 Today, we're going to build on the knowledge that you have started learning. The last time we met, discrete probabilities. We're going to use some of the concepts, but we're going to introduce new things today as well. The other new thing that we're going to be introducing from now on, going forward are the tables. Every week when we meet, you need to make sure that you have tables with you. And I also need to mention this, the 1610, 1510, your tables looks the same. And the majority of the time I will be using your tables as a demonstration. 1501, your table is, some of them are different to the tables that we're going to be using because yours, your probabilities are cumulative. But I will explain those tables to you just now. Okay, so that is for today. Let's get on with it. So today we're going to be learning the basic skills to bilomial and Poisson distribution. And the session plan for June, or not May, to change that for June is as follows. The following week, we're going to be looking at continuous normal distribution. Then we're going to look at sampling distribution and then look at confidence intervals. And the last one for June, we'll be looking at hypothesis testing. So this will be the session that we will go through. And then we will be most likely we will be left with at least two more skills sessions. And then we will start to split the groups into two, the 1610, 1510, and 1501 separately. But we will first finish all the other thing and then we start working on exam preparation. But that will be somewhere around in September. And we're going to, after June, after the two last sessions in July, we're going to take a little bit of a breather, a break. And then we're going to start with the exam preparations and other things. But we will see how far we get. Okay, so this week, any question and comment, any query? How is your assignment going? Are you coping well? Let's talk about that for about five minutes. So far. Good morning, everyone. For 1501, we are on assignment three. So the assignment is okay. I'm just waiting to finish this session. Then I'll be able to tackle the binomial and Poisson questions. Oh, so we are almost right on par with what you guys supposed to be doing for your assignments. Remember that 1610, you are going to start with that assignment now in June. So we would have already completed what you need. And in June, we would have already even completed everything for assignment three as well. So you should be able to do that. Okay, that's great. How are you guys doing? Are you coping well? Good morning, everyone. Morning. Unfortunately, I didn't start assignment three because I had a general and I couldn't manage to attend two last classes. You need to catch up on the recordings. They are already uploaded every week. After we've completed the sessions today, it's Sunday. By Tuesday next week, you should get this recording as well. So you need to catch up on those two other sessions so that you are able to start with your assignment three. Okay, that's it. If there are no other comments from other people, you need to talk to me. I need to know how you're doing. If you are doing this face-to-face, I will be looking you in the eye and seeing how scared or how afraid. Thanks, Table. So yeah, we need to do this as often as possible to just check in and check and see where people are at. Okay, so let's do stats. Let's look at stats. At the end of the session, you should be able to learn what the basic skills in terms of answering questions relating to binomial distributions in terms of Poisson distribution. Like I said, we're going to be using two tables. So there will be two tables that we're going to be looking at and you need to know how to read those tables as well because it will save you time. But not only know how to read the tables, but also how to take the formula and calculate what is required. Okay, so a binomial distribution is one of the discrete distribution because it comes from a random discrete process as well. And with a binomial distribution, because of the random discrete variable, remember what a discrete variable is, your whole numbers counting. So it is a sequential number of identical trials that you will have and will resemble the trials as your sample N. For example, when I toss a coin 15 times and creating that event, the 15 times that I am creating those are my trials. That I am creating and out of those trials, there will be some outcome and those outcome I can use to calculate the probabilities. If I own a warehouse and I produce light bulbs, if I take 10 light bulbs from the warehouse, those 10 light bulbs are my trials. That I will take them through some quality assurance to see if they are good to be taken to the market or not. So those 10 are my trial and I can calculate the probability because out of the outcome that will come from there, whether it's of good quality or not, that will help me create a probability that I can use. Okay, so with binomial distribution, there will always be two outcomes. One outcome for success and one outcome for failure will always refer to those two. Most of the time when we end, I will say probability of success of probability of failure, but you must always remember that probability of success might mean something. You might not be given a question that says calculate the probability of success, but they might say calculate the probability of a light bulb not being good, a bad light bulb that didn't pass the quality assurance that that is a probability of a failure. If they say calculate the probability of a good light bulb, that will be a probability of a success because that was successful. So you need to read the question and make sure that you understand whether you're given the probability of success or probability of failure. But there will always be two outcomes that you need to be calculating or you can use to calculate the binomial distribution. In terms of the probability of success, it is also denoted by the letter pi and the probability of failure is one minus the probability of pi. So remember one minus it's a compliment. So your probability of success is a compliment of the probability of failure or the probability of failure is a compliment of the probability of success. And these two events that you owe, the two categories that you would create, for example, needs to be mutually exclusive and they also needs to be collectively exhaustive. Like for example, on a coin, you have a head and a tail. Both of them are mutually exclusive but they are also collectively exhaustive if we combine all of them. And your events needs to be independent. One cannot influence the other. They need to be independent from one another. So this is binomial distribution in a nutshell. You do get questions in your assignment that asks you about basic properties of a binomial distribution. You should be able to learn and know those because I've just given you everything you need to know about binomial. Those are your properties of a binomial distribution. And if they ask you a question like this, it refers to the statement that we just went through. So let's see if we can answer this question. Which one of the following is not a property of a binomial distribution? Number one, is that true or false? Let's go back there. It is our first point. Remember that number of trials should be identical to N. So each experiment has N trials. You need to talk to me because these are your questions I'm asking you. That's true. Yes, thank you. The N trials are independent of each other. And this is the last sentence I said before we moved to this