 To your other session of the exam preparation, where we do revision. Do you have any comments, question, or query before we start with the session today? Any questions? Nothing? Next. Next. We're ready for discussing assignment two while doing a revision. Okay, so we can start. So, assignment two dealt with study unit four. In study unit four, you learned about the basic probabilities. So what you need to know about the basic probability is that you need to be able to define the concepts of the basic probability with which intent say, you need to be able to know what probabilities are, and what are the types of probabilities that you can get. So for example, a simple event or a joint event, and you also need to be able to calculate those probabilities. For example, calculating the probability of a simple event a is observation satisfying that event divide by the sample space, or divide by how many there are. Calculating the probability of a joint event a and b, it's given by observation satisfying that joint event divide by the sample space. Also, you need to also remember that sometimes on your exam papers or question papers, this probability of a and b can be written as a union. So it will be the probability of a and b. They represent one and the same thing. Just as in the nutshell, that's what you need to know. You also need to know that a probability has a complement because the sum of all probabilities are equals to one. So if the sum of all probabilities are equals to one, therefore, it says, if I have a probability of a, so here I can just say the sum of all probabilities, I'm using a letter a, but it doesn't mean what I'm going to explain now, it means that is only one probability. So therefore, it means the sum, the probability of a has a complement, which is one minus the probability of that complement. Or we can say the probability of a complement of a is the same as one minus the probability of a, which in 10, what I just wrote there, is the same as say, the probability of a plus the probability of a complement are equals to one. Those are the basic things that you always need to remember when we talk about basic probabilities. But then you also need to remember the following, that if event a and b are mutually exclusive, therefore, it means they cannot happen at the same time, then the probability for mutually exclusive events, for mutually exclusive events, the probability of a and b will be equals to zero, because both of those probabilities, they cannot happen or those events cannot happen at the same time. So that you need to remember. But if they are not mutually exclusive, then the probability of a and b will be the observation satisfying that event divided by how many there are. That's one. The other thing you need to also remember is, you can either find the probability of a or b, because sometimes you can find either all. So that will be the probability of a or b happening will be given by the probability of a plus the probability of b. We call this the addition rule. Plus the probability of b minus the probability of a and b, because in a, there is a and b and in b, there is a and b. So we need to separate one from that. That is for natural or normal probabilities. If events are mutually exclusive, if they are mutually exclusive, therefore the probability of a or b will be given by the probability of a plus the probability of b, because the probability of a and b are mutually exclusive. They are equals to zero. Also, you must remember that the probability of a or b is the same as the probability of a union b can be written as such. So only for mutually exclusive events, the probability of a or b is equals to the probability of a plus b. The other thing we also need to remember when we deal with probabilities, there is your conditional probabilities, the probability of something happening given that another event has already happened. So the probability of a given b will be given by the probability of a and b divided by the probability of the given which is b. That is for your normal probability. So what happens if event a and b are independent? Then when event a and b are independent for independent events, it means event a and b have no influence on one another. Therefore, it means the probability of a given b will be given by the probability of a because whether b has happened before has no bearing on what happens to the probability of a. So for independent events, your conditional probability of a given b will be the same as the probability of a. Then we also have what we call a multiplication rule. It means if we want to find the probability of a and b, given that we have a conditional probability, then the probability of a and b, the probability of a and b, let me rewrite it nicely. So the probability of a and b is given by the probability of a given b times the probability of b, and that is what we call a multiplication rule. You also need to remember that. For independent event, that multiplication rule, the probability of a and b will be given by because we know that for independent event the conditional probability of a given b is the same as the probability of a. Therefore, the probability of a and b will be the probability of a times the probability of b. That is only if and only if your events are independent. So it means you need to read your question carefully and understand what you are given in relation to your probabilities. So let's recap for the last time. Remember that you need to know how to describe your probability, define and describe. You need to be able to know how to define your different events that can happen. How to calculate the simple event, which is the probability of a, how to calculate the joint probability of a and b. Now, here I'm going to repeat. You need to be able to calculate the joint event, which will be the probability of or the observation satisfying that event divided by how many there are, which is that part. If and only if they tell you the following. A event a and b are mutually exclusive. We know that the probability will be equals to zero. Therefore, it means you cannot calculate it. You just say it is equals to zero. However, if they also tell you that event a and b are independent, you cannot use this. You will have to come and use the joint probability. You use the independent formula, which is your multiplication rule. So you need to be able to identify in the question. Are they telling me that event a and b are mutually exclusive, or are they telling me that they are independent? If they are not saying all those things, then this formula applies. For the probability of a or b, you also need to remember the following. If they're asking you to calculate the probability of a or b happening, you need to be able to know that it's an addition rule. If they tell you that event a and b are mutually exclusive, you need to know that you need to use only the probability of a plus the probability of b. You need the formulas. Usually, you will get as part of your exam paper thing here, but I hope they will send you in advance the tables all on the day of the exam. But it should be almost similar to the one that I shared with every one of you, except those who joined our sessions late or they are not in my group. Probably you should have already received the same PDF from your Twitter as well, even if you're not in my