about events. What is it that when we talk about events? I've already explained that an event is when something is happening. And when something is happening, it creates within that there are outcomes which are things that you can count or look at or observe from that. So, when an event is happening, each possible outcome that comes out of that event can be used to calculate the probabilities. Like the sun is coming up. So, an event will be the sun rising. The outcome is the sun has risen. That will be the outcome and we know that that will be, for example, let's use a coin. It's easy when you use something that is practical. When you have a coin, a coin has two sides, has a head and a tail. When I create an event, it's when I start to toss the coin. When I'm tossing the coin, I'm creating an event. And there are two outcomes that can come out of that event. One outcome is the coin can either land on a head or a coin can land on a tail. And those are the outcomes, the two outcomes. So, when I have one coin or when I have a coin and I'm tossing a coin, I'm creating an event, right? That event is called a simple event because it's only one thing that is happening. One event happening at that point. That is what we call a simple event of a coin. Tossing a coin, I'm creating one simple event which I will use that simple event to calculate the probabilities. And we get to the probabilities just now. Now, if I have one coin and I have a die and I toss and roll the die, I'm creating two events. Or if I have two coins, I'm creating two events because if I have two coins, one coin can land on a head and the other one can land on a tail, right? Because there are two outcomes from each one. And that is what we call a joint event. It is when you are creating an event that has, or that can be described by two or more characteristics. So, two coins will create two or more characteristics of that. Or another event is having an education and waking. Those are two events that can happen. Having an education can be one event, waking will be another event. So they will be two. So it's a joint event because someone can have an education and not have a job. Or not a job. Can have education and not be waking. And the other person can have no education but wake. And the other person can have no education and no work. And the other person can have no education and waking. And no waking, something like that. So a joint event is when you create two or more characteristics. Then we also have what we call a complement event. And a complement event is an event that is not part of the original event. For example, if I have a simple event, sun coming up and sun not coming up. Those are two things, right? So the sun coming up will be the probability of that sun or the event sun coming up. The complement of it will be the sun not coming up. So when I have a tail or a coin, sorry, when I have a coin, a head and a tail from that. A head can be one event and a tail will be a complement, a complement of a head. And sometimes a complement of a head, we can write a tail as hc or h dash. And that's how we refer to a complement event. So a tail can be written as hc because it refers to a complement of a head. Or we can use a copy at the top to represent a complement of a head. And also a tail and a head. A head can be a complement of a tail. Or we can call it a complement of a tail. So we can write it in that manner. That still represents a complement event. An event which is not part of the original event. Or the original event. Then we also have what we call a sample space. A sample space is the total. It's all of them, all the outcomes. A sample space is what we call all outcomes. So for example, on a coin has two outcomes. So a coin will have two. And I will show you how to calculate the probability of a simple event, joint event and a complement event. We also have a mutually exclusive event. And a mutually exclusive event, since we have what we call a joint event, a mutually exclusive event tells us that the two events cannot happen simultaneously, cannot happen together. So when two events cannot happen together, like tossing the two coins, cannot happen together, that we call a mutually exclusive event. So let's take, for example, a day which is a random day in 2014. And let's say A represents a day that is in January. And we know the calendars. It starts from January up until December. And we also choose B as a day that falls within February. So we have two events, event A and event B. A day in January can never be a day in February. Because if January passed, it's passed, that day has passed already. So it can never be in February. So these two events can never happen at the same time. So you can never have a day in January and also a day in February, happening in 2014. So those we call them mutually exclusive events. And when we calculate the probabilities, I will tell you more about the mutually exclusive events. So it means the two of them will never meet or the two of them will never exist at the same time. We also have what we call collectively exhaustive events. And this, at least one of the event must occur. And the whole set of events should cover the entire sample space. So if a coin lands on a tail, and the other event can be a coin landing on a head, they all complete a sample space. But at least one of them will land on a, when you toss a coin will land on either one, the