example. It's also true. It's also true because we said events are independent. So that is true. Each trial has two possible outcomes that are mutually exclusive success and failure. Let's go there. Two outcomes, and we know that they need to be mutually exclusive and collectively exhaustive, right? So that is true. The probability of success remain the same for all trials. Is that true? If I have a trial way, I have a, let's use a, let's use this example. If I have 10 barbs and 15 coins, will the probability of success for both of them be the same? If I toss 15 coins and if I have 10 light barbs and one of them, one of them, the 10 light barbs, one of them is faulty. That probability will be, the probability of success will be, they are 10, so it will be nine out of 10. That is the success. And that will be 90%, right? If I have 15 coins, which is very difficult now, let's not use 15 coins. Let's use one coin. If I have one coin and I need to know what the probability of that one coin landing on a head, the probability of success for a coin when I toss it and lands on a head will be one out of, one out of two. And that will be equals to 50%. Are those two probabilities of success the same? That's what they mean by that question. The probability of success remain the same for all trials. No, they're not. No, they are not. So that will be incorrect. The probability of success is always going to be half because there are two outcomes in terms of a binomial probability, two outcomes. So if I have two outcomes, so my N, my trial N is two and my X observation is one and my probability of success will be X divided by N and that will be one over two and that should be correct. Right. And that is the properties of binomial distribution. You just need to know these properties because sometimes in the questions that you are asked, they will ask you content theory that you need to know, but some of these things are a given because you need to know them. These are basic things that we already also learned about in basic probabilities as well, like a probability of an event A will be given by X divided by N. So you just bring in the knowledge that you've learned previously. Okay, now let's look at this one. Africa check found that the sources of fake news on Facebook are mostly ghost profiles. Suppose 20% of the profiles on Facebook are ghost profiles. Suppose further that we randomly select 20 Facebook profiles and check whether or not they are ghost profile. Which one of the following statement is incorrect? So now we can go back to the statement and learn what they're asking us. They're asking us to find the incorrect statement. What they have given us is that we know that we have 20 profiles that are ghost. Then I can call this my probability of success because when I go to the Facebook, I need to check for ghost profiles. So that defines my probability of success. And they randomly selected from N of 20. So I have my N and I have that. So if I go to Facebook to check whether or not there are ghost profiles, that will give me my two outcomes. It's either ghost profile or not the ghost profile, right? So not the ghost profile will be my one minus ghost profile because that will be my compliment of my ghost profile. So now let's read the question. We're looking for the incorrect one. The given information describes a binomial experiment with possible outcomes, ghost profile or not ghost profile. Are those the possible outcomes? Yes, those are true possible outcomes. The number of trials is 20. Correct. Because our N is 20. Number of test is 20, that is correct. The 20 trials are independent of each other. True. That is true because ghost profile cannot influence not ghost profile, they need to be. Independent, so a person cannot be in both or the profile cannot be in both as well for mutually exclusive profiles. The probability of success or ghost profile is 20 trials. You must be very careful. Incorrect. The probability of success or ghost profile is 20. That will be incorrect because a probability and a trial cannot be the same. A probability, how do we define a probability? It's between zero or one or one and a hundred percent. So that is incorrect. It should have said it is zero comma. To zero or it should have said it is 20% not 20 trials. And therefore we have found our incorrect statement, but let's move on to the next one. The probability of failure or ghost profile is 0.8. So that will be true because that will be one minus zero comma two and that is zero comma eight. So you see, you just need to make sure that you read the statement, identify what you are given in the statement and then come here and answer the question. So how easy was it fairly easy to answer? Now let's move on into how do we calculate a binomial distribution? So with the binomial distribution, there are also some of the characteristics that we need to calculate like the mean, which is the expected value, which is your trial multiplied by the probability of success. And where your n is your sample size or your number of trials and pi is your probability of success, remember that? Your variance, it's calculated by n times the probability of success times the probability of failure. Your standard deviation will be the square root of your variance. So I don't even have to repeat that. So how do we do that? So a student is taking a multiple choice exam. There are four multiple choice questions with each question having four choices. So it's four, four, four, four, four. They're confusing, right? But don't worry. We know that the student is taking a multiple choice question. So in a multiple choice question, like we have been doing, there will be one correct answer unless if they tell us that there are two correct answers in the multiple choice question, that there's always going to be one correct answer. There are four multiple choice questions and with each multiple choice question having four choices. So the number of trials are our multiple choice questions. Question number one, question number two, question number three, question number four. Those are our trials. Our choices within question number one, those are our options that we have, right? So we have four questions within the multiple choice questions. Therefore, we can calculate or we can find our probability of success and the probability of failure because one question needs to be correct. So if we need to find the probability of correct which is probability of success, I'm gonna call it pi. I don't have to write the probability of pi, it's just the pi. Pi will be one divided by four because we know that the probability is the number of outcome divided by the number of trials or the number of outcomes. And that will be what is our probability of success, 0.25, right? That will be our probability of success. Our probability of failure will be the right calculated right. One divided by four is 0.25. One minus 0.25 is 0.75. So we know that this is success that is failure. Now we can answer all these questions. Find the mean. The mean is calculated by N times the probability of success. So our N is four times 0.25. What is the answer? It's one which is equals to one. So the mean is one. Find the variance, the variance is calculated by four times 0.25 times 0.75. You can also use one minus 0.75. But because I already calculated it there, it's there. And what is the answer? It's 0.75. So this is listen. Yes. Where do we get this four? We defined it. There is our four. It's our N. It's our N. Yes. Four is your N. The same way as when we were looking at this example, remember, when we were looking at this, we