group. All Twitter's, we received this document from your lecture. So everyone should have it. But the challenge with this is, as compared to all the formulas that I gave you, as you can see, there are only two formulas there. Usually, the lecture will give you the formulas that are required or are for the questions asked in the exam paper. So for example, in that exam paper that they were writing in 2020, probably they were asking them for those two formulas. The questions were relating to those two formulas. I'm not sure if they say it will be the case, but you need to know all the formulas. OK, so let's start answering the questions. Oh, you also need to know and be able to describe what is the type of probabilities, empirical, priority, and subjective probabilities. So let's get to it. Which one of the following statement is incorrect with regards to experiments, counting rule, and assigning probabilities? What I also have in mind is the counting rules. So counting rules also are like number of ways you can do certain things. The types of probabilities, remember, you get your empirical probability, which is based on observations. And then you get your priority, which is based on observing historical events. And you also get your subjective that is based on personal opinion. So which one of the following statement is incorrect with regards to experiments? A, a classical method of assigning probabilities is appropriate when all experimental outcomes are equally likely to happen. B, an experiment with four steps and three outcomes possible for each step has 81 probabilities. So this you will use a multiplication rule and a multiplication rule says n times n. So you can just determine if that will be correct. In an experiment with eight likely outcomes, each experimental outcome has a probability of 0,18. And yeah, we're talking about the n factorials. So we're going to use the factorials to find that probability. Number D, number of permutation of three items can be selected from group of seven. So because they're talking about permutation from group of three from seven. So we know we use NPR or NPX. The number of combination of four items that can be selected from the group. Yeah, we're talking about the count in the rules of combination, which is NCR. So where N is your total number, N is your total number. Which one of the following statement is incorrect? Are you waiting for us, Lizzie? Yes, I'm waiting for you guys. Well, initially, just based on these formulas you've given, if I worked them out, and I'm probably doing this very wrong, I'm getting BSC already wrong. Those are not the right values. I think I'm also giving you the wrong formulas on those two. Yeah, that's why you have silence. Yes, that's why I'm assuming that that's why I'm going to call it the probability of X. There are eight outcomes. What will be the probability of getting one correct? OK, I think for number B, there is an error. OK, let's start with number A. Is A correct? A classical method of assigning probability is appropriate when all experimental outcomes are equally likely to happen. Is that correct? That is the definition of classical method. That's correct. You need to know that a study of a probability is a study of chances that an event will happen, or there is a likelihood of an event happening. And that is your classical probability method. I think number B is the answer that they gave there. It's a mistake, because this is a multiplication rule, because they want to know the possible steps for this outcome that it would be, but because they gave you so you have four steps and three outcomes. So it means for every step, how many outcomes? So if I do a combination of different steps, so we can even draw a decision. So that will be n times, I'm going to say n times n, because then it says four times three will give me 12. Lizzie, can I ask something? With regards to B, aren't we supposed to be using the first counting rule, which is the power rule? I think so, wait. Yes, we can. So if we use the n to the power of x, therefore there are four steps to the power of three. What do you get? I think so, yes. That's 64. Aren't we supposed to be doing three to the power of four? We go. Then we get 81. So three to the power of four. It's 81. That gives you 81. OK, so therefore number B is correct. And number C, you just need to apply the basic probability function. In an experiment where eight are likely to happen, so therefore it means it's one over eight, an outcome. Each experiment outcome, so each it means one, each experiment outcome will have. So the chances that one outcome will be drawn from that will be one over eight. What do you get? Yeah, 0,125. 0,125. Therefore B is correct. And D, the number of permutation for the three items that can be selected. So n is seven. So you just need to say seven p. And they say three times, so that is three. On your calculator, you do have an NPR function. Look for it. It's sometimes written in orange. You will either press second function or you will press a shift button. So you will first press seven and then press. If I'm using a case, I'll press seven, shift. And you will press whatever the button the NPR is on. And then you press three and then say equal. What do you get? 210. 210. So then it means this is correct. Then the other, sorry, the other one says the number of combination. Our N is eight. Our combination, R is four. So also look for the sign. So you just press eight. And you will press shift. If you're using a case, you'll shift. And then look for the NCR function on your calculator. And press four and press equal. What do you get? 70. 70. That will be equals to 70. And that is the incorrect one. Yay, that is the incorrect one. Two, the distance from school to home. You are given the distance and they're asking you. Determine the probability that in a week or event W or caring, choose A or because letter W is an event that the distance traveled from school is less than three kilometers. So if I look at this, it is in order. So if it's less than three kilometers, therefore it's all of those. So in order for you to calculate the probability of W, event satisfying that, divide by how many there are. How many are there? How many kilometers do we have? You need to count the blocks. And how the observation satisfying how many events are in there. Three over 10. There are three events of less than three kilometers. And there are 10 outcomes. What is the probability? So that will be A. Three. But I want the answer. You just give me A's and B's. That is 0 comma? 3, sorry, 0 comma 3. The A is the right answer. Now we get to the other probability. So we need to test each statement and check which one is incorrect in relation to some of the probabilities. So given that A and B is equals to 0 comma 4 and the probability of B is equals to 0 comma 5. If A and B are independent, then the probability of A will be equals to 0 comma 8. So go back to what we have learned. The probability of A and B for independent event is the probability of A times the probability of B. If I'm given the probability of B and I'm also given the probability of A and B, I can find the probability of A because they told you that they are independent. So I can use that formula. So if I rewrite this, therefore the probability of, I will move A divide because I need to divide this side by probability of B to get rid of it. And when you get rid of it, since I don't have enough space, I'm reciting, then that will be the probability of B. If I want, I can move also this to the other side. So I can have space, the probability of A. So I can just substitute the values, 0.4 divided by 0. That is 0 comma 8. 