 Just waiting for the recording to start. Okay. Welcome to your session nine. Can you believe that that we already have nine sessions? Today, we're going to continue with study unit five. We're going to look at answering questions on all understanding and unpacking binomial and Poisson distribution. I also included some of the things that I think I've left them out. But I think they are very important for you to know, like the counting rules and certain definitions when it comes to the probabilities, but it's not a trance mesh. I also during the past few days, I have been sending you WhatsApp messages to remind you that today, you need to bring statistical tables that we're going to use to answer questions on the binomial and the Poisson distribution. On Saturday, we can look at the table in detail. Today, I just want to lay the basics and give you a feel in terms of how you can answer the questions using those tables, and then on Saturday, we will do it in more detail. If you don't grasp it today, do not worry, do not panic. Saturday, we will do that in more detail when we do activities and exercise, and then you can see how you use those tables. Okay, so by the end of the session today, you should learn about the counting rules, how to calculate them, and how to apply them to answer some of the questions. You should know how to answer questions or be able to describe the properties of a binomial distribution, and also be able to calculate the probabilities of a binomial distribution. Similar, you should be able to know the properties of a Poisson distribution, and also be able to answer the probability questions when it comes to the Poisson, or you should be able to calculate the probabilities of a Poisson distribution. So some of the concepts that I left out, when we were discussing chapter 4, which is basic probabilities, so I went back and I looked at assessing the probabilities, as I thought maybe if we would have discussed this in more detail because we only describe what the probability is. But sometimes with the probabilities, there are different scenarios that you can use to assess the probability of an event that you are calculating. There are three approaches and statistics to assessing the probability of an event. We first have the priori method, which is based on the prior knowledge of the process. So if you already know what the process looks like, you are able to calculate the probability of that. That is what a priori means, and it's the same, you will be calculating that probability based on the number of events that satisfy that outcome divided by the sample space, and that's the other thing that you need to know. We also have what we call the empirical probability, which also tells me that you will just be calculating the probability of an event and that is happening. So to do that, you will use the number of outcomes that satisfies that event divided by the total sample space. Both the priori and the empirical probabilities, when you calculate them, they are what we call classical probabilities. We've calculated them by means of calculating simple probabilities and also calculating joint probabilities. Those, we assume that all outcomes are likely to happen from that event. So it means when tossing a coin, we know that for sure it will either lend on a head or a tail, and that is just a classic, classic probability case that you will have. But we also have another method, which we call a subjective probability. Here, it means as a researcher, you are going to base the probability on your experience, or your past experience that you have, or your personal opinion, or in terms of a situation that happened in the past. But that is subjective to you as a researcher. It's not subjective. Another researcher might use their own experience, which will be different to your experience. Whereas with the priori, everybody would have experienced it the same way because that thing will happen in the same manner. Like the sun will come out, we all go into experience it in the same manner. Everybody will experience that the sun will come out or will see the sun coming out. And that's how you will differentiate between a priori and a subjective as well. And those are the things that you need to remember when you do the basic probabilities as well. So in terms of today's session, we're going to start off with counting rules. So what counting rules does help us in a way to determine the number of ways of doing something. And those are the counting rules. So they give us the number of possible ways of doing things. So the first rule of counting rule is the power rule. And this rule says if any one of the key different event is mutually exclusive and it's also collectively exhaustive event, and that event can occur in each or in any of the end events or end trials that are happening, then we should be able to calculate the number of possible outcomes that happens from there. So for example, let's say we have a die and we roll the die three times. We know that a die has a number of outcomes. We know that the die has six outcomes that can happen and it can fall on a one, two, three, four, five, six. So there are six ways that a die can have. So since we know that it has six sides and when we roll it three times, then the possible ways that any of the number can appear will be six to the power of three. So our K will represent the number, the different event that can happen and three will determine, oh, sorry. K will determine the outcome and N will determine the number of events that will happen. So we know with this example, our number of events, we are creating three events because we go into roll a die three times. So there will be three events. But we know that with our K, which are our possible outcomes, we know that there are six outcomes that can come from a die, from one die. So therefore it will be six to the power of three and that will give us 216 possible outcomes that can happen from that die. And that is counting rule number one. Counting rule number two, it's a multiplication rule. With counting rule number three, it says if there are K events, the K events, if I have K events and I divide those K events in terms of the number of ways that I'm going to do those events. So if I have the first event or the first trial, I will name it K one, then the second trial will be K two and the third trial will be K three up until K equals to N because there are possible outcomes of N from those events. So to calculate the number of possible ways I can do several things in a K way, then it will mean I must multiply all the events together. So it will be K multiplied by K, multiplied by K, multiplied by KN and that will be a multiplication rule. So let's say for example, if you want to go to the park, eat at the restaurant and see a movie. So I have one event going to the park, second event visiting a restaurant, third event seeing a movie. So those are my K events. My trial one is going to the park, trial two is the restaurant, trial three will be seeing a movie. So there are three parks, three, four restaurants and so let's say there are three parks in the area, four restaurants in that area and six movies that I can choose from. If I want to know how many combinations I can do in terms of visiting the park, the restaurants and the movies. So therefore it means I'm going to say three, multiplied by four, multiplied by six and that will give me the number of possible ways I can go to the park, the restaurant, the movie. So there are 72 different ways to go to all those, to