 I hope the recording has picked. It seems like it. Okay, so welcome to another session where today we're going to continue. We looked at study unit, study unit one, two, and three. We did the revision on that. So today we're going to look at study unit four and five and we'll see how far we get with answering the questions. For every bit of the study unit, I'm tempted to do a recap on some of the concepts that you need to remember when you go and answer the questions. But I also am tempted not to because it will save us a lot of time. So we will do the recap as we look at the questions so that then it makes it easy for us to continue as quickly as possible. Today, since we also need to use study unit five, we will have to have the tables. So I hope you would have downloaded the tables and have them ready some way. Tables are part of the notes that I've shared as well. So everybody should have the same table because we're going to use them. We're going to work through the same tables. So study unit four talks about basic probabilities. You need to always remember with basic probabilities how to define it. You need to also remember how to define what is a simple event, what is a joint event, what is a complement of an event. You need to know the basic properties in terms of probabilities that the sum of all probabilities are equals to one. You need to know the addition rule and you need to know the multiplication rule. You need to know your conditional probabilities. And also, not only that, you need to know that probabilities you can summarize them in terms of a VIN diagram or a cross-templation or a decision tree. You need to know those as well. You also need to know about mutually exclusive events, exhaustive events and so on. But on top of all the probabilities that we deal with, you also need to know the counting rules because they are also part of the section where we deal with probabilities as well. So you need to know your counting rule, your factorial combination, permutation and the multiplication rule and how to calculate each and every one of them and identify when you read the question, when is it a factorial, when is it a multiplication, when a multiplication rule and when it is a combination or a permutation way. The permutation clearly there is an order and combination, there is no order, things like that. You need to know all those, how to define each and every one of them and that is that you need we will talk about study, you need five later on. And you need to know that probabilities are always between zero and one. So therefore it means if it's a decimal, it is a probability. If it's a whole number, it is an event that you can use to calculate the probabilities. All right, I hope those are the things that makes it easy for you to identify as you read the question. So I expect you to have already went through study unit four and then we can answer the questions. Okay, so the first question on here it talks about the counting rules and that's what I was talking about when I said you need to always remember the counting rules and know how to define also the probabilities. Okay, so let's answer these questions and learn ourselves of what we need to always know. Which one of the following statement is correct with regards to experiments, counting rules and assigning of probabilities? A, an experiment described by a sequence of four steps with three outcomes possible for each step one and step two. Four outcomes possible for step three and two outcome possible for step four will have a total of 24 outcomes. So you need to be able to look at that question and ask yourself, when I'm given these outcomes and they're telling me that there are four steps and each step has some outcomes out of it. What is it? Is it a factorial? Is it the multiplication? Is it a combination? Or is it the permutation? Or is it a probability question? So you need to know which one to use on there. Which formula to use as well, remember? Factoria. It will be n factorial. Multiplication. It will be m times n, depending if there are five. So it will be m times n times p. If there, sorry, if there are three, it will be like that and so on. And then combination, remember? The formula for combination will be n, c, r and for permutation it will be n, p, r. You need to be able to identify which one of them to apply on these questions. Number B, the number of combinations of six items that can be selected from a group of eight items is number C, the number of permutation of five items that can be selected from a group of eight is. And in an experiment with six equally likely outcomes, each experimental outcome has a probability of n and that is what we call assigning probabilities. We know that when we assign a probability, an experiment will have the observation satisfying that event divided by the number of the total outcome or the sample space. And that is the formula that you would use when you assign probabilities. So you will need to know how to calculate that. A relative frequency method of assigning probability is appropriate. When the data is available to estimate the proportion of the time, the outcome will okay if the experiment is repeated a large number of times. And yeah, it's more about the definition. Is this a definition of an experiment, a definition of an observational experiment or subset? So let's get to the question. Which one of these is correct? So number A, which formula are we going to use? Is it the factorial, the multiplication, combination or permutation? And if when you choose which formula to use, please remember to tell me the answer as well on how do we calculate it. Number A, how do we answer A? Nobody, no one. Nada, next. Is it going to be a factorial? A factorial is if they have given you one value, let's say in a race, there are seven positions. How many number of ways can I win a position? Or can I get a medal in a race? So if only seven people are getting a medal out of their own race, there are seven positions that get awarded the medal. How many number of ways can I get a medal? That is if you want to use factorial because then you say seven factorial that for it will say it's seven times six times five times four times three times two times one will give you the answer. Or you can use your calculator to calculate seven factorial where on your shop calculator, it will have an N with a exclamation mark. When you are using a casual, it will have an X with an exclamation mark. That is the function that we will use. So with this question, because they had given us four steps with three outcomes, then we're going to use a multiplication rule. As we're going to say it will be, they say for with three outcome possible outcome for step one and step two. So step one has three outcomes. Step two has three outcomes and step four has, step three has four outcomes and step four has two outcomes. So we're going to take only the number of outcomes and multiply them. So step one and two, they said they've got three outcomes. So it's three times three times four times two. What is the answer? Am I alone today? The answer is 70. 