 Welcome to your second session of the exam preparations. Now, our questions, there might be some way they are missing some headers and all that don't panic. I will use an alternative question paper for that or we'll skip that question and use another alternative question paper for that paper. Okay, so let's start with question one. Which one of the following statements is correct with regards to experiment counting rules, assigning probabilities, number A, an experiment described by a sequence of four steps and four outcomes possible for each step. Step one, three outcomes possible for step two and two outcomes possible for step three and step four will have a total of 24 experimental outcomes. Number B, the number of combinations of five items that can be selected from a group of eight will be 6,720. Number C, the number of permutation five items that can be selected from eight items are 56. G, in an experiment with seven equal likely outcomes, each experiment outcome has a probability of 0,14. E, a classical method of assigning probabilities is appropriate when data is available to estimate the proportion of the type the outcome will occur if the experiment is repeated a large number of types. Number one is your multiplication rule. Number two is combination. Number three, permutation. Number four, normal probability outcome satisfying the event. And number four is just an explanation or a definition of what classical method is. So which one of these statements, so it means for A up until D, you need to do some calculations, right? So, you know, for this one you will say it's three multiplied by two, multiplied by two. How many steps? Four outcomes from step one. Okay, so I'm missing four. Four multiplied by three, multiplied by two. Because there are three outcomes and there are two outcomes for step three and step four, so the two again. So you need to calculate that outcome and see if it gives you 24. And remember we're looking for the correct statement. Number B, you need to go and calculate the combination on your calculators, you know, the formula is NCR. So we know that N will be the bigger number and R will be the smaller number. So yeah, they give you eight and five. So therefore it means it's 8C5. Step number C is permutation NPR. So it means you need to go and calculate 8P5. And step number four, you just need to do one divided by seven. Because its outcome satisfying is just X divided by N. Have you calculated or are you still doing it? Yes, I got the correct answer is B because I mean it's C because it's 6720. Are you saying it's B or it's C? You were cutting off? No, it's C. I was saying it's C because 8P5 gives me... Wait, Lizzie. But I did 8P5 and it gives me 6720. So I don't know if the B will be correct. Okay, wait. No, wait. Okay, give me your answer for A. Let's go through each one then. Give me your answer for A. What is 4 times 3 times 2 times 2? It's 48, right? Therefore it means this is not correct because it says it's 24, right? Okay, then we move on to the next one. What is 8C5? No, 8 combination 56. Oh my God. It equals 256, therefore it means B is incorrect, right? Then move to the permutation. 8 permutation 5 is 6720. 6720 and it says it's 56. So therefore C is incorrect. Then move to the next one. 0.14 correct? 0.14, so therefore the answer is correct. So we know that those ones are incorrect. So the answer is that one. Because on this question, we don't have the statements that are supposed to be there. I don't want us to waste time trying to figure out what they're trying to ask you to do there. For now, we're going to skip it. We can always come back to those kind of questions later on. Let's move on to question 3. I'm already giving you answers. So question 3, consider. Sorry, guys. Maybe it's immersive like these days. I'm sick all the time. I must take a holiday. Consider two event A and B. The probability of A being equals to 0.5. The probability of B being 0.22. And the probability of A and B being equals to 0.11. Which one of the following statement is correct? And they gave you some statements here. So you need to use the information given to answer this. So we can go through each one of them. Each statement then do the process of elimination, right? To see which one is the correct one. So event A and B are mutually exclusive. If events are mutually exclusive, we know that the probability of A and B should be equals to 0. Are they equals to 0? And the statement that it's given. No, that's false. That is not equals to 0. So this one is incorrect. So we move on to the next one. Event A and B are independent. So they are telling you that if event A and B are independent, then the probability of A given B will be equals to 0.5. So you need to test that. To test that if those three events are independent, then the probability of B given A should be 0.05. Let's see. The rule says, remember, for independence it says, if event A and B are independent, therefore the conditional probability of the given event will not have any bearing on the probability of that event you are looking for, right? So therefore the probability of A given B will be the same as the probability of A, or the probability of B given A will be the same as the probability of B. And here they gave us that event A and B are independent, therefore the probability of A given B should be the same as the probability of B. Is this the same? What is the probability of B? It's 0.22. 