head or tail. Let's look at that example of collectively exhaustive events. So we're going to still stick to the 2014 kind. If we say A will represent an event that a day we choose, it's a day that falls within a weekday. So weekday is from Monday to Friday. That's weekday. And B will represent a day that falls over a weekend, which is Saturday and Sunday. And C represent an event that a day we choose will be in January. And D will represent a day that we choose will be in spring. So now A and B can also be collectively exhaustive events because the calendar has seven days of the week every month. So that can be, we can choose any day, either from January until December. A and C and B and D are also collectively exhaustive events because they create this sample space. They include all the days of the calendar, right? Because January, we have weekday, we have a weekend. And spring, we have a weekday and a weekend, a day that is in spring. They all complete the calendar of 2014. They all covers the entire sample space. However, they are not mutually exclusive. Only A and B are mutually exclusive because A, a day in a weekday can never be on a weekend. What else? Only that. And January and spring, when is summer? Summer starts when? I don't know, but spring and January are also not mutually exclusive as well. Okay, so that is collectively exhaustive events. They need to be, they need to cover the entire sample space. Event A and B are collectively exhaustive and they are also mutually exclusive. So a sample space, I told you that it is a collection of all possible events. So a coin, it's a sample space on its own, it's got two outcomes. A die, it's also a sample space because it's got six outcomes. A deck of cats, it's got 52 cats in it. They all make up a sample space. So now how do we calculate these probabilities? A simple probability, we calculate the simple probability based on a simple event. We know that we're going to be using, however, the classical probability to assess this. So it will be event satisfying the event, the number satisfying the event divided by the total. So if I have a coin and I know that my event, it's either a head or a tail, but I need to find the probability that the coin will lend on a head and that will be the number satisfying. There is only one outcome that the coin will lend on a head and there are, how many in total? There are two. So that will be 0,5. 1 divided by 2 will give you 0,5. And that is a simple event. So let's look at another example. So in this example, I have what we call a contingency table. And a contingency table, it's a cross tabulation table, which shows you the rows and the columns. And in our rows, it shows in this company, the number of people that were promoted and those that were not promoted, the number of men who work in this company and women who works in this company. And this company has a total number, which is our grand total of 1,200 employees. So the number of males, there are 960, the number of females, there are 240. Those who were promoted, there are 324. Those who were not promoted, there are 876. Those we call them simple events. Men, women, promoted, not promoted are what we call simple events. If I need to calculate the probability of a person working in this company and they have been promoted, then it means I'm going to take the number satisfied that promoted divided by the grand total, which is my sample space. And that will be for promoted 324 divided by 1,200. And that gives me 0,27. And we will continue using the contingency table because it's easy to represent probabilities on that. That is a simple probability. Then we have a joint probability. Now a joint probability is event satisfying joint events. We calculate it from one or two events happening. So using the same contingency table, if I need to calculate the probability of men and being promoted. Now I need to also state the following. Sometimes in your modules or somewhere, the end can be written as the probability of A and B. It means joint probability, or we can write it as the probability of A and literally end B. So the end there will represent intersection. So if I need to calculate the probability of a joint event of men and promoted using the same table that we had. So I know that this are men. This are promoted where they both meet the men and promoted. This is where we calculate the joint probability. So to calculate this, we say it's number satisfying the joint event divided by the grand total. The joint event there are 288, the grand total, which is our sample space is 1200. And to calculate them, we say 288 divided by 1200 and that gives us 0,24. That is how you will calculate joint event by using the event satisfying divided by the sample space. We'll do more exercises just now. I'm not going to ask you to calculate that. Sorry, can I just ask you a question if you can just go back to the previous page. So you wrote it at the top, there was the P and then there was a bracket and then there's an A and then there's an N and then there's a B. So that N is not necessarily, it doesn't have the same meaning as the calculation where there's an X over the N. No, there's no connection between those two. It's just the way you write it. Yeah, that's the way you represent a joint probability. It means one thing. So it's joint probability of A and B. Later on, we're going to introduce the addition rule and you will