72. 72, which makes this is incorrect because they said the total is 24. Moving to B, B says the number of combinations. So already would be they have given you a hint in terms of which formula you will have to use. You will use a combination. A number of combination of six items that can be selected from eight groups. Always remember, the bigger number will be equivalent to your sample space, which will be your N and the smaller number will be your R for your X. So therefore it means for this question, you will answer it by using your N is eight combination and your R will be six. And if you didn't know, do you know how to use your calculator to calculate combination, which is the shortcut. I just want to stop sharing. I want to toggle between. So I only have at the moment open. I have my sharp calculator open. So those who have a cashier calculator, you will have to look at your ratio. You will press. I'll tell you where is the cashier calculator. Your combination function, it's on the division side on your cashier calculator. So let's first deal with the shop. Since I have it here open. So for a shop calculator, your combination is on button number five. Those who have a shop calculator or the financial shop calculator, it will almost be on the same function. So because it's written in orange, we're going to use the second function. For those who are using a cashier, you will use the shift. So we first press the N, which is the value that we put first, which is eight. So you say eight and you go second function and you press the NCR and you press six and you press equal. And that will give you the answer of 28 and they said it is 20,000. So it means that is not correct. Let me know if you don't know how to use your calculator. It's very important that you know how to use your calculator. So do number C. I'm not going to calculate it for you. So it is permutation. So the formula for permutation is that. So it's NPR. So also do likewise and calculate and tell me what is the answer that you get. Sorry to disturb. My calculator is giving me one. What calculator are you using? I'm using a shop EL531WH, the advanced DAL, the old one. Yeah. So do you have NCR? Are you able to see where your NCR is? Which number is it? N is the function NCR. Yeah. Number eight, yes. Is number? Eight. And what color is it written in? Is it written in orange or is it? Yeah. It's written in orange and then on the right hand side there's an SX. Okay. No, it's fine. So press eight and then press the orange button on your calculator, which I'm going to assume it's second function. So you will press second function and then you will press that button that says NCR. You will press the function where you see that NCR and then press six and then press equal. Okay, there's something. Let me try and set it back to standard mode because... No, it should not be in standard mode. It should be in your normal mode. So take your calculator back to normal mode. If it's in standard mode, you need to take it back to normal mode to normal. So you will press mode zero and it will go back to normal mode and follow the same steps as I have shown you. Okay, let me not take time for everyone. Let me try and figure it out. Do you have the NPR? Yeah, NPR is number nine. Okay, so you do the same with NPR. So for this one, it will be eight, eight second function and then you will press the function that has the NPR and then you will press it says five. So this one will be five and then you press equal. Do you guys have the answer for number six? It gave me six seven two zero. Zero. Six seven two zero. Six seven two zero. So which means this is incorrect. So the answer should be six thousand seven hundred and twenty. Are we all winning? Yeah. On a cashier, the buttons are multiply and a divide. So they just below the delete button and the AC. Yes, on a cashier, it's on the NCR, it's under the divide on the button divide and the NPR is on the multiplication and you use the shift in state of second function. So for those who are using a cashier, you will press, for the first one, you will press eight and you will say second function, not second function shift, you will say shift because we're using a cashier, you will say shift and you will press the division button and then you will press six and it should give you 28. For the second one, you will say eight shift and you will press the multiplication button and you will press five and the answer should be six seven two seven. Let's see, I've got a question. How do you know out of the two numbers which one comes first? So you've got six items and is it always the bigger one first? Remember the big one needs to be N. N is big. R or X is small. So the bigger one. So you'll always remember that the bigger one between the two, you look at the bigger one, it will be your N. Okay. Okay. So D, yeah. And then D, in an event with six equally outcomes, each experimental outcome has a probability of zero comma one four. Now, if there are six and say the the outcome has the same probability, therefore it means we're going to use the probability function to calculate. So we know that there are six outcomes. So outcome one, outcome two, outcome three, outcome four, outcome five, outcome six, regardless of what they have. So these are the outcomes of this sample space. To calculate what is the probability of getting a one, it's X over N. Calculate the probability of getting a two, it's X over N. Calculate the probability of getting a three, it's X over N. So there is only one outcome that you count, so therefore it will be one over six for every one of them. Calculate the probability that an outcome comes out from a three will be one, because there is only one three. For a four, it will be one. For a five, it will be one. For a six, it will be one. So it will always be one over six, one over six, one over six, one over six. Hence they say in this experiment with six equally likely outcomes, each experiment will have the probability of this. So what will be the outcome of this likely experiment? One over six, which is equals to zero comma? One over six, one seven. So which means that is also incorrect. The last one it says it just gives you a definition of the experiment anyway, because an experiment is a relative frequency method of assigning probabilities when a data is available to estimate the proportion. Because when we calculate probabilities, remember probabilities are like real proportion, because the proportion of selecting a value of two is one over six, which is zero comma? One seven, that's a proportion. Estimating a proportion of the time of the outcome will occur if the experiment is repeated. And if this gets repeated over time, you can also calculate those proportion of probabilities. So this is just a definition of experiment. So you need to go and read how do you define what an experiment