0 comma 8, which means A is correct. Moving on, if event A and B are mutually exclusive, then the probability of A and B is equals to 0. Is that true? Mutually exclusive events means they both can't happen at the same time. And if they can't happen, then the probability of a joint event will be 0. So that is correct. If two events are independent, now we back to the independent. If two events are independent, do you still remember? It means the probability of, the event A and B have no influence on one another. So if I need to calculate the probability of A, given B, what will be that probability? If I need to calculate the probability of A, given that B already happened, what will be that probability if both of them have no influence on the other? It's A. It will be the probability of A. So therefore, this is the incorrect one. If two events A and B are complementary, so it means A is a complement of the other, of B or B is a complement of A, then A and B must also be mutually exclusive. Is that correct? Yes. Think about this as a coin. Think about it as a coin. If A represents the head and B represents a tail, they are a complement of one another. Can they be mutually exclusive? Can a head be a tail and can a tail be with the head? No. It cannot, right? So it means they are mutually exclusive. So that is correct because it cannot happen at the same time. Liz, I don't know why there's something. I know this is statistical language, but they really confuse people. So they could have actually said, then A and B must be different. Then that would make a lot more easier things. That's right. We use it as statistical language. Yes, because we need you to understand statistics language. And then when you go present your statistical output to your business, you talk business language. No, I will tell them it's different. Okay. So if two events, A and B are complementary, then the probability of A plus B equals to one. That is correct because the sum of all probabilities should be equals to one. C is the incorrect one. So here we have a sample of 800 Lenas in the table, which one we need to find which one is incorrect. So we have the table. There are no totals. So what I will suggest before we even start looking at the questions because anyway we're dealing with probabilities, probably we will have to calculate the probability at some point. There is also a missing value there. So we need to also complete that table. So what we can do is calculate the total there and calculate the total there. So quickly do the total before we go to the table. And you can just check if your answer for the total should give you, they said 800. I'm going to assume that that should give you 800, the total when you get to the end there should be 800. So the total for high school row is 320. So then you can calculate the total for Prema, which will be 800 minus 320. 480. That is 480. And then now you can take 480 subtract 240, subtract 168. Calculate 72 plus 48. 120. And 168 plus 112. 280. 240 plus 160. 400. So if we add all of them, they should give us 800. Okay, so now we're ready to answer the questions. I just want to go up. I don't want to get to the answers. Okay, I think that is enough. Okay, number eight. The event, high school and greater than six kilometers are independent. That's what they want you to validate. So how do we validate that events are independent? We validate this. You can go and find whether the probability of A and B is equals to the probability of A times the probability of B. If they are the same, so you can go find the probability of high school and greater than six. I should have written this one outside so that then here I can write the formula for the things we given. So here we know that the probability of A and B for independent events is the same as the probability of A times the probability of B. So we can find the probability of A, which is the probability of a joint event high school and greater than six kilometers. Let's calculate that probability. What is that probability? We need to go to high school and greater than 68, which is that. So it is 160 divided by 800. What do you get? Zero comma two. Zero comma two. Now we're done with that part. Let's go calculate the probability of A, which will be the probability of high school. Probability of high school is irregardless of whether they travel whichever kilometers. So they will be 320 divided by 800. What do you get? Zero comma four. Zero comma four. Now go calculate the probability of greater than six kilometers. Probability of greater than six kilometers is given by 400 divided by 800, which is this column. What do you get? Zero comma five. Zero comma five. Now we are ready to validate to see if the probability of high school times the probability of greater than. We need to validate that the probability of greater than six kilometers equals to the probability of that. So we need to take zero comma four multiplied by zero comma five. How much do you get? Zero comma two. Zero comma two. They are the same. They are equal. So the probability of high school and six kilometers is equal to the probability of high school and six kilometers. And therefore they are independent. That is one of the equation you can use. Remember, I just used one of them. I used this one for independent. You can also use this. You can also go and check the conditional probability of A and B, which you will use this formula to go and check. And you will use this formula to go and check. The reason why I didn't want to use this formula is because then I need to find the probability of A and B by using the normal probability question, which because I'm given events. And I need to divide that by the probability of B, which also. And then calculate the probability of a given. And then also calculate the probability of A and B. So I just use this one. You can also use that formula. So choose whichever one you want to use to validate your answer. Okay, so that is correct. Moving to B, event less than three kilometers and greater than six kilometers are mutually exclusive. I don't even have to go and calculate that or check because they are saying event three kilometers, which is that and event greater than six kilometers. So that event and this event, they say they are mutually exclusive. Can they the first question I need to ask myself is can they happen at the same time? No. No, so that is correct because they cannot happen at the same time. So they are mutually exclusive. The probability that a random randomly chosen learner is in high school and travels more than six kilometers from home is zero comma two. So here they want you to calculate the joint probability of the joint probability of high school and more than six kilometers from school, which is greater than six kilometers from school. We did that in A as well already. We did that in A already. Zero comma two. We did find it is zero comma two. Yes. I didn't even have to go and scratch your head again on this one. The probability that a randomly chosen learner travels less than kilometers, three