 How are you are you able to hear me? I can hear you. I miss Lisa. I can hear you too. Okay. I can hear you, too. All right, then we are hearing each other. So we can start with today's session because I keep on asking new questions and you guys are not answering me. So I'm not sure whether you hear me or not. That's why I was asking that because the next part of today's session was to ask you any comments, question, query, how are you feeling? How have up to so far, how have you experienced the tutorial session? Amazing. Okay, so I only have an Indian on the call because I think everybody else is scared. I don't know of what. Anyway, so today we're going to start a new section, so which is study unit five, which is discrete probabilities. And then on Saturday we're going to do the activities on what we've learned today. So what we've learned last week and on Saturday as well, we need to carry through those concepts into today's sessions. If we remind ourselves, we know that there is an event and with an event there is an outcome and with an outcome we are able to calculate a probability from that event which comes from the sample space. We also know that probabilities are always between zero and one. You must always remember that. They are between zero and one. And we also know that the sum of all probabilities is equal to one. And if so, then the probability of A, if A is an event, the probability of A is the same as one minus the probability of A complement. Because we've learned that an A complement is a value that is not part of A, but it's also part of sample space. So it means an A complement and an A make up your sample space. So knowing all those things, we're going to apply the same concepts that I just mentioned in today's session. So today you're going to learn about the probability of a distribution of a discrete variable. And we also know and remember the discrete variable we've learned in chapter one or study unit one. It comes from a counting process. Remember that it's a quantitative measure which comes from a counting process. You're going to learn how to calculate the mean of a discrete variable, how to calculate the variance and the standard deviation of your standard deviation. Remember the mean is a measure of central tendency. And we need the mean in order for us to calculate the variance. And remember that the standard deviation is the square root of your variance. Then we're also going to learn how to calculate the probability of a discrete variable. So a discrete probability distribution, like I said, it's going to be calculated using a discrete variable. And we know that a discrete variable comes from a counting process. Therefore, a probability of a distribution of a discrete variable is a mutually exclusive listing of all possible numerical outcomes. And we are able to calculate the probability of occurrence associated with that outcome. So it means if I have a die and I toss a die, and if I say the die means two, how many times did that die when I toss it will fall on a four. If I toss it the first time and then it doesn't fall on a four, then I record the number of times that it does not fall on a four, which are my counts. I will record until I am complete with my observations. And then I will group them and say how many times did it fall on a four, zero times, four times, one time, three times, two times, and if I render that project for a week that I observed it for a week, so I will record how many times when I did that process fell on a particular thing on those. Then once I have the frequency, then I calculate the probability or I calculate what we call the relative frequency. And those relative frequencies are the same as my probabilities. The table you are looking at right now has your x values, which are my number of outcomes. Those are my number of outcomes for a particular day, for x number of days. And this table shows us the interruptions per day in a computer network. So if I work in an IT environment on a particular day, I can look at how many times a day do we get interruptions. If we don't have any interruptions on that day, then the outcome is zero. If I get one, the outcome is one. If I get two, interruption, maybe one in the afternoon, one in the morning, then there are two, like that. And then I bring this table. What you don't see on this table are your frequencies because they are irrelevant for the purpose of this discussion or for the purpose of you doing the discrete probability. With the frequency, we are able to calculate this probability or we can call it the frequency, relative frequency. When we were doing the frequency distribution table, we would have called this the relative frequency. But in this instance, we call them the probability. This is the probability that 0.35% or 0.35% that there will not be any interruptions per day. Lizzie? That corresponds to that, yes. Did you switch it off? Yes, yes, thanks. Okay, so everybody must always remember when you join, you must switch off your video. Okay, so continuing. These are the probabilities, those are the outcomes. So when you answer the question also in the assignment or in the exam, for discrete probability, you will be given a table with your X outcome, your X outcome. So it might say X. That will refer to your probability. Is there a question? Okay, that will refer to a probability. So your P with the bracket X or it can be Y, so they can give it to you as an X and a P, Y. So you must know that one is the outcome and the other one is the probability. And that is the table that you use