is, what is an observer, what the types, different types of probabilities you get, what is the end event, what is an outcome, all those things you need to know them because they can be asked in the exam as one of the options. So the only correct answer here is E. Moving on to the next question. Consider an experiment with four outcomes with associated probabilities, the probability of event one, probability of event two, probability of event three, and probability of event four. Which one of the following outcomes does not meet the basic requirement of assigning probabilities? Now you need to think carefully when you look at this. Ask yourself this question, knowing the basic concepts of probabilities. What is the most important thing that you always need to remember and validate? If you have different probabilities, the sum of all probabilities should be equals to? That's your question. The sum of all probabilities should be equals to one. Now looking at this, they are asking you to do that. Just act. So it means the sum of all these probabilities because it says consider the experiment with four outcomes with associated probabilities. Which one of the following outcome does not meet the basic requirement of assigning probabilities? You can do the addition. Let's start with A. A, how many probabilities? So the sum of all those probabilities are equals to? One. Are they all equals to one? So the sum of all these probabilities is equals to one. And the second one? It is equals to one. It's also equals to one. What are they trying to get to on this? Because all of them are equals to one. Oh, because we're looking for the one that does not respond. Okay, right. My bet. And number C? Equals to one. It's equals to one. So the sum of all probabilities equals to one. Number D? 0.8. The sum of all probabilities is equals to 0.8. And number E? 1. The sum of all probabilities is equals to? 1. Equals to one. So which one of these is incorrect? Number D. Question three. Consider two events A and B. And they say the probability of getting an event A is 0.55. The probability of getting event B is 0.22. And the probability of A and B is 0.11. Which one of the following statement is correct? So we're looking for the correct statement. Now we need to validate each and every statement. All right. So we start with statement number A. What do we know about mutually exclusive events? If event A and B are mutually exclusive, the probability of their joint event A and B will be? 0. It will be equals to 0. From the statement that they have given us in the beginning, are they equals to 0? No, they're not. No. So therefore, that is not correct. Therefore, it means event A and B are not mutually exclusive. Remember, the tricky part here is to trick you in terms of you not knowing what to do with the statement. You just need to remember the basic concepts that you know and validate or confirm if that statement is true or not. We know that from the given statement, the probability of A and B, which is the joint probability of A and B, is 0.11. Therefore, they are not mutually exclusive. So it's incorrect. Number B, if event A and B are independent, then the probability of B given A is 0.55. If they have given you the probability of B given A, you can calculate that to validate it. First, they also say if event A and B are independent, when event A and B are independent, what do we know? We know the following. If A and B are independent, the probability of A given B will be given by the probability of A because they are independent. B has no bearing on what happened to A. Given that it happened, it doesn't have any bearing on what is happening through the probability of A or to the event A. Because we know that the probability of B given A is equals to the probability of B. And this is what we discussed. If you know this, for independent events, the conditional probability of the probability of A given B is the same as the probability of A. The probability of B given A is equals to B. That is if event A and B are independent or not. Let's go there and check. They are saying if event A and B are independent, then the conditional probability of B given A should be equals to 0.55. Is that true? Let's go back to the statements given. What is the probability of B? 0.22. From what do we know in terms of the independent of events? We know that the conditional probability of B given A is the same as the probability of B. So, that is not correct because the probability of B is 0.22 and that is incorrect. With basic probabilities, you need to know your probabilities. When you go write the exam, it's easy because I don't want to say this because these things are recorded, right? We can discuss that after. But it's easy to remember all this because it's part of your notes. You get me. All these things are in your study guide. There is a decision tree in your study guide that explains all this nicely and it tells you for independent events, they give you those things. So, if you can have next two summary notes and practice with those summary notes, in the exam, it will make it easy for reference because you will remember that because you have seen it somewhere before. It will help you, right? You can have a discussion offline. Okay. So, maybe you do understand what I'm trying to tell you here because some of the things I can say them online, when it's a recorded session like this. These things, they get watched by very important people. So, let's move on to C. C says the probability of A or B is equals to 0.66. So, this is what we call an addition rule. What do we know about the addition rule? This is what we know. The probability of A or B all can use this function. So, if you don't know how to write it or it's union, remember that. Then I'm interchanging with symbols or is the same as union. So, it's the same as the probability of A plus the probability of B minus the probability of A and B. You are given all the values. All what you need to do is substitute into the formula, calculate and check if that is correct. Let's do that. What is the probability of A? It's 0.55 because everything is given to you in the statement. Go back to the statement with the probabilities. Plus, what is the probability of B? 0.22 minus. What is the probability of A and B? 0.11. Calculate. What is the answer? It's 0.66. It's 0.66. Therefore, it means C is correct. In the exam, you will stop right there and move on to the next question. Number D. D says, if A complement is the complement of A, then the probability of A complement is equals to what is the complement of A? What is the probability of A complement? It's 1 minus the probability of A. What is the probability of A? 1 minus 0.55 and that is equals to 0.45 and D will be incorrect. That's how you will validate the questions and find out whether they are correct, the statement that they have given you. Since I've done A, do B do E. What is the probability of B complement? I think it's 1 minus 0.22. It will be 1 minus