see we will change the end to an O. But it means one thing. It means the probability of a joint event of A and B. Okay, okay. So nothing to do with this N just below the X and the calculation. The N. Yeah, yeah, that's one. No, this N. Sorry. This N, remember? This N is your sample space. Okay, yes. Is your total outcomes, right? The grand total. Yes, it's your total outcomes, which is your grand total. Your X represent just the number satisfying is an X outcome. So X is a placeholder. Yeah. I was just getting confused because there was an N there and it was an N there. Oh, actually, this N is not supposed to be an NN. It's like an U that looks, not the U, something like this. Okay, okay. I call it an N, but it's sort of a U that is upside down. Okay, and it doesn't look like the N. Okay, thank you. Thank you so much. Okay, and in a nutshell, what I just showed you was that on a contingency table, you are able to find that on the total, where you calculate the total of the cross-tabulation, that's where you calculate your simple probabilities. Inside the table of your cross-tabulation, that's where you find the joint probabilities. And your sample space will always be the probability of a sample space will always be equals to one, because the sum of all probabilities should be equals to one. So just to recap and to sum up and introduce you to some of the concepts, including what I just explained summary so far. Probability is a numerical measure of the likelihood that an event will happen. And the probability of any event must be between zero and one. Remember, we dealt with this in the first page. An impossible event is the one that has the probability of a zero. A certain event is the one that has a probability of a one. And a 50% will be a probability of a 50-50, a half, right? And we also know that the probabilities can always be between zero and one. What we also know from what I just explained with the contingency table is that the sum of all probabilities of a sample space should always be equals to one. Especially for all mutually exclusive and collectively exhaustive events should always be equals to one. So if I have event A and event B and I calculate the probabilities, the sum of both of them should be equals to one. So the sum of probability of A plus probability of B should be equals to one. Then it means if I'm given the probability of an event A, I can find the complement of that probability. And a complement, like we said, is defined as the event consisting of all the sample points of A. A complement event is an event that consists of all the sample points that does not include A, right? Because it's the other event. In order for us to calculate that probability, so if I'm given the probability of a head and I know that it's 0.5, in order for me to calculate the probability of a tail or to calculate the probability of a head complement, which is a tail, then that probability will be equals to one minus 0.5, which will be 0.5. I know that it doesn't make any difference. Let's say I'm calculating the probability for a die. So we know that if the probability that a die will land on a one is equals to one divided by six, which will be equals to what is one divided by six. It will be zero comma. Let's take out our calculator and calculate that probability. That will be one divided by six, which is zero comma one seven. Zero comma. Zero comma one seven. That is the probability. In order for me to find the probability of a complement of a one, so the probability of not one, that will be everything else. So when I roll a die, what will be the probability that it's not going to land on a one? It means it's two, three, four, five, six. So it's five. So that will be one over, sorry, five over six, right? And you can calculate that. And that will give you, because I want to show you the complement. Five divided by six will give you zero comma eight three. Zero comma eight three. So in order for us to find the complement, we just say one minus zero comma one seven. That will give you the same thing. It will give you, because not one is a complement of one. That will give you one minus zero point one seven. It equals to zero comma eight three. Zero comma eight three. So that is the complement. So now let's look at more probability questions. So let's look at addition rule. And addition rule is a probability of, it's a probability law of disjunction or what we call an addictive rule. That is when we want to calculate the probability of either events happening. What if we want to find the probability of event A or B happening? That is given by the probability of event A happening plus the probability of event B happening minus the probability of a joint event, which is the joint event of A and B. If and only if A and B are mutually exclusive. So if they tell you that A and B are mutually exclusive, therefore it means the probability of A and B cannot happen at the same time. Then the probability of a joint event will be equals to zero. And probability of A or B happening will be given by the probability of A plus the probability of B. This is only applicable for when events are mutually exclusive. So always remember that. Then we also have what we call a conditional event. Conditional events are events that are happening given that another event has already happened. So if we need to calculate the probability of event A happening given that B has