the three parks, four restaurants and six movies. Fact rule, it's called the factorial rule. So with the factorial rule, it says we should be able to calculate the number of ways we can place certain things or arrange certain things in order. So if I'm running in a race, I know that a race has position one up until position seven if there are seven lanes. So how many number of ways can I be in any of the position? And that is what the counting rule is. And we use the factorial on your calculator on Saturday, we're going to look at the factorial on your calculator. Some calculators have it as an inventory and those who have the sharp calculator, you will notice that it is somewhere on the button number four and it's written in orange. Some people who has a business and financial calculator, it will be on an AMRT, it's written in orange. Those who are using the cashier, I think on your one, you should look for an X factorial. It should look like an X instead of an N, but on Saturday, we will work through the calculator and I will show you how it looks. Sorry, my pen is not writing. So on some calculator, it's an X factorial. What does mean is it says from the first, the total sample, which is if I have seven, it will say seven and the next one will be seven minus one, which is six, the next one will be six minus one, which is five, the next one like that and it will go on until you get to one. So let's look at this example. If you have five books that you need to put on a bookshelf, how many different ways can these books be placed on the shelf? So we want to know whether can we put them on the top, the bottom, how many number of ways can we put those five books? One on top of the other or something like that, but we have five books that we can put on a bookshelf and to do that, we can calculate it by saying five times four, times three, times two, times one, and it will give us 120 possible ways of placing a book on a bookshelf. And that is factorial rule. What's the rule? It's called permutation. With permutation, there is an order in terms of how you arrange things or how you do things. So permutation is the number of ways of arranging an object selected from an N object in order and we use the formula. NPX, which is equals to N factorial divided by N minus X factorial. So N will be the total number of objects and your X will be how many number of ways you will want to do those or number of outcomes that you want to place those objects or arrange those objects. So since here you need an order, it means there will have or you will have to have defined the actual number or shown a preference in terms of the number. You have five books and you are going to put three on the shelf. How many ways can the books be ordered on the shelf? So we know that we have five, which is our N and we have three books, which is our X. To place those books on a shelf, so we use the N factorial or we can use on your calculators, you do have that function NPX. So some calculators, you can look at it. It is written in orange and it is NPR or NPX. In order to answer this question on your calculator, you will need to press the value of N, which is five, and then press second function or shift depending on your calculator. So the sharp calculators will press second function. The cashier will press shift and then you will press the button where the function NPR is at and then you press three and then you press equal. And you can tell me how much do you get? Five, second function. 60. NPR, five, oh three. And the answer is 60. So you will place those different items. 60 different possibilities of placing those five items on the bookshelves. The last rule is combination. So with combination, order does not matter with this. So we just want to know the number of ways of selecting X object from an object irrespective of the order and we use NPR. Similar to what we had with NPR, you should look at your calculator, you should have NCR. So let's look at an example. How many five books are you going to select? Three, you have five books and you are going to select three to read. How many different combinations are they ignoring the order in which they are selected? And that is a combination. So on your calculator, similar. This is your N and three will be your X. So you will start with the five and then you will press the second function or you will press the shift for Casio. Then you will press NCR button and you will press three and press equal. And how many number of ways? 10 ways. And you will find that there are 10 different ways that you will place or you're going to read those books. As you can see, the same question, but asked differently. So we still have five books, three to be placed on a bookshelf. We get 60 different ways because there is an order in terms of how those books needs to be arranged on the bookshelf. With the second one, it says you have five books, three to read. How many different combinations ignoring the order? And when we ignore order, we use combination and the answer is 10 different ways. And that is combination, permutation or also counting rules in general. On Saturday, we can do more exercises or activities relating to this. I just wanted to highlight the counting rules. And with the counting rules, the combination we're going to use today. That is why I wanted to introduce it today, not when we did the study unit four, because when we calculate the binomial probability, the function or the formula for the binomial distribution to calculate the probability of a binomial distribution, it uses the combination as part of the formula and then it also uses the probabilities as well. But we will look at the formula later on. Oh, now let's talk about binomial distribution. A binomial distribution comes from a sequential number of identical trials, which are your end sample trials that you will have. For example, when you toss your coin 15 times, you are creating a binomial distribution because in any way, when you do that, a binomial distribution by its nature, it says by. So therefore it means there are two outcomes. Therefore there is a success or there is a failure. In terms of a binomial distribution. When we look at the probability of a success, we denote it by using the symbol pi symbol and the probability of failure, which will be the complement of a success is one minus the probability of success. So your probability of failure is a complement of probability of success, which is one minus the probability of success. And also what you need to remember and know about binomial distribution is that since there are two outcomes, those categories or events that you are creating needs to be or the outcome that comes, they ought to be mutually exclusive and also they also have to be collectively exhaustive. So they should include part of the whole sample size. And the events have to be independent. So one cannot affect the other event while they are happening. So events should be independent. They have to be mutually exclusive and collectively exhaustive. So it means they must make up the entire sample space combined. Then you have the probability of success, which is pi and the probability of failure, which is a complement of the probability of success, which is one minus the probability of success. And because we know that there are two outcomes and the event needs to happen in a sequential form. Okay, so those are the properties of a binomial distribution. With a binomial distribution, like with discrete probabilities that we did, we are able to calculate the mean, the standard