to answer questions. So do you want to answer or ask a question? And please guys, make sure that you have muted your mic as well. Only unmute if you want to ask a question. Let's say using the very same table that we have, we can now then go and find or calculate the mean the standard deviation and the variance of this discrete variable. And to do that, let's say we want to calculate the expected value of a discrete variable. And to calculate that, we use this formula. The mean is equals to, which is the population mean. So we call it the mean is equals to the expected value of X observations. It's given by the summation or the sum of your observations multiplied by the corresponding probability. Remember we had the X value and we had the probability. So this formula says we're going to take the observed frequency or the observed outcome, multiply it with its corresponding probability. These are my observed outcomes. These are my corresponding probability. So because it's the summation, I'm going to do 0 multiplied by 0.35. And I get the answer plus 1 multiplied by 0.25 plus 2 multiplied by 0.20 plus 3 multiplied by 0.1 because that cross points plus 4 multiplied by 0.05 plus 5 multiplied by 0.05. And I get the answer. If I add all these values, they will give me my mean. If I add all the probabilities, they will give me one. You don't have to add the outcome. You cannot add the outcome because these are discrete variables. When we add all the probabilities, the sum is equals to 1. And if I add all the summations of our observation multiplied by its corresponding probabilities, I will get my mean, which is 1.040. Any questions? Then I will give you an exercise if there are no questions. Based on what I just said, let's do this exercise. The first thing that you need to recognize with this table is they say, suppose the distribution of the number of people per household is as follows. Now, you're going to ask me, why doesn't it start with a 0? It doesn't always have to start with a 0 because now we're counting this probably they never found a household where there are no people in it or there's no persons living there. So, they had one person living in that household, two people, three people, four people, five people. Then they calculated the proportions of those. How many number of people are there? Then they calculated the probabilities. Now, all you need to know is the table is not complete. We need to complete this table before we can calculate the expected mean. So, the first thing you need to do is to find the value of x. And how do we find the value of x? Anyone? 1 minus all those values. It's 1 minus all the other values because the sum of all probabilities should be equals to 1 and x is the complement of all the other values. So, it means you can start now, calculate what x is. So, x will be 1 minus into bracket 0.25 plus 0.33 plus 0.17 plus 0.15. If you calculated them. 0.1. Let's start with the one in the brackets. 0.9. 0.9. So, that will be 1 minus 0.9 and your x will be 0.1. 0.1. x will be equals to 0.1. So, now, since we now know what the value of x is, we can replace it there. It's 0.1. Calculate the expected frequency or the mean or the expected value. So, the expected value, remember, is the sum of your observe or your outcome multiplied by its corresponding probability. So, you're going to take 1 multiplied by 0.5. What I always do is write this table and then here I just do x multiplied by px so that I can just say 1 multiplied by 0.25 is 0.25. 2 multiplied by 0.33 is 0.66 and that's how I will answer them. And then I can come here and say my expected value is 0.25 plus 0.66 plus. So, you would have already calculated the rest of the table. I'm just giving you hints so that we don't waste a lot of time. So, complete the whole table and calculate the expected mean. Sorry, ma'am. Can you repeat that that expected value is... So, in order to answer the expected value, we need to multiply the observed, multiply by the probability and add them together. So, you could also just add them there. You don't have to come back down here. So, you can add them already on there, from the table, because this will be the sum of your outcome multiplied by corresponding probability. So, you say 1 multiplied by 0.25, it will give you 0.25. 2 multiplied by 0.33, it gives you 0.33. And then you go to the next one, plus, because that is the summation, plus. 3 multiplied by 0.17 and you will get 0.51. Once you are done, then you can calculate, add them all together and find the answer. Okay, thank you. Are you done? Okay, I see five people are liking Hendrick's number. So, what is 4 times 0.15? 0.60. Plus, 5 times 0.1. 0.5. 0.5. Add them together. 2.5 plus 0.66, plus 0.51, plus 0.60, plus 0.5 is 2.5. So, your expected value is 2.52. We are going to use this value. Don't lose it. We are going to use it again when we do the exercise. Don't lose the value. Okay, so now let's calculate the variance or the standard deviation. So, the variance of a discrete variable using the population parameter sigma squared is given by your outcome minus your expected value, which we calculated. We know how to calculate it. So, minus your expected value squared times its corresponding probability of the outcome. So, it's the sum of your outcome minus the expected value squared multiplied by the corresponding probability. So, if I come here to the exercise that we just did, it will be, because we have the mean, it will be 1 minus 2.52 squared times the corresponding probability, which is multiplied by 0.25. Plus, then you