already happened, that is given by the probability of A and B divided by the probability of A of B. It's always divided by the probability of the given. So it's the joint event of A and B divided by the probability of the given. Or we can write it as the probability of B given A will still be the probability of A divided by the probability of a given and the given is A. If we need to apply what we call a multiplicative law. From the conditional probability is because we want to be calculating the joint event. If we want to calculate this joint event, then we are going to apply what we call a multiplication rule for A and B. Why we say that is because then we need to multiply the probability of B with the probability of a given. And that's what we do. So the probability of A and B will be given by the probability of A given B times the probability of B. So if and only if A and B are independent. So now if A and B are independent, then the probability of A given B will be equals to the probability of A because B whether B has a cat has no bearing on what is happening on A. Because they are independent, so they don't influence one another. They are not related to one another. If event A and B are independent, the probability that A given B will be the same as the probability of A. Then it means for our multiplicative law, it will state that if we want to find the joint probability of A and B, it will be the same as the probability of A times the probability of B. You need to pay attention to the following. In the beginning, we spoke about joint event. We said the probability of A and B is outcome satisfying divided by the sample space. Now, if they didn't give you the outcome satisfying the event, but they tell you that the probability of A is this and the probability of B is this. Then you need to calculate the probability of a joint event. You need to know that now you are looking at independent event. How do I know that I'm looking at independent event? If they give you the probability of a boy and a probability of a girl, those two has no bearing. Whether you are a boy or you are a girl cannot influence the two because you are not related. You're not from the same group. You are independent from one another. That is independent events. And if they give you that, you just need to know that those are independent events. Then I need to calculate the joint probability of A and B and treat it as the probability of A or the probability of boy and the probability of girl. That is independent events. So you need to be able to read the question properly. So when two events are independent, even only if the probability of A given B will be the same as the probability of A and the probability of B given A will be the same as the probability of B. And this is only if events A and B are independent when the probability of one event is not affected by the fact that the other event has already happened. In summary, we looked at now. Independent event, which is when the occurrence of an event has no effect on the probability of the other event happening. We also looked at mutually exclusive event. Remember with mutually exclusive event, it means that when the one event happens, it precludes the occurrence of the other. So it means they cannot happen at the same time. And the probability of both of them happening at the same time will be equals to zero. We also looked at mutually exhaust or collectively exhaustive events, which is exhaustiveness of events, which have a set of events representing all possible outcome of a sample space. We also looked at the law of disjunction and the law of conjunction, which is the addictive flow and the multiplicative flow. When events are mutually exclusive, then it's the disjunction rule that we all know that we use. And when the events are independent, then we are applying the law of conjunction. Okay. Now let's look at how we answer questions based on your psych 3704 based on the information that I just said. We will go through them together. I expect you to answer some of them. And for some, I'll give you the hints and then we continue from there. Let's look at question one or exercise one. Just give me a second. Okay. So in the population, there are 450 people of whom 150 smokes. What is the probability of randomly selecting a non-smoker? So now, based on the information given, remember we always apply the problem solving framework. New men's prompt, problem solving prompts. So what is the question asking us to do? They want us to calculate the probability of selecting a non-smoker. What have they given us in the question? They've given, they told us that the total, which is our sample space, which is our N, is 450. They also told us that we have event smoker from those 440. So our event smoker, which will be our X for smokers is that and we can also just represent our smokers with an S and say our event smokes is 150 and we're going to represent our non-smoker SNS, which is our complement of our smokers, right? Based on the information that we were given. So now, how do I answer this question? What are the things that I need to enable me to answer this question? Oh, easy, we can calculate the probability of a smoker, which will be our X divided by N or the number satisfies smokes by N and if this is our X, calculate and tell me what is the probability of smokers? So we're going to substitute 150 divided by 450 and that is, what is the answer? 