 Michael Jordan is a very famous and successful basketball player, in fact the most successful North American athlete of all time. He holds the NBA record for the highest regular season scoring average and highest playoff scoring average. He came from humble beginnings and to get where he did he needed to be focused and to work hard, but to do this and above all else he needed to be resilient. To be resilient means to be able to withstand or recover quickly from difficult conditions. He once famously said, I've missed more than 9000 shots in my career. I've lost almost 300 games, 26 times I've been trusted to take the game-winning shot and missed. I've failed over and over again in my life, and that is why I succeed. He recognizes that failure is a necessary part of success. Without failing you can't learn, because it's when we're confused, when things go wrong, when we hit a problem and don't know how to fix it, the real, deep and powerful learning begins. To help illustrate this idea, it's useful to think of this as a visual metaphor called the learning pit. First we're faced with a problem. Given a task introduced to a new idea, we may feel apprehensive or worried that we may not be able to do it. It's at this point that we have to leap into the pit. The only way to resolve the problem is to move forward, jump in and have a go. Sometimes this can take guts and determination. When we first face obstacles in a task and we find ourselves at a dead end, we're at the bottom of the pit. You may feel that you don't want to speak out in class with fear of getting the answer wrong or are so confused you don't know how to start. It is at this point that we must move through this sense of fear and begin to try. Once you start applying different approaches or techniques, you begin to learn. You'll make mistakes, but once you have, you'll be able to think about what went wrong and how to fix it. This way you will develop a much deeper understanding of what you're learning. The learning pit is about learning to know what to do when you don't know what to do. When you're finally out of the pit and see what you've achieved, you'll feel an overwhelming sense of achieving the satisfaction. But even more important than this, you will have developed resilience, which will continue to help you long after you've left school. The next time you face a learning problem, go on, jump in. That's what I have been trying to do. Michael Jordan is a very famous... Oh, sorry. That's what I've been trying to get you to understand. That you need to jump in and start doing the work. And at some point when you are struggling, you can also even reach out and ask for help. So on my UNISA, I have posted the same information about the learning pit, but it's a different metaphor of the learning pit, where on there they told you that once you have slide in into the pit, where you are now in the confused mode and you are struggling, you can also reach out to people. Being your peers or your lecture or me as your e-tutor for help so that we can help you understand the concept even much better. So yeah, jump in. Ask if you are stuck. And we're here to help you. That's how you learn. And even this, you can also apply it when you are at your workplaces as well. So you will find that you are stuck with problems, challenges at your workplace. If you don't know how to solve them, ask. So the concept is the same. So yeah, let's do steps. So today we're continuing with the, sorry, I need to, yeah. So we continue with discrete probabilities that we started yesterday. So some of the concept that we used yesterday, we will still continue to use them because they are the building blocks towards the whole module as well. So also the things that we did with the basic probability remember where we understood that probabilities are between zero and one and the sum of all probabilities should be equals two to one, things like that. We will still continue with those concepts. And also remember yesterday we introduced some of the concept where we said some way they can use web phrases for mathematical science. You need to know what at least mean, what at most mean, but at the later stage I'm going to share with you all those things. So today we're doing two concepts, the binomial and Poisson from the discrete probabilities because also they come from a discrete variable. What are we going to learn actually today? So the first hour of today I've planned it out that we do binomial distribution. So we'll do the concepts and then we'll do the activities and then we do a whole lot of other exercises, but all of them are guided. So we will do for that one hour we concentrate on the binomial. Then we take a 10 minutes break because it's a two hour session today. We have to take break because we cannot stay on our screen for that long. And then we come back for another almost an hour. We do the Poisson probabilities also for an hour. What you are required to have with you is the tables. So your tutorial letters, at the back of your tutorial letters, somewhere you should have them ready. I know that if you registered in this semester you might not have the printed study material. But it is on my UNISA, so probably you downloaded it, you stored it somewhere. We are going to use that. So make sure that you have it somewhere ready for use. I will show you how to use both of the tables. It makes it easier unlike calculating with the formula. But sometimes you also need to know the formulas because they might ask you certain things that are related to the formula, but not actually that are related to how you calculate things, but maybe substituting values onto the formula and see if you understand what the formula looks like. And when we do the exercises you