0.22, therefore it means it is not correct, right? Because it should say it should be equals to the probability of B and 0.05 is the probability of A. So it's not correct. I'm going to skip number C. I'll go to D and E. I'm just doing this so that you are able to know when you get this type of questions in the exam, how to tackle them, right? So if A complement, or if A is a complement of A, which is A complement, then the probability of A complement is 0.78. Is that true? Is the probability of A complement? It should be 1 minus the probability of A. So you do the calculation. Will you get the probability of A complement? 0.78. No, that's false. It's also not correct because the probability of A complement is 0.45, right? Because A is 0.55. You just say 1 minus 0.55 and that will give you 0.45. So it's not the same. Probability of B complement, the same will happen. You need to just calculate the probability of a B complement. By 1 minus the probability of B, which is 1 minus, our B is 0.22. And that will be 0.78, I think. Yes. So now what's left is C. The probability of A given, oh, sorry, A or B. So you need to go back to the formula for this one, this. You need to go to the formula. You need to go and calculate that if the probability of A or B, the formula says it's the same as the probability of A plus the probability of B minus the probability of A and B. And then you go and do the calculations. The probability of A is 0.55 plus the probability of B is 0.22 minus the probability of A and B is 0.1. The answer is 0.66. 0.66. So then for it means this is the correct statement. This is the correct statement. Okay, also similar. Who can tell me what are these values that they gave us here? Are these events or probabilities? They are probabilities. There's a probability. So anything that it resembles a decimal when you are busy with probabilities, it means a probability. So since they gave you the probability, you don't have to go and use formulas. You still need to apply the formulas where they are necessary, but you can also answer some of the questions directly. However, there are some things that are missing. What is missing is your total column, right? Because on your total column, you will be able to calculate what is a simple probability because these have joined probabilities that they gave you. So which one of the following statement is incorrect? So you need to find the incorrect one. Number A, you need to calculate the probability of cisgender, which is cisgender will be the first column. So it means if you have calculated the total there, you'll be able to state whether this is correct or not. Otherwise, cisgender is the probability of 0,76 plus 0,7 plus 0,3. I'm just giving you one example. You will have to do the rest on your own. So is number A correct or incorrect? You just need to do the calculation quickly. Yes, it's correct. It's 0,86. The probability of low? It's correct also. It's correct because you have to add all these values here. The probability of cisgender and low is a joint probability. It's correct also. It is correct because it is just the joint probability of cisgender and low. The probability of cisgender or low. This one, you need to do some calculations. Remember, you will have to find the probability of cisgender plus the probability of low minus the probability of cisgender and low. That's the calculation you need to do. You know that the probability of cisgender is 0,86 plus the probability of low is 0,76 minus the probability of cisgender and low 0,76. The answer is 0,86. The answer is 0,86, which therefore it means this is the incorrect one. The last one, it says events cisgender and low are dependent. So you need to go and check that. You need to go and check that the probability of cisgender, let's use the name the way they have them. Cisgender given low, it should be, it should not be equal. You have to go and prove that it's not equals to the probability of cisgender. For you to prove that they are dependent. So let's see the probability of cisgender given. So we need to first go in because this one we know the probability of cisgender is 0,86. We just need to find out what the probability of cisgender given low. So that you will find by saying the probability of cisgender given low will be given by the probability of a joint probability of both of them. Cisgender and low divide by the probability of the given, which is the probability of low. So now to test that our joint probability of cisgender and low is 0,76. The probability of low is 0,76. Therefore, the probability of cisgender given low is equals to 1. And we have proven that because if this is equals to 1 and the probability of cisgender is equals to 0,86. Therefore, they are not equal. Therefore, it means they are dependent, which means that is correct. Because if they were equal, we would say that they are independent from one another. Good happiness. Are we good? Yes. Let's move to the next question. Same information. Now they want you to calculate the probability that they feel moderate level of being marginalized. Given that randomly selected individual identifies as transgender, what is the probability that they feel moderate level of being marginalized? So you need to calculate the probability of moderate given that the person is a transgender. So you have to write the formula, which is the joint probability of moderate and trans divided by the given, which is the probability of a trans. And then just substitute the values. What is the probability of moderate and trans? 