deviation and the variance of, oh, sorry, I had some exercises in between. We are able to calculate the mean, the median, oh, sorry, the mean, the standard deviation and the variance of a binomial distribution. And calculating the mean of a binomial distribution, we use mu, or we can refer to it as the expected value, or ex, and it is the same as your n, which is your sample space, multiplied by the probability of success, which is your pi. So to calculate the mean, you say n multiplied by the pi, where n is your sample size, pi is your probability of success or the probability of an event of interest, where calculating the variance, which is sigma squared, we use n times pi times one minus pi, so it is n times the probability of success times the probability of failure. And that will give you the variance. The standard deviation is the square root of your variance. So you just take the square root of your variance, we'll give you your standard deviation. To my questions now, which one of the following is not the property of a binomial distribution? Each experiment has n trials. That is a property of a binomial distribution because we said it has identical trials. So it has your n trials. The trials are independent of each other, that's true. So that is correct, that is correct. Why? Because events have to be independent. Each trial has two possible outcomes that are mutually exclusive, which is success and failure. That is true because we said there are two outcomes and it can be an outcome of a success or a failure and those needs to be mutually exclusive. The probability of success remain the same for all trials. The probability of success, it says, it remains the same for all trials. It cannot remain the same for all trials because if for example, I have, if for example, I have a multiple choice question and it has one answer, which is correct and the other one is incorrect. So let's say there are five options here. One of them should be correct. So this will be one over five, which is equals to one divided by five. So my probability of success for this trial will be one divided by five, which is equals to zero comma two zero. I can have a coin which the probability of success for that coin will be one over two because there are two outcome and that will be zero comma five. So those are not the same for all trials. So the probability of success does not remain the same for all the trials. Only for the one trial that you are busy waking with. They will remain the same for that, only for that trial, but for the other trials it will not be the same. Number five, the probability of success is always, the probability of success is always half because there are two outcomes. Yes, because there is a failure and any success. So there are two outcomes. It's one over two for the probability of success. Okay, let's look at another example. So remember with probabilities as well, they can give it to you in terms of a percentage or they can give it to you in terms of a decibel. So a 20% is the same as 0.20. Africa check found that source of fake news on Facebook are mostly ghost profile. Suppose that 20% or 0.2 of the profiles on Facebook are ghost profile. Suppose further that randomly selected 20 Facebook profiles and check whether, oh, suppose further that we randomly select 20 Facebook profiles and check whether or not they are ghost profile. So therefore this is our N and this is our probability of success, which is our pie, which is the probability of ghost profile. Which one of the following statement is incorrect? The given information describes, just give me a sec, hold on for me. I am so sorry about that. I needed to go switch off the radio. Okay, so now we have, are you able to hear me? Loud and clear. Yes. So now we have the probability of success and our N. We can go ahead and answer the question, which one of the following statement is incorrect? The given information describes a binomial experience, experiment with possible outcome, ghost profile and not ghost profile. Because there are two outcomes, ghost profile and not ghost profile, therefore that is correct. It does describe a binomial experiment. What is the number of trials? 20. The number of trials is 20. The 20 trials are independent of each other. Can a profile be a ghost profile? Yes, independent of each other. They are independent of each other. The probability of success or ghost profile is 20 trials. No, no, no. That's not because the probability of ghost profile is 0.20. The probability of failure or not ghost profile is 0.8. That's true. That is correct, because the probability of ghost profile is 0.20. The complement of it will be not ghost profile. Okay. So we've learned the properties, we've learned the characteristics. So let's apply the characteristics and calculate the mean, the variance and the standard deviation. A student is taking a multiple choice exam. There are four multiple choice questions with each question having four choices. So when a student write the multiple choice question, we know that there are four multiple choice questions. That is, don't get confused. That four multiple choice questions are our end because there is question one, question two, question three, question four. But all these questions have four choices within them. Only one of them is correct in a multiple choice question. So therefore it means we can calculate the probability of success, which is x divided by n of the outcome. So only one can be correct and the rest won't be correct divided by four and that will give us what is one divided by four, 0.25. And that is our probability of success. Find the mean, let's see if you can answer that. Calculate the mean, you have your n and you have your pi. So just calculate the mean. One. Do we all agree? The same one. Okay, so let's do that. So n is four and pi is 0.25. Four times 0.25 is equals to one, that is the mean. Calculate the variance, remember you can use the check to post your answers. 0.75. 0.75. 0.75, so therefore it is four times 0.25. 0.25 times one minus 0.25. Four times 0.25, it's one times one minus 0.25 is 0.75. So one times 0.75 will be 0.75. Calculate the standard deviation. We know that the standard deviation is the square root of your variance. So it will be the square root of 0.75. And the standard deviation is 0.8866. I see they have it at four decimals. Let me also put it for decimals. And that's how you will find the mean, the standard deviation and the variance of a binomial distribution. Any question? I'm good so far. Okay, so let's look at how we find, I'm not gonna ask you to calculate this because we just did the example together. Now we're going to look at how we calculate the probability of a binomial distribution. Remember the table? Also yeah, in the binomial distribution questions, they will be asking you sometimes questions in a weight phrase or in a symbol format already. You need to just know what do they mean in terms of the weight phrase or the symbol. So I just put this there just to remind you that you still need to remember this because it's still applicable even today. Calculating the binomial distribution, we use the formula, which says the probability of a normal of a binomial distribution is given by your combination, which is n factorial divided by x factorial times n minus x factorial. So remember we did this when we were looking at the combination. So the probability of success to the power x times one minus the probability of success to the power n minus x. So if for example, you play lotto and you want to know what are your chances of winning lotto, you can first find your chances or how many number of ways you can bet a lotto ticket and then use the binomial distribution formula to calculate the chance of you winning lotto. If you know what the chances are in terms of the number, if you want to win a jackpot, so let's say the chances are for winning a jackpot is 0.30, so it's 30%. That's the chance of winning a jackpot. That will be my probability of success. And if I know how many number of balls I need or number of, what is that on the lotto thingy? 