 Okay, great. I didn't even check today. I still have six responses and six are still in progress. I've opened it up. I think I've extended it up until the 14th of November, so you still have time to practice again on the same questions. I'm not going to check the results of those who submitted. So those who haven't done it, you will, actually I'm showing it from my side. Let me go to the student view, because my view is different to your one. So there are actually two assessments. There is one for study unit five, which only has two questions, and then there is one for the online practice assessment. If you haven't done so, you can go and look at them, but this is what you're going to be looking at. And I'm not sure if you do also get the bottom part, and probably you're only going to see what you have submitted as yourself, but not the others. Since I am looking at this as I'm looking at the wrong assessment, let me go back. This is the first one. Okay, so I can't show you the result of the first one. Maybe I need to go back as myself. And let me see if I can see the scores. No, I can't see your scores. So you can go and check your distribution of your scores there when you look at them anyway. So it doesn't allow me to to view your scores, because I restricted it, because I didn't want to see how many people have submitted or look at how did you individually compare to one another. So it's of no use to me. But anyway, so to continue where we left off, I have actually, I'm going to stop sharing the online one. I have downloaded it. I was able to to log in as you can see. So I'm not going to start from where we left off. Sorry, I'm not going to start from the beginning. So if you were not part of yesterday's sessions, I am sorry about that. You can watch the recording. I will upload it later today. So we're going to start from here. That's where we ended up. So really see? Yes. Before you go ahead. Sometimes I struggle to look on my desktop. So I am using my cell phone. The reason I was asking you on the WhatsApp, it's because I was trying to check the self-assessment on my cell phone. But it doesn't give me that option. But now I have tried to look onto my desktop. It's different from what I was seeing on my cell phone. Yeah. On the cell phone, my unison works different to when it's online. Okay. Okay. Thanks. Yeah. So we already did question number. I don't know because on this one doesn't download the question numbers. So whatever the question was, but we looked at this and the following question from the assessment. Follows the same question that we did. So yeah, we went with the probability question. It's also probability questions. So the question here is asking you to calculate the probability of female or 15 and under. So you will have to go. Remember the, do you still remember the venues that we created? Let's go. I don't have them anymore. Maybe not finish this one. Remember the venues? Yes. The totals. I'm referring to the totals. The totals that we had. This was three, one, I hope, 0.413 and this was 0.587. Do you still remember that? And I'm going to just add a column here. And this was 0.221 and therefore it means at the top it's 9107. You still remember all the venues? I don't have them. I hope it was like that. So now you need to go and calculate the probability of female or under 15 and over which is this one. Female and under. Remember that is the joint probability there and then this one will be the probability of female and this will be the probability, the probability of under. So the equation requires you to find the probability of female plus the probability of under minus the probability of female and over 15 years and over. So you need to quickly calculate that. What will be, do you have the answer? Not yet still calculating. What have you got? C. I've got C, 0.525. Okay, so let's see how you work it out. I'm waiting. Sorry, my microphone was on mute. Probably to female. I've done, sorry, 0.304 plus 0.109 plus 0.112. Sorry. Sorry, we hadn't. Probability of female is, what is the value of, the probability of 1.109 and 0.112. Yes, which is 0.021. Correct. Okay, and the probability of 15 and under will be 0.413 minus the joint probability of female and over. What is the joint probability of female and over? 0.109. And the answer here, 0.221 plus 0.413 minus 0.109 is us. You have to work it out in a different way. 0.525. All right. Okay, so you have to take those two together. Yes. You have to take all of that together. All of that together minus that. Right. Okay. Okay, got it. Thanks. Okay. Then the next one. Sorry. Just again on that previous one, when we say the probability of female, that means we look at the total. We don't need to add the two. We just, once we have the totals, we can use that immediately. Yes. Remember with basic probabilities inside the table, this is where you find your joint probabilities in your totals. That's where you find the probability of the columns and the rooms. So this is where you calculate your simple probabilities. So this will be the probability of male. This is probability of female, regardless of what age they fall under. Okay. Thank you. That is why it's very important to, when you get the table like this for the probabilities, to quick, to always calculate the totals, because then you will save a lot of time instead of adding all of them and trying to figure out how you substitute them into the formula. You can just calculate the total and use the total as your simple probabilities. So the next question