0.01. Which is 0.01 and the probability of trans is 0.05. It's 0.05. Then do the calculation. 0.01 divided by 0.05. 0.2. What is the expected value of a discrete probability distribution with N8 equally likely outcomes? Okay, there's some waiting missing in this sentence. What is the expected value of a discrete probability with N of equals to 8 with equally likely outcomes? Or having, maybe they equally, equally, it should have been having equally likely outcomes. So what does that mean? So if, for example, you have eight of them, you can even start with zero or you can start with one. It doesn't really matter which one you start with. So we know we have one, two, three, four, five, six, seven, eight outcomes and the probability they've got the same equal outcome. Right? So we can use the same, say, what will be the probability of having the same likelihood, which it will be one divided by eight. Like we did with the previous one, seven. One divided by eight will give us what will be the probability. One divided by eight, zero. Zero comma one, two, five. Zero, one, two, five. This will be our X and we generated our probabilities for each outcome. Okay, so that will be zero comma one, two, five. Zero comma one, two, five. Zero comma one, two, five. Zero comma one, two, five. Zero comma one, two, five. It will be like that. How do we know that the expected value is calculated by means of the sum of your X observation times its cross-bonding probability. And you can do this two ways. Are you winning? You just need to multiply each outcome and add all of them by its cross-bonding probability and add all of them. That's the first one. The second one, or what you can do with your expected value, it most likely will work for this scenario because they've got the same outcome. It's just multiplying your probability that you have. Multiply that with the sum of your outcome. It will still give you the same answer. I got, yeah, I got 4.5. If I use the first step, E at X is equals to the sum of X into P at X. Yeah, so if you use the other step. So one plus two plus three plus four plus five plus six plus seven plus eight will give you 36. So you will have zero comma one, two, five times 36, which will give you four comma five. 36, multiply by 0.125 equals 4.5. Or you can just use the first method, which is the general method that we use by multiplying your X with its cross-bonding probability, adding the next, adding, adding, adding, adding. You will still get the same answer. Sorry, Lizzie, how did you get 36? You add your X observation. This is the sum of the X observations. One plus two plus three plus four plus five plus six plus seven plus eight. Okay, thank you. The next question. Given your X, okay, I'm going to assume that this is the whole table. Given your X and your probabilities. And they're also asking you to calculate the expected value. So you can use the same method that we did previously. It will not work, but you can use the normal formula, which is the sum of your X value times its cross-bonding probability. So you first need to find what the value of the question mark is. What is the question mark? It's 0.1900. 0.19. 0.19. So you all agree with that? So the sum of all these values should give you 100%. It would give you one. Yes. Yes, they give us one. Because the answer on number 12 is 0.19. Yes, they give me one. Yeah, but that is not the answer. The question says calculate the expected value of X, right? 0.19 is the question mark. So you still need to do the calculation. Multiplying X times 0.3. So you still need to say one times 0.3 plus two times. Let's put it in brackets. 0.19 plus three times 0.17 plus four times 0.17 plus five times 0.17. You still need to do that. 2.72. Are we all good or are we unhappy? Happiness. 2.72. So really, is there any faster method than doing anything, everything down and calculating manually on the calculator? If you want to... Oh, that's the other thing that we need to discuss. At some point, we will have to touch on this so that you guys know what to expect. And remember, you needed to go and read your document on the invigilator app, right? That your lecturer have sent to say because you're writing a multiple choice question and they gave you the process in terms of how you're going to operate that. And the app, it takes a picture of yourself and it also keeps track every 30 seconds. It takes a screenshot of your whatever you are using. So if you're talking between different things, I don't know. Because your lecture said you can try it with the trial exam paper to see how the process work. I'm not sure if he will... It's one of the questions that I still need to find out from him. Whether, when you're doing the trial exam paper, will it send him a report to show how many students it found them wanting. It found, like, whatever, the way they put it on those documents. So we still need to finalize that process. But to be 100% sure and not to be penalized for silly things or small things, I would rather suggest that you use your calculator for now. Practices using your calculator, practice doing things manually. Don't rely too much on the template that we gave you. Because remember on the template, it means you need to toggle between two screens or toggle between your MOOC exam paper and the template. What if the time it needs to