the probability of B, which is 1 minus 0.22 and the answer is 0.78, which makes this incorrect. That's easy, right? Please do not move on this slide for just one minute. You will let me know when you want me to move. You may move. Thank you. All right. Moving on to the next question. Given the table they've given us, right? The World Health Organization notes that gender identity is a social construct that varies across culture. The contingency table above shows a combination of probabilities from different gender identities and how each group generally feel marginalized within their spaces. Which one of the following statement is correct? We are looking for the correct statement based on this. Looking at this table, like I said in the beginning when I was introducing this, how do we identify whether we're given events or are we given probabilities? Based on this table that we have, are these probabilities or are these events? Probabilities. These are probabilities because there is different ways of calculating probabilities if you are given events and when you are given probabilities. If these are probabilities and we have made that observation now, these are probabilities. If we know that these are probabilities, there is no need for us to go and calculate the probability using the probability of an event. It's the same as x over n or the probability of a joint event. It's x over the sample space. We don't have to because these are probabilities already. These are calculated. We just use them in the formulas and we just use them to answer the question. Now, looking at the table, question option A says, event non-binary and low are mutually exclusive. Non-binary and low are mutually exclusive. Is that correct? What they are saying is the probability of non-binary and low for it to be mutually exclusive, it means they are equals to zero. So the probability of both are equals to zero. So number A will be correct because the probability of low and non-binary is equals to zero. The event non-binary and low are independent of each other. How do we validate that? Let's assume that A was not correct, then we move on to B. How do we validate that non-binary and low are independent? We just did that. In the previous question, we spoke about this, the conditional probabilities. So let's check that. We need to find out whether because they say they are independent of each other. What we need to find out is that the probability of non-binary given low should be equals to the probability of non-binary. That's what we need to prove. We know what is the probability of non-binary. The probability of non-binary is the sum of all these values. You add all of them. Because on this table, you need to have total. On the total, you get simple probabilities. So 0 and 0 plus 0.3 is the same as 0.03. That's what we have, 0.03. Now we need to find out if this site is the same as that site. How do we find that out? We need to use the conditional probabilities. We know that for conditional probabilities, let's write the formula. The probability of A given B, conditional probability, it says, is the probability of A and B divided by the probability of B. That is the formula that we're using. So since we have N and L, we're going to find the probability of N and L divided by the probability of L. So what is the probability of N and L is 0? It is 0. What is the probability of L? The probability of L is 0.76 because we need to also add the total here. 0.76 plus 0 plus 0 plus 0 is 0.76. Therefore, this is equals to 0 because any number, 0 divided by any number, it will stay 0. So what we do here is that the probability of L divided by L is not equals to the probability of N. So therefore, they are not independent because if they were equal, they would be independent. We know that if they are equal, they are indeed independent. I don't know how to write independent. I hope that is the way to write independent. So we've established that this side is equals to 0 because 0 divided by 0.60 and the probability of N is 0.03. So they are not equal. So they are not independent. See, how do we answer C? The probability that a randomly selected person identifies as binary or feels high level of being marginalized is 0.03. So here they want you to calculate the probability of non-binary or high, which is the same as the probability of non-binary plus the probability of high minus the probability of joined non-binary and high. Because of this all, you will have to use this formula, which is the same formula as we used previously, the probability of A or B. Same as the probability of non-binary or high, it's given by the probability of non-binary plus the probability of high minus the probability of non-binary and high. So let's find the values for high. So I can just add all of these values, calculated the total. This will be 6, 4, it will be 10, 0.10, 0.14, 0.14. And the probability of non-binary we already calculated the total. And the joint of non-binary and high is 0.3. So we'll use those three values. We'll calculate. So let's go. We have the probability of non-binary, which is 0.03 plus the probability of high, which is 0.14 minus the probability of a joint of non-binary and high, which is 0.03. And what is the answer? The answer will be 0.14. And therefore, this will be incorrect. And that's how you will validate that question. Let's see how we answer. You must let me know if you are still getting confused. All right. How do we validate D? D says the probability that a randomly selected person identifies as cisgender. Then it means simple probability cisgender, which means we need to calculate the total. That's where we will find the probability cisgender. The probability cisgender will be given here. So by adding 0.76 plus 0.7 plus 0.3. What is the answer? 0.86. So that will be 0.86. Therefore, D is incorrect. Because cisgender is on its own, then it means it is a simple probability. Remember, on a contingency table inside here, these are joint probabilities. The totals are your simple probabilities or what we call marginal probabilities. Okay. Let's look at E. Oh, why am I crossing out E? E, the probability that a randomly selected person feels high level, which is the same as simple probability of high level. So that will be the probability of high. What is the probability of high? We calculated the total is 0.14. And that is not the same as 0.1. And that is incorrect. And that's how you will validate this. For moderate, you just add all of them. That will be 0.10. For other, it's 0.06. For transgender, 0.05. The total, the sum of all values should always be equals to one. That is if I need to complete the entire contingency table. Right? Easy, right? So can I please ask, is it wise to always complete the contingency table first before answering the questions? Or you can just answer the questions without even... You can, like we did, we completed what we wanted to complete at that point, right? Yes. The other thing is in the exam, you're