49, 49 numbers. I'm just gonna use the 49 because I think the last time I remember lotto was 49 numbers that I can choose from. And x will be the number that I can bet. So I know that for a board I need to choose, is it six numbers? Those who bet lotto, you know those things. So my x will be six. So now I know that the probability of winning a jackpot is 0.30, I know that I need 49 numbers and I need to bet six numbers to win. What are the chances? So you can calculate this probability and see if you stand a chance of winning a lotto. What will be your probability of you winning lotto? You can calculate that or winning Powerball or something like that. So you can use the binomial distribution. So let's look at an example, the actual example. Oh, before we look at the example, so we used, you can use the formula to calculate the probability. The chances are when you use the formula, if for example, they're asking you to calculate the probability of x is equals to two, it's easy to calculate that probability because you just substitute the whole formula. So you can say ncr pi to the power x, one minus pi to the power n minus x, that will give you the probability of x is equals to two. The challenge is if they give you less than or equals to two. When they give you less than or equals to two, then remember that the probability of x less than or equals to two, it means it will be the probability of x is equals to zero plus the probability of x is equals to one plus the probability that x is equals to two. In the exam, it's going to take you forever to do this formula three times because you need to do it for x is zero, where you place the value of x is equals to zero, you will have to do the same, repeat it for where x is equals to one, repeat the whole formula again for where x is equals to two. If you rely on using the formula to answer your probability question. The shortcut lying on the table. So table already they calculated all the possible values of the probabilities using this formula and completed this table. So we can use the table. Now you need to know how to use the table properly. With a binomial distribution, this table is six on this past exam paper that I got this table from. It has two pages. There is the first page and there is the second page because with the binomial distribution, there are probabilities that start from zero comma zero one at the top and then on the second page, the probabilities are at the bottom. They start from zero comma nine nine and they go to zero comma five zero. And since we also know that the sum of all probabilities equals to one, so that is why on zero point zero zero one at the bottom of zero comma zero zero one, it's zero comma zero, sorry, zero comma zero one, it's zero comma nine nine because zero comma nine nine plus zero comma one will give us the probability of equals to one. But that is not of importance. What is important about using the table is the following. For where the values are small of the probability of success, you can see that the table is broken down by the probability of success and all the probability of success if they are small, then we use the n value on the left. We use the left side and it's also broken down by the x value on the left. So our table is broken down by n values and all the n values have the x values, the corresponding x values. Therefore it means if I select the probability of success of n, n is equal to zero, sorry the probability of success of pi of zero comma zero seven and I select, I want to find the way n is equals to two and where x is equals to, let's say two. So if I want to calculate that probability, probability of x is equals to two, therefore I must come to the top table where the probabilities are small and locate my probability of success of zero comma zero seven. Then I must go to the left and locate my n of two to the left. So that is my n of two. Then I also need to locate my x because it says x is equals to two and I must go to where x is equals to two and I will find my probability and that probability will be that and that will be my probability of zero comma zero four nine. Four nine. What if I need to find the probability of the probability of x less than or equals to two? Similar to what we did. Yeah, we wanted to find the probability of x less than or equals to two. You will do the same. That will be the probability that x is equals to zero plus the probability that x is equals to one plus the probability that x is equals to two. Then it means I need to add all these probabilities from two until zero. Two until zero, there are one, two, three. So I must add all three of them. So that will be zero comma eight, six, four, nine plus zero comma one, three, zero, two plus zero comma zero, zero, four, nine and that will give me my probability of less than two. Less than or equals to two. And that's how you use the table. Instead of using the formula. So with the formula, you would have substituted for zero days, zero days, zero days, zero days for that one. Then you go and do the second one, substitute the value one day, one day, one day and one day for that one. Then solve the next one and it will take you forever to do that instead of using the table. So now we covered the top part of the table. What about the bottom part? So on this probability table also at the top, there are probabilities at the bottom because at the top there is zero comma one. So at the bottom here, there are probabilities but they are not written the same way on the second page. At the top, the probabilities they are not written but they only written at the bottom. But doesn't stop you because at the first page will have n up until nine or up until eight because it could only fit only eight blockies of the n's at the top and then at the bottom it can fit also the rest of the other. But this table continues, there are multiple. So it goes up to n, I think n20 or n30, I'm not sure. But there are multiple tables. It's not like only two pages. There are three pages or four pages of the biomeal distribution. All I just want to make sure that you understand that. Even though there are probabilities at the top, also at the bottom of the same table, there are probabilities. Now I'm saying here there is zero comma nine, probability zero comma nine, eight and so forth until zero comma five, zero there. Now, why I'm raising this is because for all the probabilities at the bottom page, they correspond to the n on the right. So all these probabilities here at the bottom like those ones there. So here there is probabilities at the bottom of this page and here there are probabilities at the top of this page but I'm referring to the bottom one now. So all the probabilities at the bottom, which means the bigger probabilities at the bottom. All of them, you're going to use the n. So there is an n, you will notice that also there is an n on the right. And the value starts from n zero as you can see here. So even though this is n is 10, so here it says n is two. Remember