continue with the rest of the answers and do the same. 2 minus 2.52 squared times 0.33. So, you just continue. And that is the variance. We will look at an example just now. The standard deviation of the discrete variable, which is the square root of your variance, is the square root of your variance, is the square root of the sum of your outcome minus the expected squared multiplying by the corresponding probability. Let's look at an example, how we do that. Remember our table that we started with? The example of the interruptions? Yeah, is our table. Our x-value I just transposed it so that I can have enough space. But if you still have a table like that with your x and your probabilities, it's still fine, because you can still do the same here. You can still do your x minus the expected squared on there. And you can also do that, multiply, which is x minus the expected squared multiplied by the corresponding probability, which is that that I have. Because with this, it takes longer to calculate as well. It's easy when you're doing it this way instead of putting it on to everything onto the calculator. Then it becomes complicated. So I have my x observation and my probabilities. And I know that the formula says is the sum of x minus the expected squared. And that is what I'm going to start with before I can multiply with my probability. So I start there, I say zero. Remember, we did calculate our expected value. And we did find that our expected value if we go back to the exercise, we find that our expected value was 1,40. So we're going to use that. So our expected value is equals to 1,40. So it's zero minus the expected value, which is 1,40 squared. And I will find the answer. Go to the next one, one minus 1,40 squared. And I get the answer. Go to the next one, two, because that is two minus 1,40 squared. And I find the answer. And I do for all the table. I don't have to calculate the total for this because I'm still working out the question. You can also do this in three ways. You can do x minus the expected and find the answer and then have another block where you square that answer. But I just do it once on the calculator. Or you can do all this on a calculator once. So it's up to you find the one that works for you. So once I have the square, I know that the second part is to multiply with the corresponding probability. So I just take 1.96 multiplied by corresponding probability, which is 0.35. So it's 1.96 multiplied by 0.35. Get the answer. Then I do the same 0.25 multiplied by 0.16. 0.16 multiplied by 0.25. Then I find the answer there. I repeat until I get to the end of the table. Remember, this is the summation of all this. I haven't even touched the square root. So if I add all the values, then I get what we call the variance. So I get my variance, which is sigma. So that will be my variance, which is 2.04. If I add all these values, I take the square root of my variance. If I take the square root of my sigma square, which is equals to the square root of 2.04, then I get 1.4283. Let's look at our own example so that we can have a feel in terms of doing the real calculation. So we're going to work together because I also don't have the answers to this. Remember that we calculated, we said this is 0.1. So now with that, I need to go off. Let me just discard this. And I want to take away this because I want to create a space. I'm just going to make it smaller. I'm going to make it smaller and then I'm going to pop it right here so that we can always have the formula next to us. So the formula is there. We need to be calculating the standard deviation. We're going to leave it there. I'll fill it in as well so that we don't forget about it because I want to create space just beneath the table. Now, if I remember the formula for calculating the standard deviation says, or actually also you must fix this formula, it's wrong. You must fix the formula. It's sigma is equals to the square root of the sum of your x minus your expected squared times the corresponding probability. I didn't have to remove everything. I just thought because I cannot write the sum there, it will be on top of the the square root as well. I will fix it and recent you the slides again. Okay, so we must remember that that is the formula. So since we know that that is the formula, we're going to start with the one inside and square the answer. So we're going to start there and say x minus the expected value squared. And remember that we did find the expected value. We did find the expected value. It was 2.52. That's what we did to calculate. I'm just going to write it there so that I can have order when I write. Okay, so now we need to say one multiplied by 0.25. Sorry, we don't do that anymore because we are answering this question here. X minus the expected value squared. So this will be, I'm going to write it for one, two, and then the rest we're just going to fill it up together. So you need to work it out for me. One minus, so it will be one minus 2.52 and then you press squared. What is the answer that you get? We'll have to work it out together so that everybody can be on the same page. And 2.3104. Sorry? 2.3104. Yeah, we're going to keep it to four decimals. Always remember when you're waking up, just keep your answers to at least four decimals so that when we get to the final answer, we didn't lose any of the decimals. So this is 2.31. I hope I'm hearing you correctly. 