kilometers from school. So regardless of whether they are from any of the high school or primary. So we just need to calculate the simple probability of less than three kilometers. So less than three kilometer, there are 120 learners out of 800. What is the probability? That's zero comma one. Zero comma one, five. The event, the event, three events less than three kilometers between three kilometers and kilometers. And greater than kilometer are not complementary. So that event and that event and that event are not complementary. Is that statement true or false? That will be false because less than three kilometers is a complementary of between and greater than. Between and greater than three kilometers is a complement of greater than and less than. So if I'm on this one, those are my complement for the less than. If I'm on the greater than, the between and the less than are the complement. Between and less than are complement of greater than and greater than six kilometers. So they are complementary events because if I take all of them, divide by the total, I will get one. The sum of all of them will, the probability of all of them will give me one. So they are complementary events. They are so that not makes it incorrect. Okay, so I hope you did copy those values. You have them really because I think we're going to use them again on this next one. So on this next one, since I don't have all the values, you have them. One, twenty, two, eighty, four hundred. One, twenty, I only remember this is four hundred. I'm going against this is two, eight. Go there. Oops. Oopsie daisy, four eighty, three twenty. That is four eighty. And three twenty. And answer here, it's eight hundred. Okay, so reading the question, it says, consider the table. Calculate the probability that a randomly chosen learner is in high school or travels three kilometers from home, between three kilometers and six kilometers from home. So they are asking you to find the probability of A or B, which we know. And they are not saying anything about mutually exclusive. So we know that the probability of A plus the probability of B is equals to minus the probability minus the probability of A and B. So we need to calculate all of those events. So it means we need to calculate the probability that a learner high school or travels between three and six kilometers, which then is given by the probability of high school. Plus the probability of between three and six kilometers minus the probability of high school and three and six kilometers. I'm running out of space. Anyway, I hope you are able to see what I wrote there. Which then means the probability of high school, regardless of where they are, the three hundred and twenty divided by eight hundred. Plus the probability of between two hundred and eighty divided by eight hundred. Because these are events minus the probability of a joint event, which will be that one, which is one, one, two, divide by eight hundred. Which is the same as three twenty plus two eighty minus one, one, two, if you want to divide by eight hundred. Zero comma six one. Zero comma six one. Therefore D is correct. Sorry, Lizzie, just go back to the question. So when you have an overall global events driven question, you obviously work with all the totals in the table. Yes. Okay. So remember that probabilities. Maybe it's something that I forgot to also mention there. Remember that probabilities are in decimals. So if here they gave you decimals or percentages or percentages. So which when it's when it's decimals, we usually call it probabilities. When it's percentages, we say proportions. So but they mean one and the same thing when you come to this. These are events with events. We always need to make sure that we divide them by the total so that we get them as decimals or probabilities as chances as a relative. Or we also call them relative frequencies as relative frequencies. Probabilities. Or we call them really relative frequencies. So. These are events events you always divide by the total. The grand total. To get them to decimal points. Okay. Same table. One twenty two eighty and four hundred. One twenty two eighty four hundred. What did we have? Four eighty and three. This should be eight and right. So the next question they are asking you is. How late the probability that a random chosen learner travels between three kilometers and six kilometers from home, given that they are in high school. So remember the probability of a given B. It's given by the joint probability of A and B divided by the probability of the given. Which is B. On this one they say calculate the probability that they travel between three kilometers and six kilometers. Given that they are in high school. So it means we need to find the joint probability. Joint probability of three. And six kilometers. And high school. And we need to divide that by the probability of high school. Joint event three kilometer and high school is one one two one one two divide by eight hundred. That is the joint probability divide by. The probability of high school. Is those one three. Three twenty divide by eight hundred. Remember the math. Math. This is division. So we keep the first fraction. We change the sign of the division to a multiplication and we swap. Eight hundred comes to the top and three twenty goes to the bottom. And eight hundred and eight hundred consoles and you are left with one one two divide by three hundred and twenty. So in a way if you are able to remember you can say is the joint divide by the simple. That's the easy way to remember that joint divide by simple. What do you get zero point three five zero point three five. So which is D is correct. Okay. And that is the probabilities. Now we need to move on to the next section which is the next study unit. So in the exam you all you will always get at least one question relating to this section. Or two. There might be two questions on probability because it's two questions per chapter. Or per study unit. So at least you will get two questions depending you will never know which one you will get. On this one you are getting more of the calculations and definitions. There was only one way you define something but you'll never know in the exam you might be asked to define something as well. So you just need to also know how to define some of the properties of a basic probability. So now let's move on and deal with study unit five. So study unit five study unit five is discrete probabilities. So now with discrete probabilities there are three things probably probabilities. There are three things. One, you need to know the basic concepts of discrete probabilities. Therefore it means you need to know how to calculate the probability using the table where you are given your x and your probability. You need to know how to calculate them. You need to know how to calculate the expected mean which is the sum of your probabilities times your x. Your observations so that times the probability and in them. You need to know how to calculate the variance or the standard deviation which is the sum of your observation minus your expected. Mean squared times the probability something like that. Or and you need to know your standard your variance and then you need to know your standard deviation which is the square root of your variance. You need to know that that is the basic one first