will see what I'm talking about as well. And then you need also to have your calculators ready with you. So no argument about that this is meant. We need to always keep on calculating. Please remember the sessions are interactive. When we do the activities I expect you to tell me how you answer the question so that I can see if you're struggling with how you answer this question. Even if you don't understand how to get there, stop me. Let us help you be on the same page with everybody. Don't leave the session with being confused. After today's session we're going to introduce a new table. Every week we're going to be introducing new things, new things, new things. So you need to make sure that you stay on top and you know what we are doing at that point. So when you go write the exam, let's say for example, your exams are structured in a way of your study you need. So you're not going to get confused when you write the exam. You just need to be able to identify key things from your questions that tells you now you're doing a binomial, now you're doing a Poisson. Since from chapter 4 or study unit 4 where we dealt with probabilities, you will be hearing a lot about probabilities, probabilities, probabilities, but you need to know which probability are we referring to? Are we referring to the basic probabilities? Are we referring to the discrete probabilities? Are we referring to binomial probabilities, Poisson probabilities, normal distribution probabilities, sampling distribution probabilities, and also when we get into the confidence intervals and the hypothesis, you need to know all this because all of them we're going to be talking about the same terms, but you need to know where and how to apply those terms to different components. Okay, so let's do what we're supposed to be doing for today only, which is binomial and Poisson. At the beginning of the session, this is one hour, we're looking at the binomial probabilities. So in your module, you also required to know the content because questions in the exam, they can come as content or they can ask you to do some calculation. So you need to know some of the properties and basic knowledge of things. So for example, with the binomial probability, you know that it's from a discrete variable and it's a sequential number of identical trials that are happening. So for example, what we mean by this is if let's say we are in a process to count how many number of interruptions that happens in a day. So there might be different types of let's say we are tossing a coin and we count how many number of times it lands on a head or a head, okay. And if we are three multiple points, therefore it means the process will have three outcomes from there. But the number of times that we toss in that, for example, it might say we tossing it for 20 times. So we do this process, we repeat it 20 times and that is your number of tracks and they have to happen in that sequential way. So usually when we work with binomial probabilities, there are two outcomes because it's a bi-nomial. So the bi tells you that there are two outcomes of this process. So the bi tells you that there will be a success and there will also be a failure. So for example, when you toss a coin, the success, if my success is a chain, the coin needs to land on a tail. If I toss the coin and it lands on a tail, it's a success. If it lands on a head, it's a failure. So there will always be only two outcomes. And the probability of a success where it lands on a head, oh, sorry, it lands on a tail, that probability of success is denoted by the type V, which is that sign that looks like that. We call this the probability of success, the pi. And the probability of failure, it will be one minus this pi. So one minus the pi value will give us, but this is not the pi, the 27 over, the 22 over 7 that you know there. So we just use this to represent the probability of success. And also what we need to know that since both of these outcomes comes from a discrete process, then that process has to be mutually exclusive. You know that a coin cannot have both sides. So a head is a head, a tail is a tail. So it has to be mutually exclusive, it needs to come from a mutually exclusive process. And it should also be a collective exhaustive outcome. So it means both the side of the coin needs to make up the sample space. And when we do this tossing or the rolling of a die or counting the number of outages and so forth, all the events needs to be independent. If they are not independent, we cannot calculate the binomial distribution or we cannot put them as a binomial distribution from events. So with that in mind, let's say in the exam, they ask you a question based on what we just, I just explained. Which one of the following is not the property of a binomial distribution? You have five minutes or let's say two minutes to think about all this based on the information I just said. Remember, let's go back one slide. There should be identical trials which are created in a successive manner. The outcome of the two events, there will always be two outcomes, a success and a failure and the probability of success is denoted by a pi and the probability of failure is denoted by one minus pi. And both of the categories or the outcomes, they need to be mutually exclusive and they also need to be collectively exhaustive. And the event that is happening needs to be an independent event. With that in mind, which of the following is not the property of a binomial distribution? Thanks, Sam. Just a quick question. In the exam, would you ever be asked, do you have to basically select which one would be a success or failure? In other words, it's not necessarily that you are given success and failure, you're actually given two options