is a discrete probability question. Now with this question of discrete probability, you will notice that, yeah, you are not given the probabilities. You are given the x values and you are given the events. These are the events. So what you need to do is to calculate the probabilities and the probabilities in this instance is none other than your frequency or your relative frequencies, which are your percentages. So what you do is you can calculate the total as well on this column. So you must add all these values, add all of them, only the number of families, add all of them together and write the total. Once you have the total, I don't know what the total is. Let's say the total is 20,000. Maybe 20,000 is too high. Let's say it's 5,000. I'm just going to make an example. Let's say the total is 5,000. I'm sorry about this. Let's say the total is 5,000 when you add all of them. Now to calculate the probability you just take for the first one, you will say 27 divided by, and then you will write the answer there. You will take the next one, which is 1422. You will say 1422 divided by 5,000 and write the answer there, like that, like that, until you get all the probabilities. Once you have all the probabilities, what you need to do, because the question is asking, calculate the probability that a family member randomly owns a minimum of one car. So because it says minimum, so we say the probability that x is less than or equal one. So we want to know if a family owns a minimum of one, they can either own no car or they can own a car, because it says a minimum of one car. If they would have said the maximum, it would have been a greater than. So they cannot own more than that. So you need to calculate the probability of owning that, but those are the steps. First, calculate the total, then calculate the probabilities in terms of every car used, then calculate the actual probability of owning a minimum of one car. Can I ask a question? I don't think I understood the question properly then, because it says owns a minimum of one car. So for me, that means one car or more. So a minimum of one means at least one car or more. You are right. Yes, at least a minimum will say, yes, you are right. Yes, that's correct. You ask what's on. So it will be a minimum will be more than. Okay, thank you, because I just I know that I'm in assignment one, I came and stuck with a few questions, because I didn't read the wording correctly. Okay, thank you. Yeah, because if they would have said a minimum of one car, it means you cannot have more than one car. So a minimum, it means at one or more. Yes, you are right. Okay, thank you. Thank you so much. How are we doing? Do we have the total? Maybe we should do it step by step together, so that we don't stay muted for long. The total I got is 6484. The total of the number of families. Yes, okay. So now let's calculate the probabilities. Is it not supposed to be 67? Sorry. Yeah, 6487. 6487. 6487. Oh, yeah, sorry. 6487. Yes. 6484 is less than 27. 6487. Yes. Okay. Okay, so now let's calculate the probabilities. 27 divided by 6487, we can leave it to four decimals. Let's leave the answer to four decimals. 27 divided by that gives you 0.00. So we leave it, let's leave it at four decimals. Even if we leave it at four decimals. Then 1422 divided by 487. What do you get? 0.2192. And then the next one? 0.4417. 0.1417. No, you got 0144. Should it not be 0.4417? Yes. Okay. And then the next one? 0.6769. 0. 0.04099. 0.0499. 0.0499. Yes. Okay. The next one? 0.0082. Okay. So now you need to calculate the probability of more than or equals to one. This you can do it quite two ways. You can either do it by adding all of them or you can do it by saying it's one minus the probability of x equals to zero. So the answer is going to be B. So the answer would have been one minus 0.042, which then gives you 0.9958. So yes, which is B. Any question? Anybody who's still unsure about what we are doing? No, that's fine. But in the feedback question, I got question B. I mean I got answer B and it's marked wrong. So when you do all the questions and then you submit for grading and you get the feedback, it says that B is wrong. I will check that one. Thank you. I will check that one. I will fix it if it's an issue with the interpretation as well. Okay. So now we move to binomial distribution. Remember, these are the same type of questions you might get in the exam. So there will be a mix. There will be some way they ask you to calculate. There will be some times where they ask you a theory question. You just need to know those. So in terms of binomial distribution, you know that the trials needs to be independent. Sorry, the trials needs to be independent. The outcomes, which are the events, needs to be mutually exclusive. And there's always two outcomes. Either it can be a success or a failure. And since there are two outcomes, therefore it means the probability of those outcomes will be one over two, which is a 50-50 chart. Then one outcome will happen. Based on what I just said, and oh sorry, the last thing that I also need to say is the binomial distribution, we use discrete variables. So it means variables that come from accounting process. So based on what I just said, which of the following is not an assumption of a binomial distribution? It must be A. It will be A because the outcomes cannot be identical. They have to be different. So for example, if you have a coin, it cannot have two heads. One must have a head and the other one must have a tail. The other side