done deal with it. You need to know all these three things because you might be asked to answer all of them. Okay, so that is discrete probabilities. Now there is also a need space so I'm going to get out all this. Then we need to know the following binomial distribution. So by no meow. You also need to know the properties of a binomial distribution. You will need to know the properties of a binomial distribution, meaning you need to know how to calculate the the mean which is the expected mean. Which is n times your probability of success remember with binomial probabilities. There are two outcomes they need to be independent with the entrails and so forth. You have two outcomes probability of success and probability of failure. Where your expected mean which is your mean is your n times your probability of success. You also need to know how to calculate the variance of that probability because that is n times the probability of success times one minus the probability of success which is one times the probability of failure. Or you need to know how to calculate the standard deviation of that which is the square root of your variance. The square root of your variance. And when you think that you are done, no you're not. You also need to know how to find the probabilities. Finding the probabilities on the table because then we have a table. Remember the table for the binomial. It has two sides. There is the table around it has the top part for. You have the top with the probability of success at the top and you have the left. With the probability, sorry with n and x value separating for each probability of success on the left. The top part of the table corresponds with the left side of the table. You also have the bottom part of the table. Because if I have, if these two tables were combined, like they could fit on one page, this would have been there. So you just remove this white area and these tables are combined. So you have the bottom part, which is the bottom. Which has the probabilities of success at the bottom and you have the right side. So the bottom probabilities correspond with your n and x on your right. So you need to pay close attention from the top part where it has a smaller probability of success. Which starts from 0,001 up until 0,45. 0,50. Whether you use the top or the bottom, the probabilities will always be the same. But up to this point, anything from the below 0,45. Your probability of success of less than or equals to 0,45. You're going to use your n and your x values on this table. That is one thing that you need to always remember. That is binomial. You also need to remember the following for Poisson. Poisson? Oh, not only. Oh, sorry. That's the other thing. Not only you need to remember that, you also need to remember that there is a formula. Because I didn't talk about the formula there. So you need to be able to calculate the probability also using the formula. Which is your n, c, r. I'm only going to use n, c, r. Because you know a way to find that times the probability of x. 1 minus the probability of n minus x. So you need also to know how to use this formula to calculate. Like I said, if you're not sure about the formulas, they are given me x. So this is the formula that I'm referring to. And this is the formula that I just gave you for the mean of a binomial distribution. Now I'm going to talk about Poisson just now. So with Poisson as well, you need to know how to calculate the expected mean of Poisson, which is your average, which is the same as your variance. You also need to know how to calculate the standard deviation, which is the square root of your average, which is the square root of your mean, your variance, and your everything. You need to know that with Poisson. You also need to know how to find the probability on the table, from the table, from the Poisson table. And you also need to be able to find the probability of a Poisson using the formula. Now I'm not going to write the formula down. I'm going to just show you the formula. I need to rotate again. Because I'm not going to write this formula for you. So you need to know how the formula for Poisson, which is e to the power of minus lambda dot the average to the power of x divided by your x factorial. The reason being why you need to know the formula in case they give you a formula is because they substituted the values and they ask you to find whether that is the correct way of writing the formula or not. You'll never know with this, but you need to be able to know how to use the formula or how to identify the formula. But when it comes to probabilities, I'm sure 100% sure that when you use the table, it becomes easy and it's quicker. So you need to be able to use the Poisson probability table. And with the Poisson, the tables are separated or divided into different averages or different lambdas or expected or variance or whatever the average or expected, whatever you want to call it. So the tables are separated by that. So you should be able to find the probability using this because for every x value, it corresponds with the probability table, the lambda table that you have. As you can see, this one where the lambda starts at 0,1 and set 1, the x value starts from 0 and set 7. One way starts at 2.1, ends at 3, the x value starts at 0, ends at 12, and the tables are. And also, depending on the questions, they will give you only the sections that you require for answering the exam question. So we might not find all the values when, if we are given bigger values that are not given on those tables. But in the exam, the values that they will give you will correspond to the table that you have in front of you. Okay, so before we answer the question actually, I also want to remind you that finding the probability regardless of which one we're finding, whether it's the Poisson, or it's Poisson, or it's binomial, or it's discrete. Remember the sign. Less than? What does that mean? Greater than? What does that mean? Greater than. What does that mean? What does that mean? At most? What does at most mean? Exceeding, greater, all those things. You need to be able to know what do they mean, because when it talks about less than, not greater than. When they talk about less than, it means it doesn't include that value. So if they say find the probability that X is less than three, then they're asking you to find the probability of X is equals to zero, plus the probability of X is equals to one, X is equals to one, plus the probability that X is equals to two. That's not include three. That's what they would have been asking you. If they say find the probability of greater than, then they will be asking you probability of X is equals to four, plus three, plus six. Now there comes a time where you will have to decide. If they ask you for the probability of X greater than three, for the binomial is easy on the binomial table, because the binomial table is divided by the value of n and the values of X. So your n, your X values will correspond to the number that you have for X. That's easy. So if they said it's greater than, it's greater than three. Therefore, it means you just add all the other values going there. So for this one, you will just add those two values that are remaining. For the others, you will add more than that. So for example, if I'm on, I'm not sure, because my tables are 10. So you