defined our N and we did find our probability of success. We did the same. So yeah, they gave us the probability of success. If they didn't give us the probability of success, yeah? We would have been given the outcome that satisfies the probability of success, which will be our X, and then we'll go and calculate our probability of success. So in this instance, they have given us the probability, they didn't give us the probability of success, but they give us our outcomes. But they also tell us that they didn't tell us, but we know that when you write an exam or when you answer questions from a multiple choice questions, one option of those four multiple choice options should be correct, right? Or it should be incorrect. And that's how you will define your probability of success or probability of failure. So now we've defined our probability of success as getting one correct. So having one correct out of four, it will be 0,25. And that's what we are using. So for multiply by 0,25, multiply by 0,75 is 0,75. Find the standard deviation. We know that the standard deviation is the square root of your variance because I don't have to rewrite this again underneath the square root. We already calculated it. I just used the answer. And what is the answer? 0,66. 0,8. 6,6. And that is your standard deviation. And that is how you will calculate the mean of a binomial distribution. The standard deviation and the variance of a binomial distribution. Then I'm not gonna ask you to do this exercise right now. Or maybe I can. Let me give you time to do this exercise because we have 26 minutes. Okay, so remember that this is the mean. This is the standard deviation. We know the formula for the mean expected value or the mean, it's calculated by n times pi. And the standard deviation is calculated by the square root of n times pi times one minus pi. You come here, we already did this because this is the same question with the ghost profile. We know that this is our n and we know that this is our ghost profile. We did calculate your one minus pi in one of the exercises and we found that it was 0,8. All you just do is just substitute into the formula. It should be easy and quick. We'll check the check for any responses if you have, do we have an answer? It should be quick and easy. It should be quick and easy. Quick, quick, quick, quick and easy, quick, quick, quick. Any answers? Nobody, no one. Okay, so let's calculate. What is our n and what is our probability of success? Am I muted? Can you guys hear me or not? It seems as if like I am. Yeah, I can hear you. It's like I'm talking to myself. No, you are not alone. Ha, but you guys are not even telling me. We're still calculating or we are still busy or... Still calculating. Ha! Okay, let's do this because I'm not sure what's so difficult with this because then when we get to the probabilities then if it takes you long on this easy ones then I don't know. I really don't know. So what is the mean? It's n times the probability of success. We already defined what the probability of success is. We already have the end. We just substitute into the question, which is 20 times. This is 0.2, which is equals to four, right? Then the standard deviation is the square root of 20 times. 0.2, we already have done that, times 0.8, we set our probability of one minus the probability of success will be 0.8. And that will be the square root of 3.2, which is equals to 1.78885, or which we can say it's 7.9. Which one is the correct answer? So easy, so straightforward. It's option number two, Lindy Wey, thank you. I know it might look like it's Greek, but try and answer the questions. And if you ask back, let me know. Don't just wait and wait and wait and wait and wait, right? So that we can unstack you and then you are able to see the light. Okay, so that is done with the properties of a binomial distribution. Now we need to go and start looking at how we calculate the probabilities. Remember this, do you still remember it? Do you still remember what it means to say exactly, to say fewer than, to say more than, to say at most, to say at least, to say no more than, to say no less than, to say between, inclusive, between, exclusive, or if they give you the sign, do you still remember that? Because with that, you will need to use the same information when you answer binomial distribution questions. Well, the binomial distribution in basic probabilities we introduced you to counting rules. So a binomial distribution is made up of two parts, the counting and the probability. So you can use the binomial distribution actually to find out what are your chances of you winning loto? Because you can use your counting rule which will tell you the number of ways you can play the numbers. So this will tell you the number of ways you can play the number, but there are 49 numbers, right? For loto, I'm not sure, I can't even remember now. This part, the second part can tell you or will help you with the probability of winning loto. And if you combine the number of ways plus the probability of winning loto, they can tell you what are your chances of you winning the jackpot or winning loto. But they will not give you the numbers to win loto. And that will be another calculation that you will need to know to predict the number, the numbers you need to win loto and take those numbers and say, what are my chances of winning loto with those numbers? And then you can calculate them, yeah, as well. Okay, so that is just to entice you to tell you how statistics is very interesting that maybe you want to pursue this field going forward as well because now there are more things that you can do with steps. All right, so the binomial distribution. So we can calculate, this is the formula that you need to use to calculate the probabilities of a binomial distribution and it is given by the probability of a binomial distribution by your combination, which is NCR, your combination times the probability of success to the power of X times one minus the probability of success and minus one. So you can write this formula the same way as that or you can use the formula. You are expected to know how to calculate this where N is your trial, X is number of interest or observations, X is your sample size or your trials and the probability of success as always which is the probability of interest will be the pi and one minus the probability of success. We know that that is the probability of failure. Okay, so how do we then use this formula to calculate the probabilities? We will get to that. You can also, for easy of saving time, you need to, you can use the table but we need to understand the probability distribution tables. We'll spend a little bit more time on the tables. Okay, so those who are doing 15 or one, your table might look different to this but probably it's the same or it might look different. It might have cumulative distribution probabilities. You will let me know when you look at your table if it looks exactly the same as this. So what happens with this? Table E6 has two sides. It has the top part and it has the bottom part. So if you look at your table, you will have two sheets. One will have the probabilities and one will have the other side will have also the probabilities. Now, at the top of the table and at the bottom of the table on the second page, those are your probability of successes. This is the probability of success for this table but it is also the probability of failure for this table. So this is a compliment, 0.90 is a compliment of 0.10, so they both are probability of successes and they both are compliment of one another