04. You will tell me if I am wrong there. Then move to the next one. We say 2 minus 0.33 and we need to square the answer. And what do we get? 2.7889. I think you'll have to type it for me. You said 2.0. There we go. Thank you. Lizzie? Lizzie? This is supposed to be 2 minus 2.59. Lizzie? Yes? You're saying there's a lot of echo. Isn't that supposed to be 2 minus 2.52? Ah yes, you are right. Thanks for picking it up. 2 minus 2.52, which then gives us 0.2704. 0.2704. 2704. Thank you, Ithian, for that. Then the next one. 3 minus 2.52 square. And what is the answer? 0.2304. 0.2304. Yes. Okay, then we move to the next one. 4 minus 2.52 square. 2.1904. 2.1904. 2.1904. Yes. Okay, and the last one. We know that this is 0.1. It will be 5 minus 2.52 square. 6.1504. Am I writing it right? Yes, Lizzie. Now we have completed that part. We need to go and multiply with... We need to go and multiply the x minus the expected point. Multiply the answer with the cross-pointing probability. So we're going to say that answer which is 2.3104. So you're going to take 2.3104 and multiply with 2.0.25. And then you're going to give me the answer for it. What is the answer? 0.5776. 0.5776. 0.5776. 0.5776. Yes. Then you do the same. Take the next one which is 0.2704. Multiply the answer with 0.33. We multiply that. What is the answer? 0.0890. 0.0890. 0.0890. 0.0890. 0.0890. 0.0890. 0.0890. 0.0890. Then do the next one which will be 0.17 times 0.2304. So we multiply that probability with the answer that we got. 0.0390. 0.0390. 0.0390. 0.0390. 0.000289. 1. To choose December. Before December okay. Now. Do the next one. The next one we take 2.1904 and we multiply by. 0.15. 0.3285. Is getting 0.0. 0.3285. The number to the right is 6, we're going to add 1 to 5, so it will be 0,3286, you must round off correctly, otherwise we're dropping off some digits and it will give you a problem when you get to the final answer. Then we do the last one. The last one is you need to do 706.1504 multiplied by 0.1, 0.6150, 0.6150, 0.6161, 0.510, I wrote it right. Okay, so now we have done everything underneath this square root. What is left is to apply the summation part of this. So once I'm done with that part, all I need to do now is just to add the summation there, therefore it means I can just add all the values together. So I'm going to say this plus, let me use the black pen because I used plus, plus, plus, plus summation. So summation means 80, you need to add 0.577, 0.577 plus, 0.0892 plus, 0.0392. Hello Lizzy. Yes. Lizzy, we have some disturbance here. There's few people that are disturbing us here. I'm not sure whether they're in class or not. There's Garabo event and then there's another one here. Remember after you have answered the question, please remember to go back and mute yourself. Thank you. Oh, now I'm muting myself. Okay, so did you add all of the values? 1.6495. You're getting? 1.6495. 1.6495. So since I have that answer, so I can come back to my formula because then I have my answer. I can write my square root and write 1.6495. Then you can go and calculate the square root of that answer. I don't know what wrong did I do but I got 2.4524. Then recalculate. Okay. If anybody else didn't get 1.6495, because we need to agree that that is the answer. Are we all in agreement with 1.6495? I'm recalculating the same answer. Okay, we'll wait. I've got 1.6496. Okay, we'll wait. I'm also getting 1.6496. Okay, let me also calculate. Remember, I don't drop any decimals. So let's all recalculate. Okay. 0.5776 plus 0.0892 plus 0.0392. I think the group is correct. I skipped a decimal by that one. By 0.0391. It's a possibility. So the group is right. Okay, so that's why I said let's all recalculate. 0.3286 plus 0.6150 equals 1.6496. Yes, that's correct. I think even if you square root that, the change is off to the fifth decimal. I don't even think you're going to see it. Yeah, I see that also. Yeah, you might not see it, but sometimes you will see it. You will feel it. If you drop decimals too quickly, you're adding up the values you're adding. Every time you're dropping, you're adding that decimal back in. Okay, so did you take the square root of the answer? Yeah. The answer is 1.2843. 2.843. And when you get it will be 4.4 because it is 4 decimals. The answer will look like that. But the other thing, you will look at the options that you are given. If the options are in 2 decimals, you just leave your answer in 2 decimals. If your options are in 4 decimals, you must make sure that you leave your answer in 4 decimals. And you will notice that from now on, most of the time probabilities or answers that you get might be in 4 decimals or 2 decimals. So you must bear that in mind. Okay, so and that concludes the probability distribution, the mean, the variance and the standard deviation. So the variance, it is that number 1.6496. That's the variance. So what we've calculated here is our variance sigma squared, which is our variance. And what we calculated here is the standard deviation. So that is the standard deviation. That is the variance. Happiness, are we good or? We are good. Yes, we are good. Are we on the same page? Okay. Yes, Lizzie. If we are on the same page, let's now calculate the probabilities. Let's look at how we calculate the probabilities from the table that you will be given. I know that you might say, but we already have the probability. What are you referring to? So I'm referring to cumulative probabilities or standard probabilities, which are the ones that you have there. So cumulative probabilities are the greater than or equals or the less than or equals or the less than and because then we calculating will be adding them up the values. So if you go into your study