take a screenshot you are on your Excel spreadsheet? Right? And it can take a screenshot of whatever you are busy with right there, right there. So please stay safe, rely on manual work. There are shortcuts, but, I mean, in the exam, they won't give you too many data sites. They will give you a small set like this, which is manageable. You can do that in five minutes, calculating this in five minutes or even less. Excuse me, Lizzie, would you mind if I ask a question? Yes, you can. So what of the PDF that has critical values on it? Yeah. Are we allowed to use that? The tables. Yes, the tables, you will be allowed to use them. The only thing that I'm not sure about is those, like I'm saying, the things like your Excel sheets, your opening of your study guide, do you understand? Yeah, no, I understand. Those kind of things, yeah, those kind of things that might jeopardize your whole exam, you don't have to, but like the tables, it's fine with the tables because you will be given tables. You need to navigate to those tables anyway if they give you. Understood. Is there a certainty that we aren't allowed to use the study guides? I'm not, I'm just asking, because there's been discussions whether or not we're not to use notes and things like that, and there's lots of mixed discussions about it. I know that there are those kind of discussions and those are the discussions that we always have privately, you know, because these recordings are made public. Of course. Of course. No, of course, I'm not trying to implicate you. Those questions, we can always answer them later when everything is done. So now let's concentrate on getting that. Right. Right. Questions. Of course. And later on we can talk. Yes. So. Thank you. Yes. A building inspector would like to conduct an inspection of 13 randomly selected new built houses to check whether or not they comply with municipal regulations. The inspector knows from the past experience that eight out of ten new built houses will comply with municipal regulation. Which one of the following statement is incorrect? And this is a binomial, a binomial question. So they given you your N, which is your random, some selected houses. They also told you how to calculate your probability, which is your probability of success. So eight out of ten. So you, you can just start now by calculating it. What is eight over ten? 0.8. That is your probability of success. Now. We will, at some point, some way, they might, you might need to go to the table as well. So let's look at all the questions that we have here. We'll use the process of elimination for each one of them. The experiment can be described as a binomial with 13 trans. Is that we're looking for the incorrect statement? That's correct. That is correct. Yes. Two outcomes are possible for each crime, complying with regulations. It's a success and does not comply with a failure. With binomial, the name says it all. By means. Yeah, that's true. That's correct. The probability that a newly built house doesn't comply with the municipality regulation is 0.8. So that's not comply. And we know that what our probability of success was 0.8. What will be the probability of failure? It will be 0.2. It will be 0.2. So this is the incorrect one. Expected value. If you didn't know how to calculate the expected value, it's n times the probability of success. So it would have been 18 times 0.8. And each inspection constituted trial with independent results. We know that for binomial, it's always independent. This is another one. There were two questions that looked like that, right? But we couldn't answer that. I don't think I have another question that looks similar to that one. But we couldn't answer. It was in the previous question papers. Just want to see if I don't have. I don't have. It is with the exponential one, which is poison. Okay, so with this one, we can also answer because there are no additional information on this screen. But when I skip that, we can always come back to them. If you if you are able to go on to my units are those who have, you can take a screenshot and share it with us. And then we will come back to those questions when we have time. So that we don't waste more time. Consider the building inspectors visit once again. Suppose now that 16. Oh, yeah, because probably that other information was related to this. Suppose now that they built 16. So we no longer have 18. We have 16 houses. Remember that the inspector knows that the past experience that 8 out of 10. And we also did calculate that it was 0,8 of houses comply with the municipal regulation. Now the question here is what is the standard deviation already gave you the answer. But anyway, you just need, you still need to know how to calculate it. So to calculate the standard deviation, which is s is the square root of n times pi times one minus. So you just need to go and calculate the square root of 16 times 0.8 times one minus 0.8. One comma six. Six. I just want to see this one question. Okay, because they didn't give the information. I can make it out as well. We're going to be skipping a whole lot of them. They didn't pick it up. And then your lecture also says there is no other way that we can get asked unless he takes a screenshot of some of the questions that keep also it. Yeah. Did I give you the answer? Now, let's consider poison distribution. With the expected number of occurrence per interval equals 410. So this is poison. So