will add all of those going down, things like that. For a Poisson, if they ask you to find the probability of a greater than depending on which table you are on, you see. So if I'm on this one, probability of greater than three, it means I'm going to add all of those ones. So if it was on this one, I will add only those one. But you will notice that the more you go down with the lambda values, the values are increasing. So if they said less than three, they four, if they gave you the lambda of 3.7, therefore it means you have to go and add all of those values. So you need to know how to define your less than your greater than. At least it means greater than or equal. At most means less than or equal. You need to know all those things. We did discuss this during our sessions. So I'm not going to dwell too much on that so that we can start answering the questions. We don't have enough time. So, oh gosh, I don't want to give you all the answers. Let's make it bigger so I can have space to write. Okay, so given the information, what are we looking for? The incorrect answer. So given the equation, what is the probability that two learners are absent? So we need to find the probability that X is equals to two on this table. You don't have to go and calculate anything here. You just go to where X is two and find the probability. Then that is the correct one. So the probability that X is equals to two because it says the probability that two learners. So it's the same as exactly. And that is correct. Sorry, I'm answering your questions now. Number B, the probability that between, oh that's the other thing that I didn't mention later on. So find the probability that between two and five learners are absent on a given day. Assume two and five are both not inclusive. So when they say that, therefore it means you need to go find the probability of where X is less than, it's between, sorry, it's between two and five. Because they say they are not inclusive, so it means we just do the less than. So if it was inclusive, we would put equal signs there so that then we include them. So find the probability that X lies between two and five. Therefore it means you need to add the probability where X is equals to three plus the probability that X is equals to four. What is that probability? It's 0.45. It's 0.45. So that is correct. The probability that at most two learners, what is at most? The probability that at most is less than or equal at most two learners. Less than or equal at most two learners. So it means including two, we need to find the probability that X is equals to zero plus the probability that X is equals to one plus the probability that X is equals to two. That is at most or less than. So you need to add all these probabilities. What do you get? 0.5. 0.5, which means that is correct. The probability that at least two learners, that will be the probability of X is greater than or equals to two learners. Since they are those ones, you say the probability of X is equals to two plus the probability that X is equals to three plus the probability of X is equals to four plus the probability that X is equals to five. That is one way of doing it. The other way of doing it, you could have just said the probability of X greater than or equals to two and that is the easiest way will be one minus the probability that X is equals to zero plus the probability that X is equals to one is the complement. So those are complement of those ones. So you just add only two things. So this is the wrong one. No, not the wrong. Oh, you mean the answer. No, not your wrong. No. So the answer here should be 0.5. Sorry, 0.3. What is the answer? 0.7. Yeah, that's what I got. 0.7. 0.7. So even here it would have been 0.7. You will get the same answer because you will say one minus 0.3 will give you 0.7. And this is 0.2 plus 0.3 plus 0.15 plus 0.0. You need to find a way of shortening your time in the exam. So this is the incorrect one. The probability that two between two and five that one was between two and five exclusive. So this one is between two and five inclusive. So it means it includes them. So you take 0.2 plus 0.3 plus 0.15 plus 0.05 which also you could have just said is the compliment. It's one minus the others. But yeah, the answer should be 0.7. Okay. Consider the following. What is the expected mean of this group? So we know that the expected mean is calculated by saying the sum of your X observation times its corresponding probability. So what you need to do is you need to multiply your X value and your probability. 0 times 0.1 is 0. 1 times 0.2 is 0.2. 2 times 0.2 is 0.4. 3 times 0.3 is 0.9. 4 times 0.15 is 0.6. 0.05 times 5 is 0.25. Add all of them. So in order to calculate the summation, we just add, add, add because that's what the summation is. Say adding all of them. 2.35. 2.35. So which is number B. And the next one it says calculate the variance of this variance. Are they the same as the ones that we did? Yeah. But anyway, it will not help us. So now what you need to do is the following. So we know what the expected mean is there at the top. Remember, we calculated it. We said it is 2.35. They are the same. We said it's 2.35 because we need that. So what you need to do here is to calculate. So let me first write the formula. So the formula we need to calculate because they say we need to calculate the variance. We need to calculate the sum of your X observation minus the expected value squared times the probability. So what I want you to do is to calculate your X minus your expected. Probably also just square the answer. So here you will say 0 minus 0.35 and equals get the answer and then square that answer. And then give me the answer there. So it will be 0 minus 2.35 which squared. So yeah, I'm looking for my copy. My thing is not doing what I wanted to. Which is equals to 5 comma and you need to write all the values. Don't run around of what is still in the problem mode. 5.5 to 2.5. So I'm only doing this part, the first part because I need to add to the next one. I'm not adding because we need to also multiply by the probability. So you need to get that minus that and then multiply by 0 comma 1 8. So multiply your X minus the expected squared. You can do it all at once anyway and multiply that with your probability. So my 5 point, let's go back minus 0.2.35 equals squared. I get 5.225 which I wrote there. Multiply the answer I get there with my 0.01. No, with 0.1, yes, 0.1. Because 0.1 is my probability, then I get 0.55225. Then this one I can add. So go to the next one. 1 minus 2.35 equals squared. I get 1.82825. You get the group, you get what I'm trying to do here. 1.8285. Then you need to multiply that with the probability of 0.2. Which I get 0.3645 plus I need to add the next one. The next one is 2 minus 2.35 equals squared. Which gives me 0.1225. And then I want to multiply that with 0.2. And the answer I get is 0.0245 plus. Then we do the next one where it's 3. 3 minus 2.35 equals squared. I get 0.4225. Multiply that with 0.3. Equals plus, no, 0.12675 plus. Need to do for 4. 4 minus 2.35 equals squared. Multiply. Oh, sorry. What do we get? 4 minus 2.35 equals squared. I get 2.7225. Multiply that with 0.15. And that gives me 0.408375. The last one plus 5 minus 2.35 equals squared. I get the last one is 7.0225. And I need to multiply that with 0.05. And the answer I get is 0.351125. Now I can add all of them together. So that I answer this question. So my variance will be 0.5525 plus. 0.3645 plus. 0.0245 plus. 0.12675 plus. 0.408375 plus. 0.35, my handwriting is that 7125. 