and you select which one is a success versus failure from your perspective. Talk to me. Which one is not the property of a binomial distribution? So we can go through them one by one. So, ma'am, could I raise the question while you're out? Anyone who wants to try? Number one. Each experiment has an each experiment has an entry. Is it true or false? It is true. That is true because there are entrails and the entrails are independent of each other. True. That's true. That's true. Each trial has two possible outcomes that are mutually exclusive, which is a success and a failure. That's true. True. The probability of success remain the same for all trials. False. That's false. It is false. And the last one, since there are only two outcomes, therefore the probability of success is always going to be a half because of the two outcomes, which is 50-50%. So that is true. Now you know the basic probability, the basic concept of what makes up a binomial distribution. Yeah, I just had a question before as well. Yes. So what I asked was, in this case, you're obviously given success and failure, but when you ask the question, are you necessarily given success and failure or do you select out of your options that one would be the success and the other one would be the failure? Most of the time, they will give you the success. Okay. And then you will know that that is the success and the failure will be the opposite. So it's just the opposite of the other one that is not given will become your failure. So if it is like they give you a positive and you know that the other one is negative, for example? Yes. So they might say 20% of the things arrive on time, that will be your probability of your success. 16% of it. Okay. All right. The 80% that will not be arriving on time. Okay. All right. Understood. Thanks. Okay. So moving forward, there is your exercise. Two minutes. Just look at it and see if you can find out what is the incorrect statement. So yeah, they gave you that suppose that 20% of the profiles on Facebook are ghost profiles. Suppose that they are randomly selected 20 Facebook profiles, check whether or not they are not ghost profiles. So 20% your probability of success, 20 your entrails. See if you are able to find the incorrect statement from there. Okay. Do we have the answer? The answer is four. Why are you saying the answer is four? Because 20 trials is not the probability. That's the number of trials. Okay. Let's go statement by statement. So statement number one. Even the information is this a binomial experiment with possible outcome, ghost profile and not ghost profile. Is it true? The number of trials is 20. True. That's true. The 20 trials are independent of each other. True. That's true because you cannot be involved as well. The probability of success or ghost profile is 20 trials. That's false. The probability of success is 20%. It should have said 20% or not trials, but 20% or 0.20. The probability of failure or not ghost is 0.8, which is the opposite, which is the complement of success, which is true. So number four is your incorrect one. So now let's continue and look at more other characteristics of binomial distribution. For example, with binomial distribution, you are expected as well to calculate the mean, which is your expected value, which is calculated by multiplying your trial, multiplied by the probability of success. Which is easy to calculate because when they give you the probability of success, if we look at the question that we had just now before this, or with this one, the 20 multiplied by 0.20. And you are also able to calculate the variance, which is your n times your probability of success times the probability of failure, which is 1 minus the probability of success. And the standard deviation is the square root of your variance, which is your square root of your n, which is your trial, multiplied by the probability of success, multiplied by 1 minus the probability of success, or multiplied by the probability of failure. So let's look at an example. Let's say I have a student is taking a multiple choice exam. There are four multiple questions with each question having four choices. Yeah, they didn't give me much information if I look at this. So what they are asking you is the following. From the statement that you have, are you able to calculate a probability of success? Then the probability of success is getting at least one of the questions correct, because you cannot get two of them. There cannot be two answers that are correct from a multiple choice question. We always have to have one correct answer. So therefore it means for us to calculate the probability of success, we can say x over n, we know that there are four choices that we're going to select from, but they can only be one answer from there. The four choices is just to help us to calculate the probability of success. And what will be our probability of success? One divided by four. Therefore, one divided by four will give us 0.25. 