you must always remember that that is your lambda. And they also say you must use your poison table on this one to find the probability. It says determine the probability that the number of occurrences per interval is at least. What is at least? Less or minus? Less or minus? Nope. What is at least? I'm sorry. Nope. What is at least? At least it's greater than or equal to. At least it's greater than or equals to three. Always remember that. How do I make you remember all this now? At least when a bus has an accident and the way the bus carries let's say it carries 160 passengers, right? And the bus gets into an accident. And the emergency personnel come on and they tell you how many people have died in that accident. It's the worst case scenario that I'm giving you. And they will say because they don't know the number because of the people that are admitted at the hospital or those that are also uncritical and those that are still en route. At the point that they were there. At least they will always use with like at least three people have been confirmed to be dead because they are not sure of if there are more than that number because they can be more, right? So always remember that that at least means it can be more, more, more, more. It can be exactly like that or it can be more than that. That's what at least is. Maybe by using that scary example will get you to remember at least. Okay, so enough with at least let's get to it. So you need to go to the poison table to know whether you're going to have to add all of them or do some magic. So we need to go to the table and go to poison. And I hope you still remember how to read the tables, right? So your poison table are split by x value by lambdas first and then for every lambda table, they are x values linking to that. So we're looking for 4.1. There is 4.1 and 3 is at this point. So it says at least so it means we need to add all of these values. Can you see how many they are? Alternatively, because we cannot go in it from probability of x is equals to 3 plus the probability of x is equals to 4 plus up until you get to the end. Alternatively, you can say 1 minus because we know that we're finding the probability it will be 1 minus the probability that x is less than 3. Why x is less than 3? It is the opposite of it is the complement of that. So therefore it's the same as 1 minus and I'm going to put it into the bracket. The probability that x is equals to 0 plus the probability that x is equals to 1 plus the probability that x is equals to 2. Those are the only probabilities you need to add in state of any the entire table. So let's go back to the table. In state of any all these values, you're just going to add only those three values and subtract them from one. So do the calculation 0.0166 plus 0.0679 plus 0.1393. The answer is 0.2238. 0.2238. No, the answer. Did you subtract it from one? No, not yet. After subtracting is 0.7162. So you're not giving me the answer, you're giving me the answer of the summation 0.2238. So subtract it from one, you will get 0.7762. 0.7762. Do you guys understand? Is that clear? Yes. Are we good? Are we happy, happy, happy? I really don't understand why I get there. Don't worry, don't worry me. Okay, it seems like because I don't have, did you guys post the questions that I skipped? Let's see if we have, I don't have anything. How about we just call it out? No, I've got more questions for you anyway. So this is the previous year's activity. So I'm not going to, because some of the questions are repeating from last year's assignments as well. So like question one, you already did a similar question, so we're not going to do questions like that. So let's skip that as well. I just want you to get more practice in terms of some of these questions that might look complicated. So which one of the following statement is incorrect with the regards to some relationships of probabilities? Sorry, I'll give you the answers. A, given the probability of A and B is equals to 0.4 and the probability of B being equals to 0.5. If A and B are independent, then the probability of A will be 0.8. What do they mean? You need to go and find what will be the probability of A given B. And if that probability of A given B should be the same as the probability of A. So because they say then the probability of A is 0.8. So we just need to make sure that is that correct. So let's go and calculate 0, a given probability given. The probability of A given B is given by the probability of A and B divided by the probability of the given, which is B, because that is the information they have given you, A and B. So the probability of A and B is 0.4 divided by the probability of B is 0.5. 