in terms of the table. On the side, you will have, your table is broken down by number of trials. So every block with the line divides the number of trials, your end, every end value starts from zero up to the number of trial that you have. So you will see that here we have two, zero and one and two. And when you go to three, it will start at zero, one, two and three and so forth. Now, on the top of the table, any probability which is less than 50% or 0.5, all this, you read the table from the site. So it means you go on from the site if you're using those probabilities there. If you're using these probabilities at the bottom of your table, this table at the top, it's cut off. So they are also probabilities there, but you need to refer to them as this one. This table here at the top is broken. So you will need to use those probabilities and place them here so that you are able to read those. If you read the probabilities that are bigger than 0.5, then we use the site. So these probabilities here at the bottom, they correspond with this going up to that. So if my probability is 0.65, then it, and my N is two, I need to use this site. I cannot come this site and look at two the site. Because if you notice, you will see that this table is upside down. Can you see that? So N2 starts at the bottom and it reads upwards. So that will be zero. So if you come this side and read from here, you will get a wrong answer because if I say what is the probability of, if my probability of success is 0.65 and my N is two, what is the probability that X is equals to one? If you come this site, you will find that that probability is 0.198. But the probability actually is equals two. It's equals to 0.500. So you need to be very careful when you read your binomial table. It's easy. You just need to understand it. We will do some activities and we can use the table. I will show you how to. Sorry, Mrs. Lizzie. Yes. Is it no way that the number can be clearer because on my side there, I don't see them very well. Don't worry about this. You don't have to see the numbers on this one. When we do the exercise, I will open up the table itself and I should be able to zoom in onto that. So this is just to illustrate for illustration purpose, right? I'll show you the tables just now. Okay. Okay, can I have a question? Yeah, so this is... I don't see this one on the left. Wait, before the question, before the question, this is for STA 1610 and STA 1510 for now, because I need to look at your table as well so that I can explain your table to you. So this is only for those who are doing the modules 1610, 1510 for now. Okay. Thank you. Okay, I will explain this 1510 table just now. I just need to open it up as well. I don't think I did have it here. Let me see. I've got your past example. If it's the same as the 1610, 1510, then we don't have to worry about it. No, it's different. So I will show you your table separately now, just now. Okay, so that calls for me to stop right here. Just gonna stop there and share my entire screen. Yes, everyone. Just give me a second. I need to... Just give me a second. I apologize for that. I just need to close some of my screen things because now I need to share my entire screen. Otherwise then the whole world will see my screen. Okay, so coming back to the table. So we explained the 1610, 1510 table. So if you look at this table, it's gonna scroll, table E7. So if you have it, it looks like that. So I'll just rotate it. I'm just gonna rotate this. So this is 1510, 1610. Your table looks like this. I explained it. You use the top with the left hand side. And you can see that there are no probabilities here at the bottom. So you just do the complement. So this will be 0.555. This will be... If I'm on this one, it will be 0.80. This had the probabilities at the bottom. So if I'm using this probability, the bigger probabilities, I will use this side. This side, I will use that. And going to the next page, oh, come on. Okay, going to the next page as well. So you need to also always remember at the top, there are probabilities. I know that this is 0.50. So I might not know what this is, but if you go to the bottom of this page, it's 0.01. So this is 0.01, that will be 0.01. So you will have to have those imaginary numbers that you write there because they are missing because it's cut off based on the values at the bottom of the table should be able to guide us. So those values at the top, the probabilities here at the top, we use the left hand side. Any value that is bigger, probability bigger, we use the other side. So like that, like that. 15.01, your table looks like this. I know that it's not visible enough, but we can try and make sense of the values that we see here. I will try and look for another table as we go along. So your probabilities are cumulative. They are also breaking down by N. Let me see if I'll have to stop sharing so that I don't expose my PC to the whole world. Let's see if I don't have another cleaner version. The problem with past example is that because they are scanned, the quality of those papers are not good. The ones that you have shown, they're the same as the text book. Do you have your text books with you, right? Then it makes it easier. I can just explain and then you look at the text book. So I found one that is at least much better. I can use that one. It's way much better than the first one that I opened. So then it's fine. Sorry about that because now the entire session is wasted by trying to find the book. Okay, so you are using accumulative probabilities. You can see that they are different. So yours, you will work differently. You will have to think long and hard when you answer questions. For example, because it's cumulative, it's broken down by N. If my N is five, and my pi, which for you, your pi will be P as you can see. So it means you must look at your module formulas as well. So your probability of success will be P will be the values at the top here. And your K is for as it will be our X, you will use K value. So if our K is equals to, let's say one. Now N, my P, my probability of success, I'm going to take the value that is visible, 0.5. Let's assume that. So we're going to use P of zero comma five, N of five, N we want K of one. So now if you look at this value here, that is the probability of K. So this value that I'm looking at is the probability of X. It's less than or equals to one, which means in this probability that I'm looking at here, there is the probability of X is equals to zero plus the probability of X is equals to one. So if I'm looking for this probability of X is equals to one in terms of what I'm looking at here, X is equals to one, then I have this value at the top, which is zero comma zero, three, one, three. And the probability where X is equals to, oh great, less than one is zero comma one, eight, seven, five. And then you can find your probability of X is equals to one, which will be equals to the probability of X is equals to one of zero comma one, eight, five minus zero comma point one, eight, seven, five, minus point zero, three, one, three, which is zero comma one, five. Zero comma one, five, six, two. Now I'm gonna use the same question, the same example that we use. So our probability is zero comma five. Our X is equals to one and our N is five. I'm gonna use the same information on this one so that I can show you that we're gonna get the same answer. It's just that the tables are different and how we use them. So zero comma one, five is at the top and it's also at the bottom but I'm going to rely on this one at the top. And remember we're looking for