this side n is two, but it starts with zero there. On the opposite side, the zero will start at the bottom because we're using the values at the bottom. So it will start at the bottom and say zero, one, two and it goes up like that. Whereas with this one, they go normal down. On this one, they start at the bottom, they go up but we use them with these probabilities here at the bottom. So if for example, our question was find the same question but in state of 0.07, they say we must find the probability of 0.65. Let's say it's 0.65, n is equals to two and x is equals to two. So we need to go to the bottom to 0.65. We know that 0.35 minus one or one minus 0.35 will give us 0.65 at the bottom. So I'm going to take 0.65 at the bottom and go to the right and go up to where n is two and then go look for x is two that and go, and that is as you can see that is different to that one so the answer there will be 0.425. Similar, if I need to find the probability of greater than, since I'm using the bigger probabilities, 0.65, I go to the bottom, go there to the right, look for two, find all of them, zero, one, two, add all the probabilities together and that's how you use the table to answer the probabilities. Any question? Before we do an exercise. I have a question. Yes. So when you use greater than, do you use the bottom table and when it's less than, it's the top, okay? When you look to determine which table, remember both tables, they're not, oh, sorry. How do I put it on the same table that you have? So I'm saying the table has multiple pages. Yes. On the first page, the bottom probabilities are not written. They are written only on the second page at the bottom, okay? But there are one page, like they make, when they did the page, let's say they folded the page so the other part went into the other side and the other one went to the other side. Okay. So when the questions say, what is the probability of greater than, irregardless of which side of the table you use or which side of the probability you use? So let me erase all this. So let's say they say the probability that X is greater than, let's say two again. I'm just gonna use two. Okay. Probability of X greater than two, but they would have told you what the probability of success is. That is the most important thing to remember. This, this is important. This determine where are you going to look for your N? Where are you going to look for your X? That tells you. So if this was 0.25, then we use the top part and the left part. So we'll use top and left, irregardless of which table. So remember the top, let's say we want N is 10. N is equals to 10. On the first part of the table, you won't find N is equals to 10, but you will find it on here. But at the top of this table here, there are no probabilities written like 0.01, 0.02. Maybe I should be demonstrating there with, let's demonstrate with the table. So that you don't see it in the PowerPoint presentation. Let's do it that way. I will have to go and open a past example. Or a tutorial, let's study guide or study guide. Let me see which one I can find quicker. I found something here. Let me share my screen. I'm gonna share my entire screen for now. Okay, so these are your tables. So I found this from last year's tutorial letter 101 anyway. Okay, okay, so this is the normal distribution table. We're gonna look at this later on in the next sessions. We, you're going to look for the table. They are always written there, table E, 3, E, 5, E, 6. They all have their own meanings. We're going to use all these tables later on. So we need to look for the one that says table E, 6. There we go, table E, 6. I just need to rotate it, rotate it again and again for the last time. So this is the table, it looks like this. And this is how you will see it. So I took a screenshot of only that top part. So on the first page, this is the probability. As you can see right here at the bottom, there are no other probabilities. But yeah, you need to know that this is 0.99 here at the bottom, 0.98 up to 0,50 because all these probabilities at the bottom, they correspond to those. All this at the top, they correspond to that. So the most important thing is this. The probability of success. If it's 0.25, we use the top table, the left. So top and left, if it's 0.25. If it is 0.65, we use bottom and right. So we'll use the bottom probabilities and the left hand side. So you will go and look for 65. So I'm going to say 0.35 will be 65 there. You will need to remember all that. So that you know that this is the column that I need to be when I calculate that probability and I go look for the probabilities. And I'm going to use my n from this side and my x from this. Do not use say 0, 1, 2, 3 by reading it from. Then you will write the answers incorrectly. Okay, so on Saturday, when we do lots of exercises, you will learn how to do that. I will look for more exercises that make you to look at those table like this. Okay, so the other part of the table. Yes, sorry. Sorry, before you go on, I wanted to check. Is it possible that you can be given who x, where there is no value? No. This dash, there's no way. Okay, that's the answer. So where it's, it means this is 0, this is 0, this is 0. All of them is just that because the last one is 0. There's no need for them to write 0, 0, 0 again and again and again and again. Okay. Because if this is 0, then the following numbers are also 0. Okay, no, no, thanks a lot. Yeah, okay, so when you use the table, you will notice, because I rotated the table, you will notice when I moved down. So you need to also be careful when you move down. Your table is rotated as well at the bottom, right? So you'll just need to rotate it so that the values here at the top, you can see that these are big numbers and you can not read the table like this. It's upside down. So you just rotate it back to the original and there is your table. So there are the probabilities at the bottom, but you will notice that when they give you the probability, when they said the probability is 0, 07 on this table and they said N is nine. You don't have the probabilities there at the top. So you can just use the ones at the bottom and look for 93, 0.93, which is that one. And go to nine. So it means I'm looking at this column and then go to nine. If they said the probability that X is equals to three, then you just go where X is three. So nine is my N and three and you just go there and that will be your probability. You just read it like that. Okay, so that is binomial distribution. So let's look at one more example and not how we do that on the table. So like I said, if you need to calculate the probability using the formula, you just need to know how to use the formula. And that is if our probability of success is 0.1, our X is one and N is five. What is the probability that X is one? We calculate that by replacing N with the five, X with one, N five, X one. Probability of success is 0.1 to the power of one. One minus 0.1 to the power of five minus one. And you calculate the rest of the formula and then you will get, so if you do the combination, you will get five as the answer and then you do the power of one is 0.1. One minus 0.1 to the power of five. It's five, it's four, so it will be one minus 0.9 will be 0.1 will be 0.9 to the power of four and solve everything and you will see that the probability is 0.32805. So X is one, let's go and find that on the table. N is equals to five and the probability of success is 0.1, let's go to the table. Three, two, eight, oh five is the answer that we got