0.8. Which is equals to 0.8. So therefore it means number A, because if my probability of A given B is 0.8, then if A and B are independent, then the probability of A will also be the same as the probability of the given. So number A is correct. Remember, we're looking for the incorrect one. If two events A and B are mutually exclusive, then the probability of A and B will be equals to 0. That is a straightforward right? It's correct, right? Yes. You agree? But it's also correct because mutually exclusive events, therefore it means the joint event will be equals to 0. See, if two events A and B are independent, then the probability of A given B will be the same as the probability of B. Is that correct? Incorrect. Incorrect, it should be equals to the probability of A. So that is the incorrect statement that we're looking for. Two. It seems like someone knows probabilities. If two events A and B are complementary, then A and B must be mutually exclusive. Yep. If two events A and B are complementary, then the sum of all probabilities should also be equals to 1. Yep. That is the definition of probabilities and properties of probabilities. Some of these questions we dealt with them during our practical sessions, like with the content revision session, question and answer sessions, we dealt with some of these kind of questions. So I'm not going to cover all of them. Let's see if we can find another interesting question, but I want to go to the one where we use tables more or actually this one is more important. Where they give you sentences, right? So that you know how to use the sign. The signs are very important. Okay. So the first one, the probability that two learners are absent on a given day is 0,20. Is that correct? Two learners absent? Yes, it's correct. That would be correct. Oh, we're looking for which one of the following statement is incorrect. So we're looking for the incorrect statement. So that one is correct. What is the probability that between two and five learners are absent on a given day? As soon as two and five are both not inclusive. What do they mean by that? It means you are only considering. Three and four. Three and four. Oh, yes. So are they, is the probability of two and five exclusive? 0.45. That's true. That is correct. The probability that at most, so what is at most? What is the probability that at most two learners are absent on a given day? What is at most? Less or equals two? Less or equals two. So it means it must include two and anything below two. So is that 0.5? Yes, it's correct. That is correct. The probability that at least two learners are absent on a given day? Less or equals two? At least it's greater than or equal. So it means anything greater than or equals to two must be included. And if they are, is that equal to 0.5? Incorrect. It is incorrect because that will be 70. What? What? Probabilities are good. It's the incorrect one. And the last one, it talks about inclusive between two and five. So it means you just do two and five, which is 0.70. So when it comes to discrete probabilities, and especially when you have tables like this, always pay attention to the sign and the weights. The inclusive, exclusive, the less than and the greater than. Right. So today's session actually went quicker than last week's session. Okay. I'm not going to ask you to do the expected value. Let's go to the binomial. So we also answered a similar question like this in the binomial. So I'm not going to ask you about this one. We can continue. Yeah. This one asks a different, even though it's the safest, the previous one, but on this one, we have more than, we've got new things. So we have eight rural schools, which is N, one out of four, which will be your probability of success will be one divided by four. What is the probability of success? 0.25. That could be 0.25. So if we know that, which one of the following statement is incorrect? So we know that this is a binomial question. Number one, the probability that at most one of the eight schools have a shortage of teachers. What is at most? Remember that at most it's greater than or equals to, right? So what is the, what is... No, don't you say it's less than and then it's greater than. At most it is less or equals to. Ah, you are so awake at 10 plus eight. Okay. So that is the probability that X is less than or equals to one. So we need to go to the table to determine whether do we want to go through all of them or do we want to only use the opposite thing? So let's go to the binomial table because it's less than, right? Sorry, let's go back there because it's less than. So we're going to only look at the probability of X is equals to zero plus the probability that X is equals to one. Those are the two values that we need. So also, oh yeah. Hence I wanted to use more exercises because we need to also... Yes, let's go like that. You need to know how to read the binomial table. You still remember that the binomial table has the left and the right values and the bottom values and the top values. So anything at the top where the probabilities are small, we use the left hand side. So the top left hand side. The bottom probabilities like here at the bottom, there are probabilities here. We use the right, which are those probabilities here at the bottom, but they are also here at the bottom. So this is 0.50. What is that? 45 would be 0.55. 0.560 and so on and so forth, right? So these probabilities here at the bottom, we use the right hand side. So you just need to always know that. So yeah, our probability is 0.25. So it is the smaller probability. So it means we're using the top part of the table. So we need to look at the left hand side. So let's go there. Our N is 8. It's also very important. So we need to go to where N is 8, which is this table that I did in N. This one, I must turn it clockwise. Here we go. So we know that at the top here, we've got all these probabilities 0.001 and so on. So we're looking for N of 8 and 0.25. So 0.25 will be, probably 0.75 will be 25. Yes. So 0.25 will be there. Even if I can go and count how many columns 0.25 is on, 1, 2, 3, 4, 5, 6. So I can count 6, 1, 2, 3, 4, 5, 6. Here we go. 0.25. So that is 0.25 and that is 8. So we're only looking for where X is 0 and 1. We just go to where X is 0 and 1. Are we happy? Are we good? Oh, good. Yes, yes. Okay. So did you get the same answer? Yes. So that is correct. The variance. The variance is just your N times your pi times 1 minus your pi. That will give you the variance. Yes, we're still on the binomial. So that will be your variance. So what is your 8 times 0.25 times 1. Times 1 minus 0.25 will give you the variance. We're going to finish at half past. Don't worry, we are almost done. 1.5 is the variance. 