writing against time. Spending time doing other things that are irrelevant to the question might take you longer. So in the exam, minimize your time. Take shortcuts, take your time. It's up to you. But make sure that you look at the time, like right now, literally, we are on question four. And it's eight o'clock. So it means we took almost... If we say we started at Korapast, we took almost 45 minutes answering four questions. In the exam, you don't have that luxury. Because you're writing out of two hours, 45 minutes, four questions, out of 25 questions, or 15 questions, you're not going to make it. So minimize the time. Think about it. If you know that you're going to work quicker, you can write the totals quickly. But the other thing is you are writing online, so you might not have the luxury to do that. Because it means you would want to copy this table again on the piece of paper and write the values again, and which won't help you that much. Okay. Thank you. Okay. Looking at the same table, I'm not going to complete all the values because we have them. Hence, I've completed them here. So we'll toggle between the two to find the values if need it. Given that a randomly selected individual identifies as transgender, what is the probability that they feel a moderate level of being marginalized? That is, you need to calculate the probability of moderate given that the person is transgender. So since we have a question, we need to calculate it based on the conditional probability of moderate and transgender divided by transgender. So what is the joint probability of moderate and transgender? What is the probability of a joint probability of moderate and transgender? It's 0.01. It will be 0.01. And what is the probability of transgender? It's just aiding all these probabilities, which is 0.05. And that is 0.05. Calculate and tell me what is the answer? 0.02. 0.02. It's option A. Now we're moving into discrete probabilities. Remember that with discrete probabilities, there are two things or three things that you need to always constantly remember, right? The discrete probabilities, you will be given a table or you might be given a table with the x values and the corresponding probabilities. You need to know how to define what a discrete probability is. You need to know what are the properties of a discrete probability and so on. You also need to know how to calculate the values or the probabilities. And remember the sign and the language. Remember, what does this mean? Oh, what does this mean? At least when I'm referring to at least or when I'm referring to at most. What does that mean? Right? What does this mean? What does this mean? What does this refer to? Right? You need to be able to know these things, especially when you work with probabilities, discrete probabilities. Okay. So this question is asking, oh, you need to also know the formula to calculate discrete probabilities, the expected value, the mean, which is the mean, the expected value, which is the mean of discrete probability. You need to know how to calculate the standard deviation and the variance. Okay. So what is the expected value of a discrete probability with n of 8 equally outcomes? There are multiple ways of doing this and there is the other simple ways of doing this. Because there are eight of them. So remember, the expected value, which is your expected value of a discrete probability is the sum of all probability. The sum of your x observation multiplied by its corresponding probability. That is that. So you can either say, what is the probability of the equally likely with outcomes? In question one, we did this. So we said the likely outcome, which is the probability of x for all outcomes, will be 1 over 8 for this because there are eight of them. So if there are eight observations, so it means we're going to say 1 times 1 over 8 plus 2 times 1 over 8 plus 3 times 1 over 8 plus 4 times 1 over 8 plus until we get to 8 times 1 over 8. I'm not going to do all of them and that will give you your probability. Or the easy way, because they have the same outcome, we can take it out as a common factor. We can take out the sum of x, p, x. We know that the probability of x is a common factor. It's the same for all the things. We can take it out. It will be the probability of x times the sum of x. What does that mean? It means 1 over 8 times 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8. Use whichever method you feel comfortable with. They will give you the same answer. So what is the answer? You can calculate 1 over 8 is the same. Those who don't have the case, you'll calculate it. You can calculate this manually and say 1 divided by 8 is 0 comma 125 and then you can just put it into brackets. So you can just say 0 comma 125 into bracket 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8. Close bracket and say equal. And that will give you your answer of 4.5. Quite easy. You can do it that way. Or if you want to follow the formula that we use, say 1 times 0.125 plus bracket which will take you forever. 2 times 0.125 close bracket plus 3 times 0.125 close bracket plus 4 times 0.125 close bracket plus 5 times 0.125 close bracket plus 6 times 0.125 close bracket plus 7 times 0.125 close bracket plus 8 times 0.125 close bracket equal 4.5. You still get the same answer. Save your time. Find the easiest way of doing things. Shorter methods or longer methods. Some of you might have templates from your tweeters that gives you how to calculate these things on your excel or somewhere. Use that. Okay. And that's how you will answer this question. Any questions? I know that I did it too quick on the calculator but I guess you understand what I'm trying to get at. There are two methods of answering this question. This will not work for other questions, especially if they have given you questions like this. Because there are different probabilities, it will not work with this method. The second method, it will only work with the first method because the probabilities are not constant for some. Only when they are constant, you can take it out as a factor and then calculate the rest. That's what we call factorization in meds. Okay. So going on to question number seven. Consider the following discrete probabilities. Like I said, you will be given it can either be horizontal or vertical. So it can be like this. X with its corresponding probabilities or like this. It doesn't really matter how the table they have given you looks like. So sometimes they might give you X, not probabilities, but the frequencies or the count. Then you can use the frequency or the count to calculate your probabilities. It might work like that where they have given you the actual values, the actual events and you need to calculate the probabilities. So here they give you your outcome and your probability. So your X versus your probability. The same question. Calculate the expected value. So you need to calculate the expected value of this question. The sum of X observation