from doing the calculation. I'm going to discard and go to the table. So we're looking for 0.1, so therefore it means it's going to be at the top and we're looking for N of five, so it means it's going to be at the top. So we go to the top and since my table is rotated, I need to fix it, rotate. Okay, let me remove my ink from my previous, okay. Now we're going to look for is five, probability of success is 0.1 and we need to find the probability that X is equals to one. How do we find that? We look for, make it bigger, we look for the probability of success first, which is 0.1, so it's at the top. So since it's at the top, it means I must look on the left hand side, go look for N of five, and I must go look for X is equals to one. So since I don't want to remember which column I'm using, I'm just going to draw a straight line there. X is one and I'm just going to follow the rule. And if you look at that, it is 0.3280081 in terms of the table because they left it at four decimals. They have it as 0.32800 and it's the same as what we just calculated with the long method. So find the one you want to use, but I will suggest that you use the table than using the formula. So let's look at more examples. So let's say we going to look at this one. So we know that our N is equals to 10. Our probability of success is equals to 0.35. I'm also going to write 0.75 because we're gonna go to the table. And our X is equals to three and our X is equals to eight for the other one. So you must remember the probability when we use the table. I'm not gonna use this one, I'll use the one that we have. So I just wrote the values so that we can use it on there. Remember that they have it as 0.2522 and for eight is 0.2816. So let's go do that. Let's go solve those. The first one we need to look for. It says we need to look for N is equals to 10. If before I look for N is equals to 10, I need to look at the table first. So the pi. So my pi is 0.35. So it means I'm looking at the values at the top and my N is equals to 10. So I need to look for N10. So since it's on the other page, I need to scroll to the other page and I need to rotate that page. So it's easy when you work with a hard copy because then you can just rotate the page. Okay, so remember we're still working with the values at the top. So since I don't have any probabilities at the top, I must get my probabilities here at the bottom. So I'm working with 0.35. So it means I'm working with 0.65. So this is the column that I need to be at. It means here it's 0.35. So we know that we're looking for N is equals to 10 and the probability that X is equals to three. So we go N is 10, X is three. Then I can just go down, down, down, down, down, down. They wrote. And that is the probability that we're looking for and that probability is equals to zero comma 2522. It's the same as what they had on their own table there. So let's look at the next one. The next one set. So with this one, it was pi is equals to zero comma 35. The next one says we need to find pi of zero comma 75 N of 10. We still use N of 10. And we need to find the probability that X is equals to eight. So based on the probability, it's zero comma 75. So it means we need to go to the bottom. Zero comma 75 is that one. So let's, don't want to remove everything. Oh gosh, it removed everything. So let's write that N is 10, pi is zero comma 75. And we need to find the probability that X is equals to eight. So go to the bottom, zero comma 75. That is the column I'm looking for. Then we go to the right and look for N of eight. That is N of eight. And we need to find, oh sorry, N of 10, not eight, 10. N of 10, N is 10. And we need the probability of X is equals to eight. So we need to find eight. So there is eight and zero comma, and that is zero comma 2816. That is zero comma 2816. And that's how you will find the binomial distribution probabilities. Any question? I'm not gonna ask you to do this exercise. We will do it on Saturday when we do the rest of the other exercise, but you can go and practice and see if you can get it right. We will do the same exercise again on Saturday. So that concludes binomial distribution. So let's look at Poisson distribution. In the next 40 minutes that is left, Poisson distribution uses a Poisson distribution uses the average of events. And we use it when we want to find the number of times an event or case in a given area of an opportunity. And that's where we use Poisson distribution. If you wouldn't notice, sometimes if you are driving, sometimes at the robots, you might realize that there are people sitting there having some papers, writing things, wearing some survey things, or orange overalls or something like that. But they are writing when cars passes by or monitoring the road of some sort. But the recording in terms of the tracks or cars that passes per hour or something like that, they do research on that. And those, because they research on the number of times an event or case in that road. When they do that research, that research usually follows a Poisson distribution because it uses also interval times because they don't do it continuously all the time. They do it, they stop, they do it, they stop. But sometimes they do it continuously so they can sit there from eight until nine and record the number of cars that passes the road. Or sometimes if you have a shop and you want to learn about the food traffic, like how many number of people do you get into your store on certain days or after or during a promotion or something like that, you can hire mystery shopper people that's come and look around in your shop and record the number of people that are in a shop at a given time. And that process, when you do it, you can use a Poisson distribution tool, calculate the probabilities of that process that you did. You just undertook. So how do we calculate this Poisson distributions? Like with the, like with binomial distribution or discrete distribution as well, we can calculate the mean, the variance and the standard deviation of a Poisson distribution. Now, the mean of a Poisson distribution, it's also called lambda. The variance of a Poisson distribution is the same as the mean of a Poisson distribution. It's the same as the lambda, is the same as the expected value. I hope you hear me. Your mean is the same as your expected value, is the same as your variance, and is the same as your lambda. All of them are equal, are the same. They are all mean and the variance. They are equal. The standard deviation, however, is the square root of your mean, your variance, your lambda of a Poisson distribution. And your lambda also, it's called expected number of events, which is your mean or your variance. Any question? There are no questions. Then let's calculate the mean, the variance and the standard deviation of this question. A local police station receives, on average, 3.5 emergency calls per hour. This calls follows a Poisson distribution. What is the mean? 3.5. The mean is 3.5. What is the variance? What is the variance? 3.5. 