1.5 is the variance. So therefore it means B is also correct. What is the probability that at least 0 is the probability that X is greater than or equals to 0. So at least 0 means all of them. So what will be all of them? The sum of all probabilities are equals to? 1, always. Always. So C will be the incorrect one because it says C is 0. The probability that at most 8, the probability that at most X is less than or equals to 8. And because we did it with 8, because this is binomial from 8 onwards. So it means it's all of them as well. So it should be equals to 1. So that is correct. The probability that at least 1, so that will be the probability of X greater than 1. So that should be correct. And which other thing we haven't spoke about? Oh, the formulas, this. If they ask you whether it's for Poisson or it's for binomial, then it means you need to know the formula. Oh, sorry. Let's write the formula for binomial. You don't have to memorize the formulas. Remember, I told you you can have your formula sheet that a summary of every study unit's formula sheet that you can keep handy to help you recognize which formula to use when you are answering the questions. They might give you the formulas. If they don't give you the formulas, they will tell you to bring your own formulas and the tables because those are two important things that you need. So for binomial to find the probability of a binomial, you need to remember the NCR, pi to the power X, 1 minus pi to N minus X. So that will be the formula to use. And when they give you information and they don't complete like on one of the questions. Remember, one of the questions was like that where they didn't fully answer all the questions. So this was probably one of the statements they was answering this question where they had have you to do NCR to the power X times 1 minus N minus X. So you just apply this formula, but you don't have to solve everything. You just do a little bit so that then you can be able to identify which one of the options are correct. It's similar to that. So it's similar to this. So knowing that this is a Poisson distributed value and that is one of the key things that you always would help when you answer the question is statements like this in the question Poisson binomial. And those things, they help you clarify which formula you need to be using. So for a Poisson distribution with the mean of three, so we know that this is a lambda. And then it says use the formula to calculate the probability and they expect you not to calculate the entire formula and find the probability, but to see if you know how to substitute into the formula. So we know that the formula to calculate the probability of a Poisson. It's your lambda X times e to the power of negative lambda. Or you can do vice versa. I'm looking at the options here. So they wrote it like this. So if I know what is the probability that three learners will be upset. So they already telling me that I am going to have to have my X probability that X is equals to three, which gives me my average, which is my lambda is three as well. So it will be three to the power of three, multiply by my e to the power of minus three. And looking at what is three to the power of three? It's 27. I think on this question there was an error. Yeah, it says calculate the probability that three learners are absent randomly from versus 27 three to the power of three is 27. If the mean is three, then it means the answer should be 27 times e to the power of negative three. And they say the answer is D on this one. So they made a mistake some question. So but those are the kind of things that you will, you will have to learn in order for you to be able to practice more. And that concludes which other question that we didn't do. I'm not sure if you had difficulties with your assignment and we have seven minutes on this question. The challenge with this is because I don't have the full question here to know what you were asked. It makes it difficult to know how to help you answer this question as well. So I think that is the only other question that we were not able to answer. So which other question you were not able to answer. If we can have this question to this, we can discuss it on WhatsApp as well. Those who have access to it and share on WhatsApp and then we can, someone can help you answer that question. And then we address this one. You can also post that one as well. Because we were not able to answer that and question 11 and question 11. It's more about binomia. So we did do a lot of other examples on the binomia. And question 30 if you want us to have a further discussion on this. Otherwise, then I'll see you Sunday when we look at assignment three. Have a lovely, lovely evening. Are there any questions? Thank you very much. If there are no questions, then I can stop the recording. Just give me a second and then we can have a family.