0.33, 0.33. That is the smokers. So now they said we need to find the probability of non-smokers. So the probability of non-smokers are 1 minus the probability of smokers because probability of those who are not smokers are the complement of those who are smoking because if we are able to calculate those who are smoking or who are smokers, then we can find the complement of them which are non-smokers. So 1 minus 0.33 is equals to 0.67, right? Did you get it right? Is there anyone who is still lost? Who doesn't know? Where did I get all these numbers and all this? That is everything that we just learned now. And those who don't know how to use their calculator, you must let me know as well so I can help you because today I'm not sharing my calculator but I can do that if there are people who don't know how. Okay. Are we good? Are we happy? Are we good all? Yes, all good? All good. Perfect. Look at the second one. Which of the following does not represent a probability? I am going to leave it to you to answer this one now. Which one does not represent the probabilities? We covered this, remember? The probability starts from the certain number, the end at the certain number and those certain numbers can be converted to a percentage if we multiply those number by a hundred. But which one of these does not represent probabilities? One, two or three? Number three because it cannot be a negative. It can never be a negative. So number three is the one that does not represent a probability because remember that the probability of an event A can only lie between zero and one or it can lie between. This is something that you always need to remember. Hundred percent or zero percent? Exercise three. If 10,000 students wrote a university admission test, 7,000 passed and they obtained 50 percent or more and 400 obtained exactly 50 percent, what is the probability that a randomly selected student will fail the test? So this is the same as what we did previously, right? It's the same as what we did. They give us the smokes and they ask us to find the complement of them. The same thing. They gave you those who passed, they gave you there and they ask you to find the complement of those who didn't pass. Now, yes, they ask you to find the complement of those who didn't pass. So they say you need to find the probability of a fail. That's what we need to calculate at the end. So probability of a fail is one minus the probability of a pass, right? Because a pass is a complement. So I'm just going to give you some hints because I want you to do this one. So therefore it means we need to calculate the probability of a pass and a pass they gave you. They said you have 10,000 students, which is your end. Sorry. My bad. My bad. My bad. My bad. Okay, so they gave you. Let's, we know that we need to calculate the probability of a pass. They gave you a 10,000 students wrote. They told us 7,000 passed. So 7,000 passed, which will be our P. I'm going to call it a pass what, right? A pass what? And we are also told that 50% of them also passed, but they passed with exactly 50%. So we have passed two. Both of them, they passed, right? Because a pass of more than 50 and a pass of 50 are one and the same thing. So now if we need to calculate pass, we take the 7,000 plus the 400. That will give us the total of a pass. Our end is 10,000. So in order for us to find the probability of a pass, we're going to use the 7,000 plus the 4,000 or the 400 and we're going to divide that by 10,000. And that the answer you get here, you go into substituted into the probability of a pass and find the compliment. So 7,000 plus 400 is 7,400 divided by 10,000. What is the probability of a pass? The proportion of students who passed the admission exam. What is the answer? The answer to the 7,400 divided by 10,000 gives me 0.74. 0.74. 7,000 pass that they obtain 50 or more and 400 obtain exactly 50. So does that mean the 70 includes the 400? The 7,000 includes 400. So if the 7,000 includes 400, then it means those who have passed, there are 7,000. Then we can treat it like that. There are 7,000 because they passed with 50 or more. Not more than 50, but 50 or more and then only 400 obtain exactly 50. So the second part of the question is the one that is very confusing. So 7,000, if I need to explain this again, 7,000 they received a pass mark of 50 or more. So they got 50, 51, 52, 60, 70%. Out of those 7,400 of them obtain exactly 50%. However, the question we want is for those who have failed. So those who have failed will be those who have not passed. So since the probability of a pass is 7,000, so we divide by the total number, which is 10,000. So it will be 7,000 divided by, it will be 7,000 divided by 10,000, which will be 0,7. So then 1 minus 0,7 will give us 0,3, which is option, which will be option 3. So it means when you read your questions as well, whether it's in your assignment or in the exam, pay attention to the additional information that they give you because now we would have made a mistake if we included the 7,000 and the 410. If the option they had, one of them had the answer of 0.66 or something like that, we would have gotten it all wrong. So pay attention to the information given. Also it's very important to always double check. After you've done your calculation, remember also on the humans problem solving, it says reflect and receive feedback. So always go back and review