must be a tail. And that means A is the incorrect, A is the correct one in this instance. Now the second one is asking Poisson distribution, which is also discrete chapter, chapter unit five. They are asking you to find the probability that no telephone calls will pass through the seed board. So what they are asking you to do is to find the probability of X is equals to zero from a Poisson distribution. Remember Poisson, you must go to the table, which is broken down by the values of your lambda, which are your means. You go look for the mean of equals to two and you go look for X value, which starts from zero, where they both meet. That will be the probability you are looking for. And this is daily because they said no telephone. So it means X will be equals to zero. And your mean is equals to two, which is your average. Is the answer D? I don't know. We just need to find out is the answer D. You can also use the formula. Oh, sorry. I forgot about some people might prefer to use the formula to calculate. Remember, you will be calculating the probability that X is equals to zero by using E minus lambda, lambda to the power of X, lambda multiplied by lambda to the power of X divided by X factorial. You can use that as well to calculate. So if you look at the formula, not the formula, but let's say we use the table. Let's go to the table to help those who don't know how to use the table. So we go to the table. We look for X lambda is two. So we know that our probability, we're looking for the probability that X is equals to zero, where the value of lambda is equals to two. So our X is zero and our lambda is two. And that will be the answer that we are looking for. It will be zero comma one, three, five, three. Actually, I had it shared. Why did I go out? Okay, sorry. And the answer will be option number E. And was it the one that you selected as well? No, I chose the, but it was because I looked at lambda one instead of two. Okay. So those who are calculating manually, so it will still work out the same because you will say E to the power minus two, two to the power of zero divided by zero factorial, which zero, this will be equals to E to the power minus two. And you should go to your calculator and see. Second function E X minus two, and you get zero comma one, three, five, three. And that's how you will answer the questions. And that will be your poison. Maybe I should also repeat this. Remember, what I'm trying to do is also give you a feel in terms of how many number of questions you might get relating to each study unit. As you could see that every time we talk about in everything, I also mentioned the study unit, so that you know that it also follows your, your exam also follows the structure of your study unit. So you'll move from study unit one, two, three, four, five, and there will be at least two questions per each study unit. And with the exceptions of some of them, where they have multiple sections, they might ask three questions and this is one of those where you will be asked the question on discrete by binomial and poison. And remember with the binomial as well, you might be asked the question to calculate the probability, not just a theoretical question like that one. Also, with the poison, they might be asking you to calculate other things other than the probability or they might be even asking you to calculate the mean, the standard deviation, and the variance. So you just need to be aware of all the scenarios. And when we do lots of exercises, they will give you more chance to look at the type of questions they really asked in the exam as well. So that will be a discrete probability which is 30 unit 5. Can I ask a question? Sorry, can I ask a question on the previous question? So here it says the mean is equal to 2 per minute. But the question asked 2 consecutive minutes, which would mean the mean would be equal to 4 in 2 minutes. Because it said 2 consecutive minutes. So that says the mean is equal to 2 per minute, which is 1 minute. But how they're asking for 2 consecutive minutes, which is 2 minutes in a row. And I think that confused me. Yeah, so the other thing is when you read the question, don't overthink most of the things that you read in the question. Identify immediately what you are giving in the question. Because at the moment, that 2 consecutive minutes is not your average. You must remember that. That just tells you that there will be no telephone in those 2 minutes that will pass for every 2 minutes that will come and go. But that is not the average of this distribution. The average of the Poisson distribution is 2 minutes. So for example, when they observed this, maybe let's say they observed this for the whole day. After every minute or every 2 minutes, there might be 3 phone calls, 1 phone call, 2 phone calls. But this is every 2 minutes. When they calculate the average, they will take the average over the period of that time. Not only looking at what happened now and what happened after this, but for the whole process. For example, let's go back to the binomial. As you can see with the binomial, let's say this was observed for 5 days or for 1 day, but for a period of time. So let's say it's for the calendar month. They might say in a day or in a calendar month, how many families had no car? 1 car, 2 cars, 3 cars, 4 cars. So this is over a period of time. It's not only based on one snapshot of a time. It might be observed over a period of time. That is why they are able to see how many pay each scenario. The same way will happen