guide, let's say you're using your study guide to do this, there is a table in your study guide, but I think there are errors on there. You can ignore that table. I'm not sure if they did fix it, but you can ignore that table and use this. As your reference, when you are going to answer questions on discrete probabilities, and this is not only also on discrete probabilities per se. Probably when we also do hypothesis testing or something like that, you might want to refer to what do they mean by this because sometimes the statement that they will give you will be in a wet phrase and the wet phrase is this last set column that I have or there will be in a symbol from it or they might be in the meaning a wedding format. So what do I mean by that? For example, a symbol, this symbol, everybody knows it because everybody on their calculator say equal, equal, equal answer, equal, equal. That is an equal sign. Nobody can say they don't know what that is. That is equal to. But in statistics, we might refer to it as exactly. If in the question they say find the probability of exactly, then they're referring to you putting the equal sign when you put the formula on or you write down the formula. This sign, the current sign to the right, sorry, to the left is less than. That will represent all values. If this is zero, it represents all values less than that that goes there. So it means it goes to the negative. If this is one and I represent the less than, it means all the values decide. But you must always remember that with a less, sorry, if it's a less than, it does not include the value that it's less than. So if I say less than one, it does not include one. It means all the other values that are smaller than or fewer than or below. So if this is a number line and that is one, so this is zero, this is minus one, this is minus two. Therefore, I'm referring to all these other values that side. I'm not referring to one and that is that less than sign. The greater than sign refers to the number or the values bigger than, but does not include that value that you are talking up. So if I say greater than one, like with less than, so if I have the current to the right and I say greater than one, therefore it means all the values bigger than one, more than one above one. Or they can say in the sentence, what is the probability of more than? So you need to know that more than means greater than. Probability of more than means greater than. Above means greater than. You must always remember that. Then we have less than or equal. So what does less than or equal mean? So if I have less than two and I have a number, mine and it started zero and one and two and this says what are the values that are less than two or less than or equals to two? When it has an equal sign, therefore it means when I add the equal sign. So remember that if it's less than, it will be those values only one and zero. If I add a less than to it, therefore it will refer to all of them. So it will refer to two, zero and one, two, one and zero. That is less than or equal. We also call it at most or no more than. So it means including that value and the less than value or the smaller values after that. So if they say find the probability of no more than, then you know that they are referring to less than or equal. If they are referring to find the probability of at most, then it means it's less than or equal, greater than or equal, which is that sign. The same. If I say greater than so remember if I have greater than zero, it means only those values doesn't include zero. The minute it says greater than or equal where they have an equal sign, then it means it's all three of them. In which they can say find the probability that at least, at least, at least, or at least, in my language we say at least. So don't correct me when I say at least. So I'm used to saying it like that. So at least if all the values greater than or equal, sometimes they might say no less than. So you need to know all these words, all these words phrases so that you can associate the word phrase with the sign or the symbol. We also have between. We have the values between. So if I have a number line, zero, one, two, three, and four. If they say between they need, okay, let me put it this way. If they say between and they don't state anything, so we say it is inclusive. If they say between and they don't mention anything else, we say it is inclusive. By saying inclusive it means it includes also those values that is lying between. So let's say it is between one and four. So let me write it correctly. If it's between one, so let's say my value of X is between one and four. We say it is inclusive because when it's inclusive, it will have an equality sign to it. And that is inclusive. So what does that mean? It means the between starts from here and it says also so one is less than X, but X is less than four. Therefore it means X is greater than one, but it's less than four. So the other one is here because it's inclusive. So this says X is any value that is bigger than one. So going there, but it cannot be more than four. So it's also any value less than four and it gets blocked here because it cannot pass this. Let's put it that way. Let's do it as a demonstration so that we can understand this. I will use different colors. So these are my numbers one and this is my four. It puts a block there. It means it cannot be beyond those two values. It's between those values. So it says X is somewhere between those two values. So already place a block there so I cannot go out. So when it is one, so