3.5 as well. 3.5. What is the standard deviation? Is your square root of your variance? Or the square root of your mean? Which is equals to? 1.8708. Which is equal to 1.8708. And that is Poisson distribution. Now let's look at how we calculate the probability. I'm not gonna ask you to calculate that. We can do it on Saturday. So with the Poisson distribution, we use the formula, the exponent to the power of your negative mean times your mean to the exponent of X, which are your events, number of events divided by the number of events factorial. Okay. And that is Poisson distribution formula. So if they say, what is the probability that, I'm gonna say two events happen in that police station where an average, they said what average of a local police receive on average 3.5 away on average is 3.5. What is the probability that X is equals to two? What is the probability that at least they receive two people? To calculate that, because you only need X and lambda, you will say the probability of X is equals to two, is equals to E to the power minus our lambda is 3.5 times 3.5 to the power X is two, 3.5 to the power of two, divided by two factorial, which is E to the power negative 3.5. 3.5 squared. 12.25. 12.25. Two factorial, it's two second function and where you found the n factorial or n factorial equal, which is equals to two. This you can also write it as 12.25 E to the power minus 3.5 divided by two. You can write it like that. And you can also solve the whole thing. Where do I find the E? On your calculator, you do have an E to the power X or something like that. And it should be written in orange. So look for E to the power of X function. That is the one that you need. And you all know where to find the negative. This is not a minus, it's the negative, the plus or minus value. So you're going to press the second function and press where you find that E thingy. So second function or shift and find the button that says E to the power X, you will see on your calculator it will have an E with the copy. And then you're going to press minus, which is the negative of the plus or minus. And you say 3.5 and you say E plus. Okay. Guys, can you please mute your your mic your mics when you join the class and make sure that you don't play any radio or any recordings or songs close by because when I publish the recordings to YouTube when there is a song playing in the background it will not publish because it's a copyright the recording will be copyright and in order for me to remove that it means I need to scrap out part of the video out so please please I beg you I beg you always check when your mics are on and mute yourselves okay I was explaining how you calculate the e to the power so you go find second function the button that corresponds to e to the power of x and you press the plus or minus sign and then you press 3.5 and you say equal and that will give you 0 comma 0 302 and then you can multiply by 12 point 12 point two five and say equal and say divide by two and that will give you the probability of zero comma one eight five zero reason why I am doing this it's because last year in some of the exam papers or some of the tutorial one one questions they gave them the answer in an unsolved manner they didn't want you to calculate the probability but they just wanted to see if you understand how to calculate second things using the but poison distribution so maybe the answer is just looking for that answer or that answer or just that answer depending on what the lecture is looking for so just be wary of that as well so that is poison distribution like with any like with any probability so let's say for example let me clear all the ink you want to calculate the probability that x is greater than or equals to two where depending because then you will need to do x is equals to two plus the probability x is equals to three plus the probability x is equals to four plus until wherever the probabilities are at therefore it means you will have to repeat this formula for two repeat the same formula for three repeat this for four repeat for all the values so in order for you to save time in the exam as well you can use a table like with binomial distribution we do have a table so let's say we want to find the probability that x is equals to two where the lambda or the average is 0.5 so we go to the table I'm going to show you the table just now you can go to the table and look for x before you look for x you go to look for the lambda because it's very important to look for the lambda on the table because your table is broken down by lambdas not the same as with the binomial this one is broken down by the lambdas so you need to find the right lambda and once you have this the lambda then you can answer the question based on the x-vein because for every lambda table the number of x's are different of x events are different I will show you the table so we look for the lambda which is 0.5 and then we look for x it says x is equals to 2 where they both meet my cycle is on the wrong plot so this is 0 comma 758 not not this one I'll fix this I will repost the correct papers okay all you can use the formula and substitute we know that our lambda is 0.5 our x is 2 we substitute into the formula and you will calculate and find the same answer so let's look at the the table how it looks this catch this okay so go into the table you need to look for poison table so this is still binomial and there is our poison so I need to rotate okay so like I said the tables are divided by the lambdas so if you look at if you look at the table the poison table is table e7 you will notice that this first table starts at 0.1 and ends at 1 and the number of x's starts at 0 up until 7 if we move on to the lambdas with the lambda table like this one starts with 1.1 ends with 2.0 you can see that the number of x's expands so we have more two more additional x values it will look like that for all of them so for every table the x values will become bigger and bigger and more and more and more so you need to look at that before you answer the question so let's say for example our average is 3.5 and we need to find the probability that x is greater than or equals to 2 remember now so you need to go to where lambda is 3.5 so this is 3.0 so I need to go to the next table okay I will rewrite all that again now just now so this will be the table our average is 3.5 and we want to find the probability that x is greater than or equals to 2 if you look at this table it's gonna take you forever because we know that this will be the probability that x is equals to 2 plus the probability that x is equals to 3 plus the probability that x is equals to 4 plus the probability x is equals to 14 so it means you will need to add all those probabilities for this question so all of this you will need to add all of them in order to answer this question it's a long long long long one alternatively you can find the complement which the probability of x greater than the probability of x greater than 2 is the same as 1 minus the complement which is the probability that x is less than 2 because I don't have to include 2 because on this one it includes 2 so the other one doesn't have to include 2 so this will be 1 minus the other probabilities which are just the probability of x is equals to 0 plus the probability that x is equals to 1 and that will be 0 comma 0 302 oh sorry I forgot minus 1 so this will be 1 minus 0 comma 0 302 plus 0 comma 1 0 5 7 which then will be 1 minus 0.3 0 302 plus 0.1057 equals 0.1359 equals minus 1 1 minus 0.1359 is equals to 0.86 for one you can also go ahead and add all of them if you have time and you will find the same answer as I have so that's how you use your table to find the probabilities instead of using the formula because if I was using the formula I would have found the probability of x is equals to 0 and calculate and find the answer which is 0 comma 0 302 and then go ahead again and calculate the same formula and calculate the probability where x is equals to 1 so that I can find the probability of 0 comma 5 7 so let's look at