with the binomial. Read the question carefully. So in the statement where it says the probability that no telephone, the last bit of it, it will confuse you, but it should not because they are not saying on average, 2 consecutive minutes. No, they are just saying for 2 consecutive minutes that will happen. So if they look at 2 minutes now, 2 minutes after that, but that does not tell you the average. And for Poisson, you need to be calculating. You need to be using the average to calculate, not just the actual value of the average. Okay. I hope it makes sense. Yes, it does. Yeah, because, okay, so we need to focus on the information that's given in the first part of the question. Yes. Okay. Thank you. But I'm not sure if you did attend yesterday's session when we were looking at the Facebook. There was a question where, some way it was asking about the ghost and no ghost. I'm not sure of no. I'm thinking of another class. Facebook, ghost, and non-ghost, where we were looking also at the probability of success and probability of failure. So for example, in the question, you might say that what will be the probability of no ghost? Oh, they might say the probability that there was no ghost account created is this. The minute they mentioned that, it's not up to you to decide whether this is a failure or this is a success. As long as they mentioned that throughout the binomial distribution, the probability of no ghost is 80%. Immediately, that will be your probability of success. Regardless of the weakest one or the strongest one, it doesn't matter. They will give you the probability of success. And then you can calculate the probability of failure, which will be the opposite of what they have given you in the statement. Maybe we will get those kinds of questions later on when we do more exercises. You will see what I am referring to. But you just need to make sure that you know, you identify what you are given based on the chapter that you are at at that point. Okay. All right. Thank you so much for that explanation. Thank you. Okay. Now we move into the normal distribution. So the other thing you will notice that you are in the normal distribution, they will tell you. Sometimes they will tell you that things are normally distributed with the mean of 65 and the standard deviation of 12. Really, you must know that this is the mean and this is your standard deviation of 12. The question is asking you approximately what percentage of students have below 50. So what they are asking you is what percentage? That's the other thing. Do not get confused with how do I calculate this percentage. Same way, it's your probability, multiply by a hundred. Remember the probability we always represent them as decimals. But if we want, we can also call them percentages if we multiply them with a hundred or we can call them proportions. They can interchange this way. What proportion of students have below 50 or what percentage of students have 50? They mean almost one and the same thing. It's still the probability that you need to calculate. So this will give you, will be your X. Remember, since it's normal distribution, we use the Z score, which is your X value, which you always given in the question, minus the mean, which will be given in the statement divided by your standard deviation, which will be given in the statement as well. So what you need to be doing is to calculate this probability. Once you've calculated this probability, then you're going to find the answer of that probability. The probability of Z equals whatever the value will be zero point from number from the table. Remember here, you'll be calculating the, oh, maybe I'm moving too fast. So the first step you do is to calculate the Z score. So you will calculate the Z value and find the Z and take your Z and go find the probability on the table. Remember that. Remember the following as well. The sign matters. The sign here says below. Remember, the probability of Z less than a value is the value we find on the table. The probability of Z greater than a value will be one minus the value you find on the table. The probability of Z lying between two values, a and b, this will be the probability of Z less than, less than b minus the probability of Z less than a. And the values you find on the table, you subtract from one another. Remember all those things. So once you'll find the value on the table, you're going to take that value and multiply it by 100 in order for you to find the percentage. Okay. Let me repeat. The question is asking you to find the probability of a value below 50. You will first calculate your Z value, which is everything that is inside the bracket. So if I look at the sign, it says the less than. So therefore, we need to be finding the Z less than that value by using the Z score formula. The X is in the question. The mean and the standard deviation are from the statement. You find the value of Z based on the sign you see, you do what is appropriate on the table. If it's Z less than a, the value you find on the table, you're going to multiply it by 100 to get the proportion, the percentage or the proportion. Okay. Okay. Maybe we should also do it together so that we don't stay silent for long. How do we substitute the values? Our X is 50 minus our mean. We already identified all this 65 divide by our standard deviation 12. And what will be the answer you get? Negative 1.25. Sorry? Negative 1.25. Negative 1.25. And remember, because we're going to the table, since we're going to the table, we always need