0.25. That is our probability of success. So we also know that we have four multiple choice questions. So our n is equal to four. So it is easy to calculate the mean. We just multiply, we use the formula and multiply by the probability of success. Our n is four and our probability of success is 25. Probably our probability of success is 25. Therefore it is 0.25 times four, which gives us one is our expected mean. Then to calculate the variance, we also do the same. So our n is four times 0.25 times 1 minus 0.25. So I can start with 1 minus 0.25, 1 minus 0.25 equals 0.75 multiplied by 0.25 equals 0.187 multiplied by four. And my variance is 0.75. And to calculate my variance, I just take the square root of my variance and I'm using a business financial calculator. So I'm going to say second function, three, which gives me my square root. So you will have to go find the square root. And the answer I get is 0.886, which means if I give it to two decimal, so I can take the square root of 0.75, which is the same as the square root of the variance. I've already calculated the variance and that gives me 0.87 if I round it off correctly. Any question? If there are no questions, here is your exercise. Okay, do we have an answer? Are we still busy? Are we still busy? Yes. Remember the mean end times the probability of success? Sorry, I didn't give you the whole statement. Yeah, I was about to ask, is this the only part of the question? Is it something else? It's related to the one that we did previously. Sorry, my bad. It's related to this. Did we do that? No, no, no, no, no. Okay, sorry. My bad. My bad. It's not related to that. Is the ASD, the ASD. I thought I had copied the link, the question correctly. Sorry, I will give you just now the values that you need to use for people with autism. I apologize for that. And there is your detailed information. Let's go back. There is 50% people with autism, spectrum disorder, and six people living with ASD. Is there anything being displayed? Is it not shared? Sorry about that again as well. Thanks, you can see it now. Okay. So I don't know if the full question is there? The full question is there. Autism South Africa has found that 50% of the people with autism spectrum disorder, ASD, struggle with social interaction. Assume we randomly select six people living with ASD. Consider the following statement A to C. Let's say okay. Yes, the expected number. So it means calculate what the expected value is. The variance calculate what the variance is. The standard deviation calculate what is there. Remember they give you the answers here. So they say the expected value is three. The variance is 1.5. The standard deviation is 1.25. Which of the statements above are correct. So you need to choose after you have calculated each one of them choose which one is correct. Are we done? Not yet. It should be easy. If you are done just say done so that I can know that so many of you have completed it. Done. Yes, are you still busy? Other people? Yes, we are still busy. Done. Okay, so we don't have enough time. Guys, you just need to substitute the values and calculate. What are you busy with? Are you done? Done. Thank you. So number one, the expected mean is it correct? Because it's just 50% times six, which is 0.5 times six. What do you get? It's correct. Three. And number two, the variance, six times 50%, one minus 50%, it's the same as 50%. So it will be for the variance. 1.5 times 0.5 times 0.5, which is 1.5. For the standard deviation, you just take the square root of the 1.5. And what do you get? 1.225. 1.225. So now come to the statement, which one is the correct one? 5. It should be 5, because all of the statements A, B, C are correct. Okay, so that is how you come here. The other basic concept when it comes to binomial distribution. So you must pay attention to the way that question is asked, because they will give you 50%, then they will also give you the six. So they will not say the probability of success is this, but they will just give you some hints, and you must always remember that a probability can be between 0 and 1. So in this instance, they will give you the probability as proportions, like in a percentage form. Okay, so now let's look at how we calculate the probability of a binomial distribution. Remember this? We covered it yesterday. So I just wanted to flesh it because you need to also remember, with the binomial distribution as well, they will ask you this kind of questions as well. So you need to know how to calculate at least exactly at most, more than between. You will have to know how to calculate all those probabilities if they give you a weight phrase, you need to know what symbol it refers to. With a binomial distribution, the formula is in this way. The first part of the formula gives you the combination of a binomial distribution, and then the second part of it, it can place the probability of it. So if you are that person who likes to bet lotto, you can take this to determine what are the chances that you're going to win lotto if you bet the numbers, and how many number of foot. So this will tell you, the first part of the binomial distribution will tell you how many number of ways you can bet lotto, and the second part of it will tell you how many or what will be the probability of you winning the lotto, and if you combine the two, they can at least give you the probability of you at least winning lotto. So with this, you're going to use the prior probability that you know, maybe probability will be you bet lotto and 30 times you win the lotto. So that will be your probability of success. So from all the times that you have been betting lotto, 30% of the time you do win some sort of money from the lotto. You can use it as your probability of success, and your probability of failure will be 70% of the time you're not winning lotto. So if you want to know what are the real chances that you are going to win that lotto, then you can take the number of times you bet lotto, the number of times there are outcomes or the outcomes of the lotto, and the number of times you will want to win lotto, and you can calculate your probability. And that will tell you how many or what are your chances of winning that lotto ticket. And that is the binomial probability. So the first part gives you the combination, and the second part gives you the probability. But both of them, when they are combined, they give you the probability of a binomial distribution. So we can use the formula to calculate the probability, or