times corresponding probabilities. You can just say the one multiplied by zero comma three. That will be X times Px. That will be zero comma three. But we have a question mark. The question mark is the sum of all these values minus them from one. That will give you what is the answer there? So one minus into bracket. If you like it, you can say one minus into bracket point three plus point one seven plus point one seven plus point one seven. Those brackets equal zero point one nine. And that is the answer here. Zero point one nine. And now multiply by two is zero point three eight. Zero point three eight. You can do it that way. And the next one, one seven multiplied by three is zero point five one. And the next one, one seven multiplied by four is zero point six eight. And the next one, one seven multiplied by five is zero point eight five. If you add all of them, that will be your outcome. That will be your summation. And that will give you the answer. So what is the answer? I must not do all the activities for you. Two point seven two. Do we all agree? Yes. That is two point two point seven, which is option number one eight. So can you see the result from how you got that question mark? The question mark, you get it from one minus the sum of all the other values. Zero point three, because the sum of all probabilities should be equals to one. Zero point three plus zero comma one seven plus zero comma one seven plus zero comma one seven. You just use the open bracket, close bracket to get that. Otherwise you can add all of them and then subtract them from one. Lucy, I don't know if I'm doing something wrong, but if I add them all up I get two point four five. When you add which one now? And the last column that you've... There's some somewhere you missed one. Yeah, somewhere you've missed something. Recalculate. You know what I've done? The first one added point zero three instead of point three. Sorry, I'm stuck. Thanks. No, Aris. Question eight, part of the discrete probability is there are two sections that you need to always know and always remember is the binomial and the poisson. Both of them, you need to know the properties of each. You need to know how to calculate the mean, which is the expected value, the standard deviation, and the variance. You also need to know how to calculate the probabilities if they ask you to calculate the probabilities. Now, with both binomial and discrete... No, binomial and poisson, you need to use the table to calculate the probabilities. I hope you have the tables with you, but we would use the table. We will need the table right now. Let's first answer this question and see if we will need the table. A building inspector would like to conduct an inspection of 13 randomly selected new building houses or new built houses to check whether or not they comply with the municipal regulation. The inspector knows from the past experience that eight out of 10 new built houses will comply with the municipal regulation. Which one of the following statement is incorrect? So we're looking for the incorrect statement. What is it that they have given us here? They told us that our N, which is our 13 randomly selected new built houses, that they have checked. They also told us that eight out of 10, which is the probability of success or failure or whatever the one that they say, the inspector knows from the past experience that eight out of 10 new built houses will comply with the regulation. So it means they have given you the probability of success, which is equals to eight divided by 10. And in terms of probabilities, which is in decimal form, eight divided by 10 is equals to 0.8. So since we have the probability of success, we have our sample size. Let's answer the question to find out which one is incorrect. You need to know the properties of a binomial. It has same tau, independent event or outcomes, two outcomes because it's binomial. Those are the properties, right, that you need to always constantly remember. They are two outcomes, probability of success or failure and so on and so forth. So let's see. A, the experiment can be described as a binomial with 18 trials. Is that correct or incorrect? How many trials do we have? What is the N? It's 18, right? So therefore, this is correct. So it's not incorrect. It is correct. So we move on. Two outcomes are possible for each trial, which there are two outcomes, comply with the regulation success and doesn't comply with the regulation failure. Is that correct or incorrect? Correct. That is correct because it's binomial. It has two outcomes. It's a success and a failure. Number C, the probability that a newly developed house doesn't comply with the municipal regulation is 0.8. Is that correct? What is a compliment of a success? It's a failure, right? That question is asking you to calculate the probability of a failure. That's not comply. If our success is 0.8, therefore the failure will be 0.2. 0.2 because 0.8 minus or 1 minus 0.8 will be 0.2. So that is incorrect. That is the incorrect statement. What is the expected value of newly built houses? So you need to know how to calculate the expected value of a binomial probability. The expected value of a binomial probability is calculated by N multiplied by the probability of success. Your N is 18. Your probability of success is 0.8. And what is the probability? The expected value, sorry. 18 times 0.8 is 10.4, which means this is correct. Each inspection constitute their trial with independent results. Is it true of false? So we know that it's binomial. They are independent, so that is correct. So those are the things that you need to, the properties that you need to remember and know about binomial. Next question, consider the building inspector again and blah, blah, blah. Now they have built 18 new houses. So this is our N. 8 out of 10, which means our probability of success is still 0.8. But we have a new N. Our N here is 18. What is the probability that only five out of, only five out of 18 built houses will comply with the municipal regulation? Now, looking at the options, it means we need to use the formula and not use the table. What is the probability? It's NCR because I'm going to use the shortcut. You've learned how to calculate NCR, right? I'm not going to use all those factorial divided by X times N minus X factorial, which this NCR is the same as N factorial divided by N times N minus X factorial. Or is it X? I can't even remember one of the two. Something like that. But the formula, I don't know it by heart, so let's not, let me not do that to myself. I remember the MCR and this will be times the probability of X times one minus probability of N minus X. So we know that we need to calculate the probability of five. Our N is 18. So it's 18. We are five. Our probability here is 0.8 to the power of X of five times one minus 0.8 to the power of 18 minus X. So calculate NCR and tell me what is the answer? Just the NCR? 8,568, right? And we can say multiply. We remove the