my X is greater than or equals to one. Therefore it means my X is any value this site, but it cannot pass the wall. It has to stop by the wall because that all tells me I have to stop right here. Also it says X is any value X is any value between less than four because this is less than or equals to four. So it starts from four and it goes, but it cannot pass one because there is my wall for one. Therefore it means my X, if I calculate the probability I will be getting all those, I will be getting all of them. That is the between. So it means X is between all of this. We will do an activity just now so that we put everything into context. Then we also have the between, but in this instance because they want to make sure that you understand the between. They will say that is between A and B in exclusive. So if they say exclusive, therefore, so in words, phrases, let's say it is exclusive, it will be like that. So using the same blocks so we can go and find our X. So we know that it has to be one and it has to stop at four. So it says it's greater than one. So it does not include one. So it's any value greater than one so it will go there. It can also go up until four for this instance for this one because it says X is greater than if I'm looking at the blue it goes there. But it also says X is less than four but it does not include four. So it's somewhere there. It can also go there. Now what we need to pay attention to when it comes to not including the other values. It's where they both are common. So we know that they are common at two and three. So therefore it means the X lies between one and four. It means it's only those two values. So we're only going to add those two values. So based on what we just linked as well. Sometimes with the between as well if they give it to you in a symbol because it's very difficult to put it in words as well because they will have to say Z lies between A and B and it's inclusive of A or something like that. So then you know which one to put the sign of equality and where you don't put the sign of equality. So let's say in this instance they say they give us this. So yeah they say X lies between one and four but it also includes four. So with that it means this line the black line starts at four because at the moment it does not start at four it will start at four. So when we look at the common it will be those ones. So it means X lies between one and four inclusive of four. So therefore it means it says it's two, three and four. So let's do the probabilities. Now we've learned a lot about the content. Let's go and do the probabilities. Let me know if you hear me because I think I lost connection. I can hear you. All right. So now let's take this table. Remember our table that we started with of the interruptions per day. Here is our table. Let's apply all what we've just learned right now. So now I'm going to just choose any value at random. I'm going to use the symbol not the phrases. So let's say we want to find the probability. Let's use a color that is visible nicely. Let's say we need to find the probability that X is equals to two. Finding the probability of X is equals to two is easy. You just go to where X is two. X is two at that point. Therefore that probability is zero comma two zero. We just take that probability. It's as easy as that. Straight forward. I'm not going to ask you to do what is the probability of three or what is the probability of four because it's easy to do. Now next, what is the probability of X less than two? The probability of X less than two, we know that less than means below. So therefore we are here at two. It does not include two because we know that it's not an equal sign there. So it starts from two but it does not include two and it's all the values below. So therefore the probability that X is less than two is the same as the probability that X is equals to zero. Start with zero plus the probability that X is equals to one. And that will be given by the probability of X is zero is 0.35 plus the probability of one is 0.5 and you add them together is 0.6. So the probability that X is less than two will be 0.60. What will be the probability that X is four? What is the probability that X is less than four? 0.05. So it's the probability that X is equal 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. So you will have to add all of them. 0.9. 0.35 plus 0.25 plus 0.20 plus 0.1. And what is the probability? 0.8. That will be 0.9. 0.6 plus 0.2. 0.6 plus 0.8. Oh yeah. What is the answer? You're all getting me the different answers. 0.9. You are not supposed to add 0.1 because you said less than one. You said less than four. So we have to calculate from. From three. Okay, okay. Yeah, that's great. Yeah. No, I think you added 0.1 instead of 0.0. I mean 0.10. Yes. 0.9. 0.9. The answer is 0.9. 0.9. Yes. Because this is 0.6 plus 0.27 and 0.8 plus 0.1 will be 0.9. Unless my math doesn't work right. So they add 0.9. We need to pay attention to the side. So, okay, alternatively. Okay, not alternatively. Let me not do that. Let me not confuse you right now. We can do the alternatives on Saturday. So for today, we do straightforward. So what is the probability? So we did the less than. So I want you to calculate the probability that X is greater, not greater than or equal, but greater than. It's greater than three. What is the probability that X is greater than three? 0.1. What is greater than anything? Above, but does not include. 