the probability of X is equals to one, where my N is five and my probability of success, which is my pi, okay. Where my pi is zero comma five. So zero comma five, it's at the top. I'm gonna go to N of five and X of one and I'm just gonna scroll, scroll, scroll, scroll. Scroll, scroll, until I get to my probability. And if you look at it, you can see that we're getting the same answer, which is zero comma one, five, six, two, which is the same as zero comma one, five, six, two. So you both going to work on your tables to answer the same questions. Right, I am done with that explanation of the two tables. How we calculate using the formula because sometimes you need to just know the formula as well. So it's easy when you calculate in the probability of exactly, but it's not going to be easy when you calculate in the probability of greater than or at most or less than equal or less than. It's not gonna be easy because you need to repeat this for every probability that you're calculating. So what is the probability of one success in five observations? If the probability of an event is zero comma one, where X is one, N is five, and the probability of success is zero comma one because we're looking at that. So since we have this information and we know what the formula is, we can just substitute into the formula. So which is our N is five, X is one and the probability of success is zero point one and then we solve this and we get zero comma three, two, eight, zero, five, right, easy. Remember that our X is one, our N is five, our probability of success is zero comma one. So let's go to the table and see if we can find this. We're going to our tables, we'll start with the 1501. We're looking for N of five, P of zero comma one and X is equals to one. Therefore they are asking us to find the probability that X is equals to one. That's what they are saying. So let's go, our N of zero comma one, oh, sorry, our P of zero comma one, our N of five and our K, which is X, I'm using K on this table, K of one, which is that. And we know that it corresponds with this. So therefore that will be equals to zero comma nine one, eight, five minus zero comma five, nine, oh, five. Because we need the exact value of this because this table is cumulative. So it means when I'm here, it includes the value from there and there. And that will be, what is the answer? Point nine one, eight, five minus point five. It's point. Three to eight. Are you sure? Is it not point zero four? I'm getting the same answer as well. Oh, sorry. Point three, eight, zero, zero. Oh, no. I mean three to eight, zero, like that, okay. Always remember to keep all four decimals. Otherwise if the options are three decimal, you can keep three decimals. So let's do the same on this one. So we're looking for N of five pi of zero comma one. And we're looking for the probability that X is equals to one. So we can increase the table a little bit for those who don't have the table in front of them. So come on. All right, so we're looking for N of five, zero comma one pi and X of five. So they both meet and that is the answer. As you can see, that it corresponds with our answer that we were looking for. The challenge with using the formula is if we need to find the probability that X is greater than or equals to one, therefore it means we need to add all of them. So there are five. So it means we will need to calculate the probability that X is one, plus the probability that X is two, plus the probability that X is three, plus the probability that X is four, plus the probability that X is five. Then it means you need to repeat all these five times. This is for one, we've done it. Now you're going to replace one with two. Therefore it means everywhere where there is one, you're going to put two. It means one, two, two and calculate and answer that. Then once you are done with two, go and answer for three. Everywhere where you see X, you put three. Everywhere you answer that. Everywhere you see X as four, you answer that and you get the answer. Alternatively, you can use your table. Alternatively, you come here. If they're asking you what is the probability that X is greater than or equals to one, you know that it is from this to there, right? And we also know that this value here at one, it includes the value from here. And we also know that this value includes the value from number two, number three and number two, number one, number zero. If I'm standing here and I subtract, so this probability of X is greater than or equals to one will be the same as one minus the probability of X equals to zero because that's the only value that is not counted in this process. And that will give us the probability of X greater than or equals to zero, which will be equals to one minus 0.05905. In this instance, it will be, the probability will be zero comma nine for zero nine. I'm doing something wrong here. One minus 0.5905. Zero point four nine five, zero point four nine five. I don't know where I'll get the nine from. Zero point four nine five, four zero nine five. Come on, witty witty. That will be how you will answer the question. So let's say it is on this side on the STA 1610, 1510. So we need to find the probability. We use the same, zero point one. And they are asking us to find this, all of them. Same step. You just say the probability of X greater than one. You can come here and add all of this. You can add all these values. That will give you the probability of X greater than one, which will be zero comma three, two, eight, zero plus, until you get two plus zero comma zero, zero, zero, four. Because this is zero. You don't have to add that. And that should give you the same answer. Yes, we got zero comma zero, nine, five. Or alternatively, we know that is the same as one minus, the probability of zero, which will be zero comma five, nine, oh five, which is zero comma four, zero, nine, five. You can use either way, but you need to know how to use your table. So when we do some exercises, you should be able to answer the questions. So yeah, I was just also demonstrating how to use your table. So I'm not gonna ask you to answer this question now, but now let's see when we look at the poison and then we'll come back to the activities later on. So we've looked at the binomial table, right? Now let's look at the poison table so that then we can go into the exercises. Poison is also from a discrete process. So we use poison distribution when we interested in the number of times an event or case in a given area of an opportunity. And what do we mean by the area of an opportunity? We mean that will be a continuous unit, no longer a discrete counting process, but it also follows from a counting process, but this one is a continuous unit or an at an interval unit of time. And here we're talking about finding the average or finding the mean of a certain event happening at that given point in time. So for example, the number of scratches on the car, we can calculate the mean of those and then we use that to find the poison distribution or the poison probability. The mean of the mosquito bites, sorry, the number of the mosquito bites on a person, the number of computer crashes, the number of things you produce and we calculate the average of those and then we use the average to calculate your probability. So how do we do all that? So with poison distribution as well, you need to know how to calculate the mean, the standard deviation, and the standard deviation and the variance. Now, the mean of a poison is your average, is your expected, and it is your lambda. So this is your expected value, is your mean, is your average, is your lambda, is your average, and is your lambda, it's all of them. Also, it is your standard deviation or not the standard, the variance. So the variance of a poison distribution is the same as your mean, your average, your expected value, so this also is your variance. Also, the square, so the lambda that we're going to be using is called the average, the expected mean, the mean, the variance, the mu, we will use the same. However, we know that the standard deviation is the square root of your variance, so that you need to know. So it means your standard deviation is the square root of your expected mean, the square root of your variance is the square, in terms of poison, because standard deviation is the square root of your variance, and we know for poison, for poison, the variance is the same as your mean, or your expected event. Local police station receives on average 3.5 emergency calls per hour. These are calls, these calls are poison distributed. Find the mean. We know that the mean is your average, find the mean, what is the mean? 3.5. The mean is 3.5. What is the variance? It's also 3.5. 