7125. And that is equals to 1.83375. 1.83375. And the answer if I look at this will be C. C is my correct answer. Lizzie, is there a calculated version of this? No, there's no calculator version. You just need to calculate the manually. You can create an Excel spreadsheet and do it on your Excel, capture the data and do all the calculations. Otherwise you have to do that manually like that. So now we move on into binomial distributions of the discrete probability. In addition to the learner transport or learner's absence, the Department of Basic Education is also interested in addressing the shortage of teachers in the rural area. Previous study has suggested that one in every four rural area has a shortage of learners. So one of our four rural areas schools have a shortage of learners, of teachers. Suppose 10 schools are, 10 rural schools are selected independent of each other to check whether or not the school has the shortage of teachers. Which one of their following statement will be incorrect? So we have our probability of success. We have our N of 10. So you can go ahead and calculate the probability of success. What is 1 divided by 4? 0.25. Our probability of success is 0.25. That is our pi, our probability of success. So one, in a binomial experiment, each trial is independent of each other. That is correct because for all binomial distribution events, for all the binomial events, they need to be independent events. They should be independent events. The probability that five of the 10 schools have a shortage. So now they want to know what is the probability that x is equals to five. This does not mean that we use that 5 divided by 10. This is binomial distribution, so it means we need to go to the table. Or we can use our formula NCR pi 1 minus pi to the power N minus x. Whoever wants to use this, go ahead, use it. But we're going to use the table. So we need to go to the table. Let me rotate it again and again and again and again and again. I rotated it wrong. So we know what probability of success we're looking for is the probability of success 0.25. So it's at the top because at the top, we always use the smaller values. So since it's at the top, I must use the left hand side. What is our N? N was 10. So I must look for N of 10. So I must use N of 10. So this is my N of 10. So I know that my probability of success is 0.25. My N is 10. And I'm looking for the probability that x is equals to 5. So now when I scroll to the next table, the values here at the top disappears. But I have the values here at the bottom. Remember, the sum of all probabilities equals to 1. So what will be my complement of 0.25? It would be 0.75. So because 0.75 and 0.25 should give me 1. So I must go to 0.75. That is my 0.25. Or if I don't want to remember that complement, I will have to go and count from the left because it's easy. 1, 2, 3, 4, 5, 6. So on column number 6, that's where I must be. 1, 2, 3, 4, 5, 6. You see, it's the same. So that is where I must be. I must go to where x. This is my x value, where x is 5. And this is where you will need a ruler because if you're going to rely on your eyes alone, you might not get the right answer. So I'm relying on my ruler as well. I'm just going to draw a line across. So we know we are on this column. So that is my probability. 0, 0, 5, 8, 4. 0, 0, 5, 8, 4. So if you would have used this, your n is 10, your r is 5, and this is 0, 2, 5 to the power x of 5 times 1 minus 0, 2, 5 to the power 10 minus 5. You should get also the same 0, 0, 5, 8, 4. You should get the same answer. OK, so we need to calculate the expected value. And our expected value for Poisson is n times pi. Our n is 10. Our pi is 0, 2, 5. Do you have an answer? That is just 10 times 0.25. 2, 5. That equals to 2, 5. Therefore, this is the incorrect answer. The probability of a school having a shortage of 0, 25 while that of a school not having a shortage will be 0, 25. That is just a complement of the other because this is the probability that we've calculated as well. One in every school has a shortage of learners. It gives us 0, 25. So that is what they are saying. 0, 25 having a shortage and the complement of it will be 0, 75, which is correct. The probability that 4 out of 10 schools have a shortage of teachers. So we need to go and find the probability of x is equals to 4. So going back here, x is 4. We just go to the one at the top of where we were. 0, 1, 4, 6, 0. Easy. Or you can use the formula to calculate. The same formula that we used there. You can use that. Oh, gosh. You can use this formula. You can use that formula also. By substituting x or r with a 4 every way. And that will give you 0, 1, 4, 5. So our answer here is C. Okay. The next one. Suppose eight schools are selected. So our N on this one is 8. Eight schools are selected, independent of each other to check whether or not the school has a shortage of teachers. Similar to the previous questions that it has suggested that one in every four schools have a shortage of teachers. So we know that our probability of 1 out of 4, which is 0, 25, is still going to use that 0, 25, which is our probability of success. Number A, the probability that at most, at most one of the eight schools has a shortage, it means we need to find the probability that x is less than or equals to 1, which means we need to go and find the probability of x is equals to 0, plus the probability of x is equals to 1. So easy. We go to the table. We look for. We still use 0.25. Now, in state of 10, we're using 8. N is 8. So we are here. 1, 2, oh, sorry. 1, 2, 3, 4, 5, 6. 0, 25. Let me just highlight that. So we set, we're looking for x of 0 and 1. So we just need to go to 0 and 1. So that will be the two values there, which is 0, 1, 0, 0, and 2, 6, 7, 0. 0, 1, 0, 0, 1, plus 0, 2, 6, 7, 0. What is the answer? 0, 3, 6, 7, 1. 0, 3, 6, 7, 1, which is correct. And yeah, they say we need to find the variance and finding the variance. We say n times pi times 1 minus pi. So our n is 10, 8. Let's get rid of 10. It's 8. N is 8. Pi is 0, 2, 5. Times failure 1 minus 0, 25, which is equals to 1, 5. 1, 5, yes. 1, 5, that is correct. Probability that at least, at least 0 school has a shortage. So we find the probability that x greater than or equals to 0. Which should be equals to 1. I don't have to calculate that. Because if it says the probability of at least 0, so it means I must do the probability of x is 0, 1, 2, 3, 4, until where it ends. So at least 0 would mean adding all of them. Because it says at least 0, 1, 2, 3, 4, 5, 6, 7, 8. I must add all these probabilities. And when I sum all the probabilities in the sample space, they will all give me 1. So I don't have to even go. So this is the incorrect question or the incorrect answer. The probability that at most 8 schools, the probability that at most 8 schools will be the probability that x is less than or equals to 8. Which means you're going to have to add. I'm also going to not do that. What they are saying I must do there. So you're going to add. Let's remove this. So when they say the probability that at most 8, we are on 8. So it means adding all of them again. So at most 8 means all of them. So it should be equals to 1 as well. Because it's all of them. It's the sum of all of them. So sum of all these values because it says less than or equal. Adding up all of them. So that will be correct. The probability that at least 1. So this one says the probability that at least 1, which will mean greater than or equals to 1. Therefore I can just say it is the same as 1 minus the probability of x is equals to 0. Or alternatively what they are asking you to do, which will take you forever to calculate is at least 1. It means all of them till 8. So it means from here to there you need to add all of them. That is at least 1. Or alternatively you can just say 1 minus whatever that is remaining is only 1, 0, 1, 0. 