0.20. Okay, you guys, you guys are on the road. So the probability that X is equals to four, plus the probability that X is equals to zero. And you was right because it's 0.05 plus 0.05. That will give us 0.1. What is the probability? Yeah, they ink all together. What is the probability that X is less than or equals to two? Inclusive two. 0.80. What's 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. And that will be 0.35 plus 0.25 plus 0.20. Zero. Which is 0.8. 0.8. Yes. Okay. What is the probability that X is greater than or equals to two? 0.4. Which is the probability of X equals to two, plus the probability X equals to three, plus the probability X equals to four, plus the probability X is equals to five, right? To five. Which is zero comma 20. Zero. Plus zero comma 10. Zero comma five. Zero comma 0.5 plus 0.5. Which is? So you are on the road. You do understand this. This has a easy, easy, easy, As long as you can remember the side, then there's no harm. Okay, let's do the between. Let's get the feel for the between. What is the probability that x lies between 1 and 3? 0.55. Okay, which is given by the probability that because it is inclusive, so it's that it's 1, 2 and inclusive of 3. So it will be the probability that x is equals to 1 plus probability x is equals to 2 plus probability that x is equals to 3. Right? And that will give us 0.25 plus 0.20 plus 0.10, which is equals to 0.55, 0.55. Five. So you are on the roll. So now I want you to calculate the same, my less than my greater. Okay, what is the probability that x lies between inclusive of 1 and less than 3? So x is greater than 0.45, but not but less than 3. 0.45, 0.45, 0.45, which is given by only those two values where the stars are. So which is the probability that x, inclusive of 1 plus probability x equals to 2, but does not include 3. So therefore it will be 0.25 plus 0.20, which then gives you 0.45. And that is how it is. I'm sorry, hello. Yes. Hi ma'am. I'm sorry. I'll take my lecture first, but I cannot repeat that question. Okay. So what is the probability that x lies between 1 and 3, inclusive of 1? So because x is less than 1, but sorry, x is greater than 1 and including 1 or it's greater than or equals to 1, but x is also less than 3. So since it includes 1, so that we will find the probability of x is equals to 1. And the one in between 1 and 3 is 2, so it will be the probability of 2. And when we get to 3, because it says it is less than 3, but it does not include 3. So it's all these other values less than 3, but does not include 3. So we don't have to add 3. So it's x is equals to 1 and x is equals to 2. Then you go and get the probability of 1, which is 0.25 and the probability where x is 2 is 0.20. And you add them together, you get 0.45. All right, 10, 10, 10. Okay. So since we've played around a little bit, let's go and look at how the questions looks like from your assignment exam anywhere where you do your practice, so that you can have a feel of this. So continuing with the same question that we had, remember in the exam, these are like the probabilities questions. You will have four statement or five statement asking you to validate those answers, whether the probability of equal, the probability of exactly the probability of less than or equal and so forth. So remember we did calculate x. We know that this value here is 0.1. So now number one, your first question is what is the probability of more than four people per household? Do you remember what more than is? Greater than. More than is greater than. Remember that? So for more than, which is greater than, then it means you're going to find the probability of x greater than four. 0.1. Which is the same probability of x equals to five, which is just 0.1. I am just going through this so that for revision's purpose, you are able to remember how we got to the answer. For those who are still lost, you can play the videos again and again and again to see where we got the answers from. Because any number above four is five, so we're going to get the probability from the probability of x equals to five. Okay. What is the probability of between two and four people inclusive? What is the probability of between two and four people inclusive? How do we write it? We say the probability of x lies because it's inclusive, so they will be equal, right? Who is? x equals to greater than or equals to two, and it will also be less than or equals to four. 0.65. Then we go to that. I say is the probability of x is equal to two. Aaron? No, it's fine. I wanted to go with that one, but it's fine. You can go ahead. Okay. No, no, no. You can go ahead. I just want to see if I'm on the right track. So it's the probability of x equals to two plus the probability of x equals to three plus the probability of x equals to four. From now on, plus my voice is dying. I'm going to keep quiet. Okay. So it's that my voice is going. 