and check that you have done the correct things. So let's look at exercise 4. Exercise 4 says the probability of an event or caring which depends on something else or caring, such as passing a test when you do not understand a course can be described as a probability of an event or caring which depends on something else or caring, even that another event or care. The probability of an event happening depending on something else or caring which such as passing a test when you do not understand your course can be described as, is it 1, is it 2, is it 3, is it 4, is it a conditional probability which is the probability of an event happening given that another event happened or is it 2, which states that the two events do not influence the other or is it 3, which states that two events can never happen at the same time or is it a multiplicative event which also states that two events can happen based on another event that would have already happened dividing it by another event that already happened. Number one. It will be number one because number one states that an event or caring given that another event has already happened which is a conditional probability. Correct. So this exercise five is not supposed to be part of the exercise. Let's get to the next one which almost look exactly the same as the past one that we just did. I'm going to ask you to answer this one without my help. If 10,000 students wrote university admission test, 7,000 passed and they obtained 50 or more and 310 exactly 50%, what is the probability that a randomly selected student will fail their test? It's the same. It's almost exactly the same as this question that we just did, right? So I expect you to be able to answer that with ease. So the answer would be then number three because it's the same question as the previous one but I just want to understand why did they not take in consideration the 300? Because those people didn't fail, they actually they passed the test, right? Yeah, remember now the 300 is also included in the 7,000 because remember in the 7,000 they passed with 50 or more. Okay, sorry. I need to read the question properly. So easily, okay. Then number three would be the answer. Yes. You actually get the same answer if you add the 7,000 and the 300 because then you get 0.27 and if you round it then it's 0.3. Yeah, but it would not be the right way of I'm telling you because the 7,000 includes as well the 300. I think the question is not very clear but for this it kind of works either way. Yeah, I've got some exam papers but they are not very dear because I want to share with you my script. Let me find another one. Another exercise because that was I thought it would have been done by now. Let me see if I can share my script. Okay. Just want to get to one of the questions but if you share my screen I can use the presentation. Just give me a second. I will show you another question which I think probably this one can give us some more in forwarding some of the things that we went through as well. So let me just go where I got it. It's from one of the tutorial letters. It's from let me just double check my tutorial letter which one did I use. 2018 tutorial letter 101. That's where I got this question from and probably I can also include two more questions and then it will take us to the end of the session because now I'm trying to face different questions. Okay. That is it. So let's look at this. If a coin is flipped three times the sample space of possible outcomes would be. So I have a coin and you flip it the first time. The first time it will land on a head because there are three, right? A head, a head, a head. The other time it can land on a head, a tail and a head. Or it can land on a head, a tail and a tail. Or it can land on a tail, a tail and a tail. Or it can land on which scenario we didn't cover. A tail, a tail and a head. Or we can have a head, a head and a tail. Which one we haven't covered? A tail, a head and a tail. What other scenario do we have? Tail, tail, head. Tail, tail, head. Tail, head, a head. Tail, head and a head. Do we have? So those could be the scenarios that we have. If we covered all of them. So head, head, head, head, head, head, head. Head, tail, head. Head, tail, head. Head, tail, head. Head, tail, head. Head, tail, head. Head, tail, tail. Head, tail, tail, head. that one doesn't have it this one doesn't have oh eight tail take there we go this one has eight eight tail take that one has eight tail take that one has so that we covered that we covered that we covered. Tail, tail, tail. So already this one is out because it doesn't cover all scenarios, right? So tail, tail, tail, tail. Triple T, triple T, triple T, triple T, okay? Tail, tail, head, tail, tail, T, T, H, T, T, H, T, T, T, H, T, T, T, T, T, T, H. This one doesn't have T, T, H, right? It doesn't have tail, tail, head. So it's out. Head, tail, tail. So we left with only two of them. Head, tail, this one doesn't have. So therefore it means option three will be the correct one. So you need to be able to know what are the outcomes that can happen from first in three points, whether those points can land on different ways. So I've covered all of them. And if you look at this, that is head, tail, tail, which it's covered like that. And then we have tail, head, tail, which is that. And we have tail, head, head, which is that. So it covers all of the outcomes. So therefore