0, 1, 0, 0, 1. That will give you at least 1, which is equals to 0, 8, 9, 9, 9. I think I even added many 9s. Which means that is also correct. Happiness. Now we move on to the binomial, the poison. Let me check how many questions are there. There are only two questions. So we should be done as well on time because I need to jump on to another class after this. Oh, these are your other favorites. My other favorites. Not really because they are writing their exam next week. So I'm just helping them. They are doing psychological research. So I'm just extending a hand. A helping hand. Suppose now you have knowledge that the number of grades, 10 absence at one school are poison distributed with the mean. With the mean of 4.2 per day, which means they gave us our lambda, which is 4.2 per day. What is the probability that at most, at most 10 grade 10 learners are absent? So the probability that X less than or equals to 10. So the 10 is this, not the 10 of the grade 10. Because that can be very confusing as well. If you get it. So at most 10 means you will have to go and add all of them, which is also, oh gosh, a very long way of calculating it. So we need to go to the table. Poison table. I have to rotate my picture. So now here is the challenge with this one because I don't think on here we have extended tables. I'll have to open another table. But in the exam, they will, they won't give you things that they don't supply you with. Let me open another table. I can just open a past exam, not exam paper. Where is the tutorial letter that we always used, which has tables in there? I think this one has a lot, a lot more than the one that we are using now. Yes, this one has a lot. So there we go. So going back 4.2. So we need to go to 4.2. So, and then the question was at most 10, 10, 10, 10, 10. At most 10. So here is 10. At most 10 means you need to add all of them. So it means adding all this. In state of adding all of them, we can say 1 minus the probability that X. It's greater than 10. So it should not include 10, but greater than 10. So therefore we will have 1 minus 0.0027, 0.0027 plus 0.09 and 3 and 1. 0.0009 plus 0.0003 plus 0.001. The other one was 0. I'm not going to add the 0. So I'm just only going to add only those because I'm calculating those ones. So instead of adding all 10 of them or 11 of them because they started 0, 11 values I can just add the remainder on the table. They should give me the same answer. So 1 minus, what do you get? 0.996. Oh, you already calculated the full answer. So 0.9996. So the answer for all the ones in the zeros is 0.004. It'll be 1 minus 0.004. If you're looking for that one. So this one, wait, I'm confused now. So in your brackets, inside your square brackets, the addition of all those values is 0.004. Yeah. So the answer then is 0.996. Yeah. 0. Which is B? Okay, which is B? Okay, the last question, then we are done. Suppose that the number of grade 8 absences follows a Poisson and the mean is 3. So now our lambda is 3. Use the formula. Now here is the tricky part. They say use the formula to calculate the probability that 3 learners are absent. 3 learners are absent. So our X is equal to 3 as well. So we know the formula. So we can write the formula for Poisson. The probability that 3 learners is given by. So the Poisson, let me write the formula e to the power of minus lambda times lambda to the power of X divided by X factorial. So, okay, so looking at the options that they have given. Let's rewrite the formula in a way that when we get to the end, it gives us the same answer e to the power minus lambda. I just rewrote them because at first I started with the e. So let's substitute the values. So our lambda is 3. Our X is 3. Our e to the power of, so because this is multiplying anyway. So that's multiplying e to the power minus 3 divided by 3 factorial. So let's look at the answers that we have here. If any of them has the same as what we have. So it means we need to do more. We need to do more. So doing more means we need to simplify this. What is 3 to the power of 3? 3 times 3 times 3 is 3 times 3. E to the power of minus 3 and 3 factorial is 3 multiplied by 2 multiplied by 1 is 6. So can 6 go into 27? 27 divided by 6 is 4.5. E to the power minus 3. Remember the multiplication. I don't know why they keep the multiplication there. So the answer is option D. So you see you need to also know how to use your formulas. Don't rely only on the table to calculate the probabilities because then you won't find the right answer. So that is D. And with that concludes our revision of chapter or study unit 4 and 5. See you on Wednesday when we do 5 and 6 and sorry, 6 and 7, possibly maybe 8. That was quick. That was a quick revision. So it means you still need to go and practice and study the last thing before you leave as well. Remember those who are on my UNISA group, you have the assignment exam type of a preparation paper or practice exam day. So ignore the one that says assignment 5. Is it a study unit 5 practice assignment? That was part of the lesson plans that we were busy with. I linked it there, but you don't have to do that one or you can do it. It's up to you if you want to because there are several, I think five questions that relate to the same concepts that we did today. They are from study unit 5. You can do that or not. But the most important one is going through the past exam paper that I have loaded on there. Those who are not in my group, I'm one minute late, those who are not in my group. I will find a way. I tried to find other mechanism. I can't find them. But I'm going to assume closer to the time. I will share the PDF format of that and then you can also do that. That is if I can find a solution for you. Because then me re-kept charging or typing that onto another platform is going to take me forever. Yes. I was going just to say try Google Forms or Microsoft Forms. Not Google Forms, Microsoft Forms. Sorry, but that takes a little bit of rework. Yeah, it will take a rework because when you answer a question, I need to give you a response to say you are correct. And if you are not correct, I need to give you a response to say why that answer is not correct and how can you fix it next time. So it will need a system like my UNISA. So I will try and find another. I do have a system that I use for the grade 12. But it's time consuming. I don't have enough time of the day to do all these things. But anyway, thank you for coming because I'm late for my other session. Thank you for coming and enjoy the rest of your evening. I will post the recordings and let you know once it's updated on my UNISA YouTube on Teams as well. Thank you. Bye. Thank you. Bye.