to leave our value at two decimals. So let's go to the table. It's in the negative side of the table. We go to table E2 and negative 1.25. So we're looking for minus 1.25. So minus 1, we will find it on the side. And minus 1.2, we'll find it on the side. So yeah, we should be looking for 1 minus 1.2. And at the top, we're looking for 0.05. Did you locate it? Oh, sorry? Did you locate it? Yes. Okay. So how we locate it? We go find 1.5. And this is 5. So that is the answer we find is 0.1056. So we go. We went and found the probability, which is equals to 0 comma 1056. We need to take this and multiply that by 100. Because it is less than. Remember that for a less than value, the value we find on the table, we use that. So multiply that by 100. And the answer will be 10.56 or 1.56 percent. But we want it as a whole number and then integer, a whole number. So run it off to an integer. The answer will be 11 percent 1. Any questions? Anyone who is still lost? If there is nobody who is lost, then we move again to the normal distribution again. Now, with the next question, they're actually asking you if the distribution of the large group of high school students is normally distributed with the mean of 55 and the standard deviation of 5. Which one of the following statements is true? Now, they want you to go through each and every statement and check if there are two important factors. So you need to use z x minus the mean divided by the standard deviation. The standard deviation they provided to you, the mean they gave it to you. Your x values are all these values here. The sign in front of every value there. You need to calculate each and every one of them. And then want to have the z value, you must go to the table to go find the probability and multiply that probability by a hundred, like we did with the previous one. So we're going to start with number one. I'm going to use the top patch. I'm going to, we're going to start with that one. So what is our z? Over. The over will be, is it greater than or greater than or equal? Greater than? Over will be greater than. So let's substitute the values. Your x is 60. It will always be in the question. Your standard deviation and your mean, so your mean is 55. Your standard deviation is five. So calculate the z value. So 60 minus 55 is five divided by five is one. Let's go to the table. We come to the table. We have to go to the positive side of the table. We're looking for one comma zero zero. Did you find it? 0.8413. It's 0.8413. Now, the sign, we said it's greater than. So our probability of z greater than one comma zero zero. We know that it is one minus the value we find on the table and the value we found on the table was 0.8413. So what is this probability? 0.1587. If we multiply this by 100, do we get 16? Yes. And it means A is the correct one, but you can do for the rest of them. So let's say we do for 45. So the only thing there what changes on what we just did for 45, we just replace this by 45 and we calculate and the sign as well will change because it says it's below. So the sign will be less than and all these values as well they will. So calculate because everything stays the same is just only the sign and the 45. What do we get? So you want the probability of minus two? No. What is the answer for 45 minus 55 divided by five? Minus two. It will be minus 2.00. So you go to the table, we go to the negative side of the table and we look for minus 2.0 and that will be the answer 0.0228. Go here, you say your probability of z less than minus 2.00 is equals to 0.0228. Multiply that by 100 and the answer will be 2.28 or 2.3, which is not the same and you can do for the rest of them and see if you know how to answer them. But I'm just going to leave it as that. And that was the end of it. The last question was did you find the exam difficult? Did you find this exercise difficult or manageable? Those who took it because this is the last question. I think it might be easier than the exam. What if I say this was your exam? Didn't have a problem. It's too early to tell. So with time, when we do a lot of exercises like this then with time you might feel comfortable and then you will take one pre-exam test for yourself to see if you will be able to manage it on time because I will create one for you, time to one for two hours, where you just make sure that you make time for yourself to sit for two hours, take that exam and make sure that you complete it within two hours and then you will look at your results because I will make sure that your results are published as well for you to be able to see how you did and with the answers as well. So much like I've been doing for the other for this exercise and assessment, practice assessment. Yeah and actually this concludes today's session. We are so early. It only took us an hour. That's good but next the next week it might take us longer because we will be doing those complex the ones that you identified as difficult chapters from your feedback. I will publish the exam later today or tomorrow morning so that you have the whole week to work on them. Those who haven't done this exercise on their own you can go back and start looking at it again on your own without looking at what we did to see if you still remember how to answer some of the questions just to repress your mind in memory. Otherwise, thank you guys for coming through. We'll see you on Friday. Enjoy the rest of the week. Thank you ma'am. Thank you. Thank you. Thank you. Sorry ma'am, can I ask a question? Yes, you can ask.