option three is the only correct answer from there. Question eight from that tutorial letter has this following. Two people, one girl and one boy will be chosen to go on a field trip from a class of 12 girls and 15 boys. What is the probability that Mary and her brother will happen to be in the same class? And both will be selected if boys and girls are selected. Both of them are are they dependent or independent events? Do they influence one another? No, they don't, right? They are independent events, both of them. So since they are independent, so if we need to, as they are independent and we are asked to calculate the probability that a boy and a girl, both of them happen to be selected. So we need to find the probability of A and B. And because they are independent event, therefore the probability of A multiplied by the probability of B. Or we can call that now we need the probability of a boy and a girl based on what we know, right? And that will be the probability of a boy times the probability of a girl, which is three. Now what is the probability of a boy? There is only one girl and one boy that needs to be selected. So that will be one over for the boy, one over 15 multiplied by one over a girl. They should only be one selected and there are 12 girls, which is equals to what is one divided by 15. One divided by 15 is equals to 0.066666. 0.006, I'm going to keep three decimals. And one divided by 12 is 0. multiplied by 0.083083. 0.66 multiplied by 0.8 is 0.005555555, which we can round it off to 0.0056. So 0.067 multiplied by 0.0883 is 0.005561, which we can round it off, which will be option one. So I see that it is 7 o'clock, 7.30, but we can give you five more minutes just to look at the last one. So with the last one, if we don't do that, you can do it and we can have a discussion on WhatsApp as well, because our time ends at 7.30. A teacher is teaching a class about probabilities. She knows the learners, oh, she shows the learners and Paul, which contains three red marbles, five blue marbles, seven yellow marbles. One of the learner, Lucy, is asked by the teacher to select one marble out of the bowl at random and to hand it to her. She shows the class that Lucy chose a blue marble. And she demonstrates to the class how to calculate the probability of this outcome. While holding onto the first marble, she asks Lucy to select another marble in the same way. She now asks that the class calculate the probability that both the first and the second marble chosen by Lucy happens to be blue. Select the best estimate out of the following options below. The first thing that you need to remember is to always look at what is given to you, right? So you're going to first try and understand the information given and from the information that is given, you need to find which formula you're going to be using based on the information, whether are you given a mutually exclusive, are they independent, and so on. That will help you to determine how you have to calculate the answer to the question. So let's look at this. I'm just going to give you the hint because I don't want to keep you wrong and otherwise we can continue the conversation on water. So since there are five marbles that are blue, but however, they select only one. So the first one and the second one already, if they're already taken out one marble, there are no longer five marbles that are in the O. There are no longer five blue marbles. They have reduced but your sample space will always stay. For the first time you draw, your sample space will be three plus five plus seven. That gives you your sample space. So that will be your sample space. The second time you go in and you select because they selected five marbles, so there were five marbles that went blue out of what is three, five. It's 10, 15 out of 15. That was that. So the second time you go, you no longer have 15 and because it's an end, they are independent. What happened before doesn't influence what is happening now. So they are independent. Then you go into select. There will be 14 in there and there are no longer five. Right. There are only four of them. I don't think also I am calculating it right because it's only selecting one more marble at a time. Okay. Let's see if it's true what I am. Give it to you now. So that will be the probability of selecting the first blue and the second blue marble because they are five blue marble. So any one of them had an equal chance of being selected to be part of this but the sample space is 15 and when they go again, there are no longer because they didn't replace it back. If they did put it back, then you would put it back in the box, in the in the back and then or in the bowl and then your sample space will still remain 15. So you need to be able to know how to answer those. Okay. So that concludes today's session. Are there any questions, comments before I close off and switch off? Are there any questions, comments, theories, anything? If none, then I am going to post on the chat the link to the WhatsApp group and I'm also going to post on the chat the link to the register. Please make sure that you complete the register and leave the session without completing the register. Since there are no questions, comments and queries and we are at the end of the session. Thank you for coming and all the best. See you save time next week. Tuesday. Thank you very much. Just hold for a second as well.