0.33, 0.33 plus 0.17. Yes, plus 0.15 and that is equals to 0.65. You know, I am so happy. I'm impressed with all of you today. I don't know what to drink today. Oh, maybe this is easy. This was the one we've been doing today. The others were stressing you. What are we doing today? It's so easy. You'll find it. I hope we can get 15 questions of this one, okay? Pardon? We wish to get 15 questions of this other exam. Yeah, because we're all going to pause. Okay. So next one. Sorry. I'm saying the previous chapters were killing us, Shem. Oh, I'm so sorry. But I thought the previous chapters are the easiest ones. So that's a discussion for another day. After we've done a couple of this and then you will come back and tell me. All right. So let's do number three. What is the probability of less than three people per household? It's 0.58. How did we find it? So the probability of less than, which then it means it will be x less than three, because it does not include three. It says it's less than. Therefore, it is the probability x is equals to one plus the probability x is equals to two, which are 0.25 plus 0.33. And that is 0.58. Okay. Any questions? We are almost 30 minutes away from being done. So let's look at two more questions or one more question. Let's hope we can fit two. Africa checks knows that around election time, the number of daily fake news posts about politicians follows the following discrete probability distribution. They give us our x outcome of those daily fake news posts and their corresponding probability. So let x be the number of daily fake news posts. Which one of the following statement is incorrect? And then they give you five. Remember also, I don't think going forward we will get that option that says none of the above. We can just scratch it out from all this. Okay. Which one of the following statement is incorrect? The first one it says, the sum of all probabilities equals to one. Is that statement correct? It's not true. It's true. The question I'm asking, is that statement correct or incorrect? It's correct. Because this is just the basic concept that we need to know. The sum of all probabilities will always be equals to one. Find the probability that x is equals to four. Now if you look at the table, there is a question mark there. You need to find that question mark, which is the probability of x is equals to four. Zero is incorrect. Zero is incorrect. It should be point three. So what will be the correct answer there? Zero point three. Yeah, zero point three. Zero point three because it will be the probability of x is equals to four will be given by one minus the sum of the probability of x 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, which then will be one minus 0.1 plus 0.15 plus 0.2 plus 0.25 which then is one minus 0.7 then equals to zero point three. This is incorrect. Probability that x is less than or equals to four. Is that statement correct? Yes, that's correct. That is correct because you just add all of them. That is correct. The probability that x is greater than or equals to four. Also correct. It is correct because there is nothing beyond four, so it will be the same as the probability of x is equals to four, which that is correct. Okay, next question. Let me see. Oh, that is the last one. We can take our time. We'll have more questions on Saturday. Oh, we can leave early actually. 30 minutes early. Here you have the same table. You need to calculate the expected value of daily fake news. Remember the expected value. I'm going to give you the formula right now because you're still familiarizing yourselves is the sum of your observations multiplied by the expected value. It's your x multiplied by the corresponding probability, the sum of x multiplied by the probability which is zero times 0.1 and then continue plus. You can write also here on the table just like that and then just calculate each one of them. We know that this is 0.3. We've calculated it previously. I got 2.5. Okay, let's see what others get. Also 2.5. Okay, let's like the answer there. Let's see if I can get more people getting it. If you get different answer, don't be scared to tell us but we have a different answer. There might have calculated wrong as well. We just want to know. Let's make sure that everybody calculates. I also have 2.5. Also get 2.5. I also got 2.5. Okay, I'm also calculating for the first time this evening. Are we all done? Let's get the answer. We can multiply zero times 0.1 is zero. Why am I giving you the answers? I said I'm going to simplify it. So that was the start. Let's do it together. One times 0.15. It stays in 1.15. 0.15. 0.4. 0.4. 3 times 0.25. 0.75. 0.75 and 4 times 0.3. 1.2. 1.2. So we can just say it's 0.15 because I'm not going to bother with 0 plus 0.4 plus 0.75 plus 1.2 and we can all sing 2.5. 1, 2, 3, 4. Let's go in harmony. 2.5. Well done. Thank you guys. But this prayer, I won't even, maybe with time, it needs practice. And then with time we will get our harmonies correctly in sync. Lizzie, we're going to be scared. Covered today. We've done the discrete probabilities. Now I need to also make you aware that next week we're still continuing with discrete probabilities. Today we only covered the basic discrete probability distribution. Next week we're going to cover binomial distribution and Poisson distribution. Now those are very little bit tricky. So the first hour we're going to look at binomial. Then the second hour we're going to look at Poisson. But they are very fairly easy. As long as you can understand the concept we use today. Poisson. Pardon? Yes, sounds like a poison. Yes. But as long as you can remember the concepts that we went through today, like the table that we went through this, if you can remember all this, it's going to go fairly smooth like it went today as well. So anyway, let's recap. You have learned the concepts of the probability distribution for a discrete variable. You've learned how to calculate the mean. You've learned how to calculate the variance, the standard deviation. And lastly, we've learned how to calculate the probabilities of the discrete variable. And with that, it concludes tonight's session. If you have any question or a comment, now is your chance to speak. Any comment or question relating to what we just did today. Okay. Next, we're going to stop the recording.