we can use a table. It's easy to use the table, but you need to understand how the table works. It's not a straightforward table. You need to know how it works. So table E6 has two sides to it. It has the top side, which I call it the top side, where it has the smaller probabilities at the top, those are your probabilities of successes. And it has, and it's broken down into the number of trials. So starting from two, so you will have N2 with its corresponding observations that are related to that trial. So within that number of trials, so let's say they did this test, they did it twice in this instance. But when they repeated, or they did it the first time, there were outcomes that came out. There was a zero outcome, one outcome, two outcomes. When they repeat again, because this is a two-time twice trial, or two trials, then when they repeat again, how many number of times they didn't come up with that outcome, or it came with one outcome, or it came with two outcomes. And that is how the table is broken down. So you need to know when you answer the question. For example, with the one that we just used, where we had our N of 6 and our probability of success of 50%. So we know that we had our N of 6 and our probability of success of 0.5. So when you look at the table, the table at the top has all the probability of successes. And it has 50% there, and it has 50% at the bottom as well. So you will have to be very careful in terms of where you want to read the table. But the values will look almost exactly the same. So now, how you read the table. So this is where it comes in. How you will read the table. At the top, there are smaller probabilities. All these values at the top, they correspond to these values on your left. So your N and your X values on your left are read with the values at the top. When you go to the second part of the table, which is the next page, at the bottom, there are complement probability of those ones that are at the top. So at the bottom, there are bigger probabilities here of successes. Therefore, it means all these probabilities at the bottom, you're going to read them with the values that are here on your right. So at the top, with the left, at the bottom, you read with the N and the X on your right. If you look on your right, you will also notice that they have N and they have X and they start from the bottom and they go up. So both tables, they work the same, but you just need to know how to find or identify which table you are using and which probability you are using. For example, let's say we want to find the probability that N is 0.92. So let's say N is 0.92. I'm starting with the complex one, 0.92. But our N, let's say our N is 4. So because the probability there at the top is 0.92, so if I go to the top, I won't find that probability. But if I come here, I will see there is a 4. I should not do that. I must go to the bottom and look for 0.92. So let's remove the N that I have done there. And so we know that we're looking for N of 4 and the probability of success of 0.92. So now we go here at the bottom, we go look for 0.92 here at the bottom. And then we also come here on the side of the table. So the bottom, you won't find that probability on that table there. You will find it on the first table because there the probability of 4 is here. If you look here is your N, N of 4. You see I'm looking on my right, even though there is the left side N. The reason why is because now from the right, the probability starts from the bottom and they go up. From the left, the probability starts from the top and goes down. So your values, your observations. So now we look at N is 4. And let's say we want to find the probability that X is equals to 3. So we want to find the probability that X is equals to 3 from the table. So we just go here and go find the probability that X is equals to 3. That is 3. And now I need to go and find the probability of 0.92 on this table. So knowing for sure that this is the complement. If I know that this is 0.92, then at the top it should say 0.08. So I can come here to 0.08 and highlight this table and also go and highlight the second row. And when they both meet, that will be my probability and that will be 0.2492. And that's how you read the table. So if the probability was 92, I will go to the bottom table, go to the right of the table and go look for the N and the X observations from there. So what happens when it was 0.8? So let's say the probability was 0.08 and N is 4 and X is equals to 3. So what you will do instead of using the left, now you go to the top, we look for 0.08, we come to this side, we look for 4 and we come to this side, we look for 3. And that will be that probability, which is equals to 0.0325. And that probability will be 0.035. And that is how you read the binomial distribution table. Okay, so let's look at more examples. I know that I'm e-ticking up into the time for T, but I will give you the exercises when we take a break. So this is the same question. It asks, what is the probability of one success in five? Observation, if the probability of an event is 0.1. So they want to know your X will be one because they want to know one success and your N is five and your probability of success is 0.1. So if you go to the table, you go look for 0.1 at the top. So 0.1, it will be at the top and you're going to look for N on your left and N is equals to five on your left and X is equals to one. So if I switch on to the... Okay, so I'm going to stop sharing the presentation and share the table just now in the slideshow. So I just want to demonstrate how you go about calculating it using your table. Okay, so this is your... This is the table. It looks like this. So let me make it bigger. I hope you are able to see