3.5. What is the standard deviation? The square root of 3.5, which is equals to? 1.8708. 1.7. Huh? You were fast. Sorry, 1.8708. 1.8708. So I'm just gonna leave it at 1.87. And that's how you will find the mean, this variance and the standard deviation of a poison. I'm not gonna ask you to calculate this because it's easy to do the mean and the standard deviation. We just did that. Now let's talk about the poison distribution formula. So with poison, you also need to know how to calculate this poison by using the formula, which is your E on your calculator. The E on your calculator. You must look for it. If you want to use an E, it's the one with an E and X at the top. You will need to know how to use that. So it's the E, which is the base of a natural log system to the power of a negative expected event, or lambda, times the lambda to the power of X divided by X factorial. And that is the poison distribution formula. You can also use a table. I'm gonna take you through your tables per se. So let's say this is the poison distribution table. So those who are doing 1501, yours is also still cumulative. This is the table corresponding to 1610, 1510. So the poison distribution table is broken down by different tables, which are broken down by lambda values. And every lambda value, a table will have its own specific number of events that corresponds with that. So your X observations. So some tables will have 10, some tables will have nine, eight, more than the others. So you just need to be careful when you're using the poison tables as well. So if we need to find the probability of X is equal to two, where lambda is, or the average is 0.5, then on the table, we just go to where the table, the lambda table is, we look for 0.5, we look for X of two, where they both meet, that will be the probability. Ignore my circle, it's circling the wrong value because it should be circling. Zero comma is 0.78. And that is the probability of X is equals to two. Or using the formula, we just substitute into the formula, our lambda is 0.5 and we substitute our lambda and our X is 0.2, two factorial, and we calculate, we will get the same answer. Let's look at the tables. We're gonna do the same on the table, so let me write these values down, X is two and our average is 0.5. Going to our tables, you need to look for, this is 1501, you need to look for poison table. So it should say poison probabilities and they, it is. Now on your table, these cumulative, as you can see, so where I talk about lambda average, you can see that yours is the mean, so this is the same as lambda. And where we talk about K, we're talking about X and your tables are also split by your lambda values. So you can see that this table has zero up until 20 and let's see the next one. It starts from, so you will need to be very careful. You need to look at the lambda values and it also is split up to 32 and it goes on and, okay, so they only gave two tables on this. Okay, so let's go and answer our question that we are looking for or we were doing. So we said X is two and our lambda is 0.5. So 0.5 and X is two, X is two is there and you can see that on this table, this is 0.9856. So we know for sure that this value, it includes zero and one values. So in order for us to find the probability that X is equals to two, we will have to say 0.9856 minus the previous value because we know that if we are here, this value includes the first one as well. So if we're going to subtract only 0.9098 and that should give us the value that we are looking for, which is 0.9856 minus 0.9098 which is 0.0755, I might be typing things wrong, let me double check, which is 0.0758. And that's how you will use your table, STA 1501, those are the tables, you can see that on this table, STA 1501, those were doing discrete probability. 1510, 1610, also the same, we go to our Poisson table, let's look for Poisson and this is our Poisson table. Like I said, it is broken down into lambda values. So we're looking for probability of X is equals to two where lambda is equals to 0.5 and we come here, we look for 0.5, lambda of two where they both meet, that is the probability, easy. The only challenge is when you need to use the formula, when you use this formula, and you need to find the probability of a greater than. I'm not gonna go through that because you already know how to do that by now. So with that, it concludes our session, we're gonna go into doing some activities or some questions and exercises. Okay, are there any questions? Is there any question or comment? Someone who's still not sure what's happening or how to do certain things? Who needs me to elaborate further? Nobody? Okay, so without that, let's go and answer this question. You can use the table, you can use formulas, it's up to you, but I will suggest that you use the tables where it's possible and only refer back to the formulas. If you see something is not working well on the question and it requires you to use a formula, but you can also practice by going through the formulas as well as using the table to make sure that you are getting the correct answers. Remember as well, you need to remember, if they're asking you about at least calculating the probability of at least more than, greater than you also need to know what they are referring to. So, exercise one. Previous studies have suggested that one in every four rural schools have a shortage of teachers. Suppose 10 rural schools are selected randomly or independent of each other to check whether or not each school has a shortage of teachers which one of the following statement will be correct. So now we need to go back and understand what the question is asking us. We need to be looking for the incorrect statement as we go along and answer the question, but what have they given us that will enable us to answer this question? Previous studies have suggested that one in every four, one in every four, which means my X on my N of this question so that I can calculate my probability of success, which is one in every four. Okay, let's do that. You don't have to label them N and X. So that will give me my probability of success because it says of shortage of school teachers. So that is my probability of success. What else they telling us that our N is equals to four? So my probability of success here will be 0.25 and N is 10 not four. Why am I saying four? N is 10. Which one of the following statement is incorrect? So number one, the probability of a school having a shortage of teachers