another example we're going to use the same example that we used with the police station what is the probability that the station will get six calls per hour so now we need to find the probability that x is equals to 6 because we just need the probability that we'll get six calls so with that easy you just go to the table 3.5 we look for six where they both meet that is the probability easy like that which is 0 comma I forgot the number now 0 comma 0 7 7 1 which is 0 comma 0 7 7 1 alternatively we could have used the formula the probability that x is equals to 6 will have been e to the power lambda lambda to the power x divided by x factorial and we'll say e to the power minus 3.5 3.5 to the power our x is 6 divided by 6 factorial so I can calculate 6 factorial is equals to 720 and then I can do the top part second function e x negative 3.5 equals 0 comma 0 302 multiplied by into bracket 3.5 close bracket not close bracket to the power second function to the power close bracket equals 0.3699 and I can just divide that by 720 something is wrong what did I do wrong 3 to the 3.5 to the power or I said to the power of 2 sorry I need to say to the power of 6 so if you don't know on your calculator some of you have x to the power of y function or y to the power of x function on the case you might have x to the power of an empty blocky and that will help you to calculate the power of a 6 so I didn't use that I use the square so I need to start again so this is not right so this should be second function e x minus uh to the plus or minus 3.5 times equals times 3 3.5 to the power so I need second function y to the power x 6 equals and I get 55 so now I think I've got it right 55.5108 divide by 720 and the answer I get now is 0.0771 which is the same as what we had there I can also do it on this calculator so those who are using case show it's easy to do I need to clear that I need to go back to the normal calculator which is 1 okay so I can use my function uh and we need e to the power so we can use uh where is e on this calculator uh it's yeah second shift to the power and I need minus negative 3.5 then I need to go down and I need to put a multiplication and I need to say 3.5 and I need to put to the power of 6 use my arrow to go down and say 6 factorial uh where is the factorial on this one oh there is my factorial it's orange so shift factorial and I just press equal and they it's my answer same same same same I'm Lizzie yes maybe for the for the folks on the on the meeting for most of you that use android you can download lots of these digital calculators on your phones if you don't have the scientific one there's actually quite a lot you can find very useful yeah it is any question because then we are at the end of the session you will need to go and practice before you start taking your assignment doing your assignment questions please make sure that you practice and practice and practice because this type of questions are similar to what you get in the in the exam in the assignment as well so I might not be doing exactly the question that you get in the exam but or in your assignment but they are all similar okay so let's do one last exercise uh for some reason I cannot clear my my slide because this is not related okay so let's end the show end the show sorry about that technical glitches rhythm it doesn't allow me to clean it up do this there we go thank you for watching me doing the technical stuff all right so let's do the last one what is the probability that the station will get four calls so we need to find the probability that x at most it's less than or equal remember that at least it's greater than at most it's less than or equal less than or equals to four therefore we need to find the probability that x is equals to zero plus the probability that x is equals to one plus the probability that x is equals to two plus the probability that x is equals to three plus the probability that x is equals to four so we'll have to add all the probabilities so from zero to four so I need to go out of the slideshow keep I need to keep that so 3.5 we need the probability that x is less than or equals to four so 3.5 was our lambda four so it means we need to add all of them so what will be the probability if we add all of them okay calculator so that will be 0.0302 plus 0.1057 plus 0.1850 plus 0.2158 plus 0.188 equals and change to decimal and then we get 0.7255 unless if I calculated something wrong there did you all get the same 0.7555 and that will be the answer to that question okay and that is how we calculate the probabilities so just to wrap up we will have learned how to calculate oh what are the counting rules we've learned what the binomial distribution is the properties of a binomial distribution how to calculate the mean the standard deviation and the variance of a binomial distribution and also how to calculate the probability of a binomial distribution using the formula or using the table and we also looked at Poisson distribution and what are the properties of a Poisson distribution how to calculate the mean the variance and the standard deviation and we know from from the calculations that the mean which is the average which is the expected event which is the same as your mu which is the same as your average or lambda is the same as your variance they are all the same and your standard deviation is the square root of your variance or your mean and we also learned how to calculate the probabilities using the table or using the formula I just want before we go just to show you the question I'm referring to to say sometimes they will just want you to not complete the whole entire calculation just I hope it is one of these questions here there we go so they just want you to know how to calculate the Poisson distribution formula so we will do this exercising class on Saturday so some you can see that this question says you already calculated the entire probability and this one says you just substituted into the formula it might be that this is the correct one and the others are incorrect or this one is also incorrect and the other ones are correct but we will look at that when we do the activities on Saturday after this then we are done for the night any question any comment any query speak now any concern any frustrations um living yes if um I were to be wait if I were to do the assignment to like right now would I be able to do it with all the information that we have yes absolutely we have covered everything you needed to know to do assignment two otherwise you can wait until Saturday because you still have a little bit of time your assignment two is due on the day first which is Monday and we have two two opportunities to finish it yeah so you can look at it now and then later on you can try again um so with the assignment two as well I think now because people are busy on there on on my UNISA submitting assignments are due and all that um you might have challenges when you submit your assignment two right your second your second attempt just give it some time or give it another day or so before you redo the assignment two again don't do it immediately on the same day okay you can you can go through your assignment two I just to get the feel of the questions um and see and then after Saturday you can go back and look at um where you make the mistakes and fix it and then reset me to remember the highest the highest marks will be your highest score will be captured so there's no harm in using both opportunities and with that anyone so I'm going to stop the recording