the table. As you can see that where I am is N is 20. So we need to go to the one where N is. There we go. So this one is the table that we're going to use. So we're looking for N. Sorry, my bad because I have a zoom to use the select key. Okay, so we know that we're looking for N is five and we said X is one and the probability of success is 0.1. So if we go to the top, we look for 0.1 and that is 0.1 at the top for our probability of success and we look for N on the left because we're using the top part N is five and our X is one. So there is our X and where they meet and that will be 0.2380. And if we go back to the table, I just want to switch back to screen so that then I don't have to go back and forth like that. You can use the presentation as it is. Okay, so if we go to the exercise, we can see that we have the same information as we had before, which gave us 0.3280. Any questions? If there are no questions, some of these are part of the slides that you have on my UNICEF. I'm not going to go too much into that detail. I want you to answer this question. Actually, not that one. This one. So let's see if you can answer this question and then we'll come back and let's make it a quarter past one, close to 20 past one and then we can come back and do this question together. So you can also take your five minute break while you're busy when you are done. So see you in almost eight minutes or so. If you have any questions, you can ask. Ma'am, I'm just going to have to be excused from the online session for now. I will watch the recording on my UNICEF later on. No problem. Thank you very much ma'am. Have a great weekend. Looking forward to our next session on Friday. Thank you. Are we all back? Yes, I am. Can I ask a quick question? Yes, you can. The table that you referenced on Tutorial 101. Is that the cumulative probabilities table? Yes, that's the cumulative. No, it's not a cumulative probability table. So it's just your binomial table with ordinary, normal, simple probability. It's not cumulative. Okay. So let's see the questions that they have asked us here. The first one, it says here the probability of all 20 profiles. The probability that all 20 profiles are ghost profiles. Therefore, they are asking me to calculate the probability that X, because they say all of them. So it will be less than or equals to 20 so that it includes all of them. So whether they are 19, 18, 17, 15, to all of them. So what will be that probability? So it means you'll have to add all of the probabilities from the table. So let's see how we do that on the table, just for demonstration papers. So I must go to things I'm not working well on my side now. What are you seeing in front of you? Do you see the presentation? No. It's all of us. Okay. We see it now. Go out of the presentation mode. So I can access the, can you see them? Yes. Are you still seeing, you're seeing the table? Yeah. Okay. Cool. It's a little bit of a delay. So we know that our n is 20 from the question that they asked. And we also know that our probability of success is 0,20. So we're going to look for 0,20 and it is here at the top of the table. Therefore, it means we're going to be using the left side. Since the first part of the table does not have everything, so you can go to the second part and you will see that your second part when you do this on online, it looks very different to it. It's upside down. So you can rotate it. You can do the rotate, rotate, rotate because then it means to, you need to use this side. You cannot read it the way it looks. So I'm going to rotate it. And here it also gives me all the ends and it ends at 10. So I need to go to 20. So it means 20 will be the next slide, the next part of it. So if I go to this side, I must also rotate this so that I can read it. So there is your n is 20. At the top, we know that our 20% and our n is 20 and it says the probability of all of them. So it means the probability, all of them. If you add all of them, they cannot be equals to 0. So if we go back to the presentation, it makes it very difficult now to navigate between the two. So yeah, it said the ghost profile are equals to 0. So it cannot be equals to 0 because for all profile, they have to be equals to 1 because the sum of all probabilities will be equals to 1. So what will be the probability that no profile are closed? Therefore, what they are asking you here is to find the probability that x is equals to 0. So if we go to the table, if we go to the table, we go look for the probability that n is 20, x is 0 and that probability is equals to 0 comma 1, 1, 1, 2. So if we go back to the question, maybe I should not go back to the presentation mode. We can leave it as such. That will be the correct one. So what is the probability that only dating profiles are ghosts? So here they are asking you to calculate that the probability will be, what is the probability that x is equals to dating? Because they say only dating people are ghosts. So if we go to this table, so we look at the probability that x is equals to 13. So if we come here, we look for x is equals to 13. And since this is the last part, and that will be equals to 0. And that is the probability that x is equals to 13. And when we go back here, it says what is the probability that only 12 profiles are ghosts? So it means the probability that x is equals to 12. So we go to the table and we go look for the probability that x probability that x is equals to 12. And the probability that x is equals to 12. If I look at this, it will be that one, 0 comma 0. Let me remove the other writings. x is equals to 12. And that will be the probability that it's 0 comma 0 1. And do you know which one is incorrect? We will really answer that question actually with number one.