is 0.25 while that of a school not having a shortage of teachers is 0.75. So here they're asking you to validate if the probability of success is 0.25 and the probability of failure is 0.75. What is the probability of failure? The probability of failure will be the complement of the success. So is number one correct or incorrect? It's correct. Number one is correct because our probability of success is shortage of teachers because that's what we are proving. So this is 0.25 and the probability of failure will be those with no shortages, which will be 0.75 and that makes that statement correct. The two occurrences for each trial are a school having shortage of teachers and a school not having a shortage of teachers. Are those possible outcomes? Yes. Yes, those are the possible outcomes because we also said if this is the outcome and this is no shortage, therefore it means those two are also the two outcomes that is true. You must ask if you don't understand what we're talking about as well. The expected value of the schools with a shortage of teachers is 2.5. So yeah, they're asking you to calculate the expected value, which is n times pi. It's 2.5. It's 2.5 because here we calculating our n is 10 times 0.25, which is equals to 2.5. The probability that only two of the 10 schools have a shortage. Yeah, they're asking you to find the probability that x is equals to two. So you can use the formula, which is your nCR pi of x, one minus pi of n minus one. Or you can go and use your probability of x is equals to two, where your pi is 0.25. By going to the table, let's go to the tables. This is a binomial question, right? So we need to go to the binomial tables. And on the binomial table, we're looking for n of 10, probability of x is equals to two, and p on this instance is 0.25. So let's go find that n of five, we're looking for n of 10. So we need to scroll, scroll, scroll until we get to 10. Scroll, scroll, scroll, and your table doesn't have 10. But probably on your one, because this is from the exam on your print or your textbook, you should have n of 10, 11, 12 going forward. So I can't help you on this one. So you need to look at your tables that you have right now and go for n of 10, let's assume that this is 10 there. You will use k of two, and then you will use p of 0.25, you will be at this. To find this value, you just need to subtract the one at the top, right? So but you need to use the right table. Do that while I explain to the 16, 10, 15, 10. So here we go also to the binomial table. Now, we know that this is a binomial table and we also are looking for n of 10, so n of 10 is here, but we need to be using the values at the top because the values at the bottom are bigger. So we need 0.25. So what is 0.25? What is the complement of 0.25? 0.75, we need to be somewhere on this column here, right? On this column, because then at this point 0.25 is there. So I've got that, can just remove this. So we know that n is 10 and we are in this column. So we just need to look for 0. And only two, I just passed it. And the answer is 0.2816. Yeah, 0.2816. 0.2816, which is 0.25. 0.2816. The probability that only three of the 10 schools have a shortage, then we look at the probability of x is equals to three of the same. So you just go back to your table. So those who are using 15 or one, you need to go find the table n is equals to 10 for you to be able to answer that question. So yeah, we just go to the one at the bottom because we're looking for three. And it says 0.25, so 0.23, which is incorrect. So we know that all these are correct and this is the incorrect answer. And that's how you will answer the questions. Okay, so now let's go to the next question. The average number of adults with AESD consulting with a neuropsychologist, Pate Day, is poison distributed. So they sometimes give you those hints that you know that you are working with poison, right? With the mean of 1.5. So we know that this is our lambda, this is our mean, this is our expected value. Whichever they're gonna ask, that's what we know. What is the probability that on any given day, a neuropsychologist will consult with one adult? Okay, what is the probability that they will consult with one adult? And these are the answers that we have. So the first thing we can do is we can use our table because it's easy to use, right? So let's go to the table. We are looking for poison of 1.5 and X of one. We go to our table, we start with 1501. We go to poison and we're looking for the average of one. Our lambda is 1.5 and we're looking for the probability of X is equals to one and X is equals to one is at that point and that is the value. So therefore, we can say zero point, I will say the probability of X is equals to one will be given by 0.5578 minus 0.2231. And what is that probability? 0.5578 minus 0.2231 is the probability of X. 0.3347, 0.3347, right? Let's look at 1610. Go to poison table. We're looking for lambda of 1.5 and the probability that X is equals to one. So we're looking for the table where the poison value is 1.5 average and X is equals to one. And the answer we get is 0.3347. Now let's go back to our question. So we're gonna go back to our question and we know that this probability is 0.3347 and you get all these options and you panic because at the moment you're going to choose option number five, right? Need more information to calculate the probability, probably because you can see that there is no option for none of the above and all that but you forgot about this. Nope, you can't do that. You will need to go manual right now. That is why it's very important to know how to calculate these things also manually. We know that the formula is your E to the power of negative lambda, lambda to the power of X divided by X factorial. Looking at the question, I just need to substitute into this formula. So E to the power of negative. If I, you can see that my formula and this formula are written vice versa. Don't do worry about that. We can fix that later on 1.5 and our average is 1.5 to the power one divided by one factorial. Or we can write it as 1.5 to the power one E to the power negative 1.5 divided by one factorial and E to look exactly the same as the question we are looking for. And that's how you will answer the questions. Let's move on to the next one. Suppose we have knowledge that the number of grade 11 absence at one of the schools are Poisson distributed with the mean of 2.5. What is the probability that at least two grade 11 lenders are absent on a given day? So we can state what we are given. We know that we need to find the probability that X is greater than, I like that 11, greater than or equal, right? It is at least, at least is more than or equal. Greater than or equal to, we are told that our mean is 2.5. Now we need to go and find this Poisson. So you don't even have to use the formulas. You can use the tails. So what does that mean? This will mean it's a very long calculation that you need to do. But what else do I know? You can calculate this two ways by saying the probability of X is equal to two plus the probability of X is equal to three plus the probability of X is equal to four plus until you get to the end of the table with the probability that they have. You can do it that way or you can come and look at this and say, what is the complement of this view? So the complement of this X is equal to or greater than two will be one minus the probability of a less than. Less than two. What does that mean? It means one minus the probability of X is equals to zero plus the probability of X is equals to one. That is that. That is what