bracket and put multiply by 0.8 to the power of five, multiply by one minus 0.8 is the same as 0.2 to the power of what is 18 minus five? Sorry, my sister, my apologies. Where you say 0.8? Is it not supposed to be 0.2 to the power of five? Well, but our probability of success is 0.8, right? Yes, yes. Yes, so 0.8 is our probability of success. One minus the probability of success is 0.2 because it's one minus 0.8 will be 0.2. What is 18 to the power 18 minus five is 18. So look for this answer. No, my apologies. No worries. We will blame it on half past eight. Which is option? Option one. Really, could I please ask now? The formula, the other formula you wrote on top there is the ncx equals to n factorial divided by x factorial and then open bracket n minus x. Okay, I'm trying to use that formula on my calculator and it shows a. No, no, no formula on the calculator. The only thing on your calculator you need to calculate is the ncr. You just need to calculate ncr which is 18. Okay. And five. What calculator are you using? I'm using acacia. Acacia. So you say 18 shift and then you go look for that button that has ncr on top and you press five and you press equal. And that should give you the answer for the ncr. That's the only thing you will need the calculator for the rest of them. You can use your mental calculations in your head. Okay, thank you. Because you can see that this they're not asking you to give one single answer. They're asking you to just simplify the formula. Right. So it means we need to just do each part separately. Okay. Hmm. Consider blah, blah, blah. Suppose that is the same question but they have given us the new n and we still use the probability of success of 0.8. The question here is calculate the standard deviation. You need to know what is the formula for calculating the standard deviation. So the standard deviation for binomial is given by the square root of times pi times one minus the pi. So you just substitute and calculate n 16 times your pi 0.8 times one minus 0.8, which is equals to one point six, six zero. If you are using a sharp calculator, you will have to do this step by step. So you can just say 16 times when eight times into bracket one minus when eight or you can just say times zero point two because it's easy to calculate that and you get the answer of 2.56. So the answer of 2.56, that is what we call the variance. This is the variance. It is sigma squared. That this answer, just this answer inside. Once you have that answer, then you just press the square root of the answer and it will give you the standard deviation, which is option B. On the case you have a fraction thingy or the thingy, you can use your square root and it will give you a blocky and then you just put the values and press equal. And that is the easiest calculator to use, especially when you're doing calculations like this. It saves you time. So if the question was calculate the variance, you know that the variance is just the value underneath the square root. You remove the square root, it will give you the variance. Okay. Moving on to the next question, question 11. We are now in Poisson, the same with binomial with Poisson. You need to know the properties of a Poisson. You need to know how to calculate the expected value, the standard deviation, the variance. More especially, you need to know how to calculate the probability. You can also use tables or you can use formulas. So consider Poisson distribution with expected value or the expected occurrence per interval equals 10.5. Calculate the probability that the number of occurrence is exactly nine. So our x is equals to nine. So what they're asking you to calculate is the probability of and this is Poisson. So we're going to start with e to the power of negative pi, negative lambda. So our lambda is 10.5 and lambda to the power of x divided by x factorial. So e to the power of negative 10.50 times 10.50 to the power of x or x is nine divided by nine factorial. So you might ask yourself where is on the cashier, a factorial looks like that on a sharp calculator, a factorial looks like that. So you will have to go and find where your factorial is at. On a cashier e, because we have an e as well. On a cashier, let me open my calculator quickly. On a cashier, your e will have an e with a block. You must look for it. I think it's on top of lin ln. You must look for it. It's written in orange. On the sharp calculator, your e is on top of lin as well. It is written as e, come on, s, e, x. So here is when we are using formulas to calculate. Okay, so yeah, so because e is written in orange, we're going to press the shift or second function. First, so we do the first patch, second function e and it is e and you need to use the plus or minus button. Plus or minus button, not the negative, but the plus or minus button, which is on the sharp calculator, it is the value there at the bottom. On a cashier, it is the minus. On a cashier, you will also, you have to use the value in the bracket of the minus. That is the negative that we are looking for. So here we'll press negative 10.50 and I'm going to do equal because I'm calculating this first part and I'm going to multiply the answer I get with 10.5 to the power. I need the power. So on this calculator, my power is this x to the power of 9 and I must do equal because then it's multiplying what is at the top and then I must divide everything that is at the top, divide by 9 and my factorial, I need to find where my factorial is at. It's on button number four, it's written in orange, so it's 9 factorial and I press equal and the answer is 0.1177. Can you see how complex is this? Easy. Those with a cashier calculator use your fraction thingy above, so you will say you will press the button of your fraction and then you will press the shift. You will press shift and then you will press the E button and then you will put in the E button, you will put the negative and then 10.5 inside there and then you go out with the arrows. I wish I have my calculator open. Can I open my cashier calculator quickly? So I can show the others. It's complex doing it this way, so I have to. Okay, take two minutes, go to the bathroom, go drink water. I'm going to be with you just now. I need to switch off because then I need to stop sharing. Hello, how's Lizzie? How's Lizzie? Can you hear me? Oh gosh. Okay, my bad. I had put on the mute and I've been explaining all these things to myself then because I was talking to myself. Okay, I apologize but I hope you saw the steps. I can start again to explain again because I was speaking to myself. So I'm saying you need to put the fraction button. Then you need to press the shift and then press the link and press the negative and then you press 10.5. You don't have to put a zero because 10.5 zero is the same as 10.5 and you use the left arrow so that the flicker can move from a five to the bottom next to the line and you can press the multiplication or you can press the closed bracket, the open