it means. So if I can write a shorter version of the long calculation to a small calculation, I should be able to do that as well. So let's go ahead and see if we can answer this question. So we need X of greater than two and lambda, which is the average is 2.5, all right? Go to the table, I'll start with STA 1501. Okay, lambda of 2.5, it's here. We know that we're looking for the probability that X is greater than two, which means we're looking for all these probabilities from here going there, all of them. We can add them all up, but we don't, we can't add them because they are already cumulative. You can say from here, how you will answer the same question. You will say one minus the probability of X is equals to one. Because we know that this is cumulative, X is equals to one at this value includes the probability of X is equals to zero, plus the probability of X is equals to one. That's this value. Maybe we don't even have to write it in that manner because you can see that it says summation. We can also even say it is less than or equals to one, but when you write it that way, it means you need to add both of them. We don't want to add both of them. We only want the value at X is equals to one. We only want this value. So that will be one minus 0.2873, which is equals to one minus 0.2873, which is 0.7127, 0.7127. Those who are doing 1510, 1501, we're gonna do the same. We're going to look for 2.5. 2.5, that's our 2.5, so our lambda of 2.5, and we need to find the probability that X is greater than or equals to two. And we search, it is all of these values, right? So you can go and add them. 0.2565 plus, that is the first method that I gave you, 0.2138 plus, until you get to 0.002 up until you get there, because this has zeros, you can add the zeros to the values, they won't make any difference. And the second method I gave you, I said it can be the same as the complement, which is one minus the probability of X less than, less than two, which means is the same as one minus, I will need to say those two values because the less than two are those two values, 0.1 minus the probability of X is equals to zero, plus the probability of X is equals to one, which means one minus 0.0821 plus 0.2052, which is one minus 0.28783, right? Which is equals to 0.7127. Happiness, happiness, it's a day. What, so any questions, any comments, any queries? Let's go back to the other question that we didn't answer. I'm gonna give you time to answer this question on your own. Let's see if you can answer that. We are told what the average is, which is our lambda. What is the probability that the station will get six calls? It will be the probability of X is equals to six. What is the probability that at most six calls per hour? That will be the probability of X. It's less than or equals to six. Oh, sorry, four, which is the same as the probability of X is equals to zero, plus the probability of X is equals to one, plus the probability of X is equals to, to plus the probability that X is equals to three. Come on, that is the alternative. That's the probability that X is equals to R, probability. There is no R actually in this instance. For 1610, yes, there is an R. X is less than or equals to four. It's one minus the probability of X greater than four. It depends. If greater than four only has two values, then you can use that. For STA, 1501, what they are asking you, which is the probability of X is greater than four, or less than or equals to four is the same as the probability of X is equals to four. That's the easy part. For you on your table, that will be easy. Otherwise, oh, you can use your formula. Remember the formula to calculate the probabilities, the probability of X is equals to e to the power of lambda. If you rely on formula, lambda to the power of X, expect to get choose whichever one. You can post your answers on the chat. You can see that some people already are posting, means things are becoming clearer and easier for others. So let's look at the answers. So the first one, before the first one, what was the probability is 3.5. The average is 3.5, so that is very important. So we're going to use the same table. This is for STA, 15019, 3.5. And we know that we're looking for the probability that X is equals to six. So we go to X, that is the value there. So for you, because that is a cumulative value, it includes the previous ones. So you're going to say 0.9347 minus 0.856, which should be equals to, which is equals to 0.077. Right, easy. 1501, we're looking for 3.5, and we're looking for the probability that X is equals to six, which is easy for you, because we just go to where it is six, and that is equals to 0.0771. Easy. The next question, which you can answer it using either the first option or the second option, or those who are doing 1501, it should be easy for you, because we're looking for the probability of at most four for the same. So let's go and see. So since we're looking for at most, this is for STA, 1501, the probability of X greater than or equals to, oh, sorry, less than or equals to four, we're going to 3.5, we're going to four. We know that when we are at this point, it includes all these values. It's all the values less than. So which is the same as the probability where X is equals to four, it is the point that. So the answer there is 0.7254. Let me see what you guys had to those who answered the question. Yeah, someone said it is 0.188. So that is for 1510, 1610, yours will look different. So we're still looking for lambda of 3.5. We are, what are we looking for? The probability that X is less than or equals to four. And I gave you the option to accept 3.5 and four. And there we are, right? Nope, we are not there. It is not that. That person who used this because they used the shortcut that I gave for, I said this is only applicable for 1501 if they're using the table. Otherwise they can use the formula to substitute and calculate. So 1610, 1510, you need to follow the right instruction. So this, we can add all these values. So it means because I'm not gonna repeat all this that I wrote there again. So you just need to add all these values. 0.0302 plus 0.057 plus. 0.02 plus 0.057 plus until you get to 0.1888. Or alternatively, you can do the inverse of it. But you can see that it's also very long. So I think this one minus the value of this will be longer than if you only add the four. So the second option is not viable in this instance. So this option you should not use. We're gonna rely only on the first option, although it's what 1501 can use that. So let's see what is the answer when we add all of them. 0.0302, 0.05717. 0.1850 and 0.2158 and 0.1888. I come back here, I'm gonna write the values here. 0.0302 plus 0.1057 plus. 0.1057 plus 0.1850 plus. 0.2158 plus 0.1888. And that is equals to 0.7255. And we know this one was 0.0771. And that concludes today's session. Please make sure that you complete the register. The link is shared in the chat. Any questions, comment, Piri, before we close off the session. Okay, if there are no questions, you have learned how to do some basic calculations relating to binomial distribution and Poisson distribution. Remember also, you can also use the formula or the table, but it's always easier to use the table, but it does not say all the time you should be able to know how to do the calculations, especially using the formulas as well. Please make sure that you understand your tables and be able to know how to use your table as you move along. And that concludes today's sessions and enjoy the rest of your day. Thank you for being part of our session. Thank you, Sissi.