bracket and closed bracket at the end but I'm going to use the multiplication 10.5 and we press the power which is this button that will give us the power and is to the power of nine and you use your down arrow to go to the bottom and then you need to press nine and shift and your x factorial is on the x to the power of a negative where you press the x factorial and then you press equal. So you can see that this equation is the same as this equation that we have written which is the same. Those who are using a sharp calculator you also get the same question. The only trick here is for every step that you do for calculating the e to the power of a negative you need to press the equal press the multiplication button and press the next button the next step and say equal always remember to press the equal side if you don't press the equal side what your equation will do will apply the bot mass rule when you are just continuing and punching in the values without pressing the equal side and it will not calculate correctly especially if you don't use the brackets. So pay attention to how you work with your calculators especially for the sharp calculators because then the steps you need to calculate them individual unless if you have the right calculator way it also creates a fraction calculator like this. Can I ask you to just do the Casio one more time just go through the steps I'm not getting it right on my computer on my calculator. On your Casio. Casio. Okay thank you. Start. Press the fraction button. Press the shift button yes shift okay and press the learn where there is the e to the power of the blocks right you see that function yes press the button that has that function press the minus not the not this mind not the negative button not the minus the subtraction the negative with a minus inside the bracket press that and then press 10.5 and go to your arrows and press the left the right arrow not left right arrow which is this arrow yeah once you have pressed that button then press the multiply which is the multiplication and then press 10.5 and want to have pressed 10.5 press the shift button and then press the X with the power of a box actually we didn't have to press the shift press the shift again we need to take away the shift and then you press the box and the X with the box and then press 9 and then go down arrow and then press 9 again and then go shift and press the X factorial where these X to the power of negative one press that button it will create the next vector and then press enter okay so okay thank you yes alternatively you can also use the table so you need to go to Poisson tables and remember your Poisson art are split by probabilities right by the lambda values 0.1 0.5 these are your Poisson tables so let's see if they get to 10 if they don't get to 10 then you can use the Poisson tables so they do get to 10 but they stop at 10 not 10.5 hence we had to use the table of the formula so in the exam they won't give you questions like that anyway where you can use a table so you can see that this is just 10 is not 10.5 if it was 10.5 like we have with the rest of them like 9.5 and 9 we would and if you use just 10 you won't get the same answer as you would have okay so it gets to 10 and then it goes to 20 so you can use the table but if if the values are like less than 10 you can use the table like 1 and from 0.1 up until 10 and 20 you can use the table to get your probabilities okay so let's see we only left with seven minutes and we are on question 11 I don't know how many questions are in this I think we will be almost we are almost done there are only two more questions there were 18 questions okay maybe we'll be done with this now consider a Poisson distribution with the expected number of occurrences per interval being 0.43 so our lambda now it's easy to use and they say at least what does at least mean this is when now you need to remember it's at least less than or at least greater than or is at least greater than or equal or at least less than or equal but this at least it's greater than or equals to three therefore it means there are two ways you can do this you can go to the Poisson table and add the probability of x is equals to three plus the probability of x is equals to four plus the probability of x is equals to five plus up until you get to the end of the table because if we go to the four point what did it say four point three that is our lambda 4.3 then it means we need to add all this up until 15 you can either do that or you can say is the same as one minus the probability of x less than three which means it's one minus the probability of x is equals to zero plus the probability of x is equals to one plus the probability of x is equals to two which makes it easy for us to calculate the state of adding from three until 15 we can just add from zero until two and subtract from one so let's wait so you need to have come on why is it going oh you need to have your tables ready in the exams close by where you can use them so the probability that we're going to add are only those three let's add them say it is one minus open bracket 0.0136 plus 0.583 plus 0.1254 close bracket and equal and you press the sd change the answer is 0.8027 so it means if you add all this values here they will give you 0.8027 let's go to our thingy she's option a is our answer if you want to rely on formulas you will need to calculate for each one of them using this formula which will take you forever right whereas you can just come here and use the probability already calculated add them together and calculate the only thing you need to always remember is the formula what is at least how do i calculate the shortcut of that at least especially if the table is big if the table is short at least it's easy to calculate if there were only five so it would have been three four and five that you are adding together but because there are 15 so it's going to take you forever so use the complement of of the at least it's a lambda okay and that's how you will answer that question in the next two minutes that concludes today's session let's see if we can answer the next question okay so in terms of this one they've given us the lambda which a customer service representative notes that the service notes that since moving most of the service online the average front desk traffic or front desk visit is currently 1.7 per percent per hour so we have our rate which is our lambda of 1.7 so we go to our table and we look for 1.7 and we find 1.7 okay which one of the following statement is correct we need to validate each one of them okay so the probability that no one visits the front desk it means the probability that x is equals to zero what is that probability so you go to the table and go and look for don't don't get trapped by that it's zero comma zero zero no go to the table to validate because the table is there with all the probabilities so the probability of no one visiting is zero comma one eight two seven so that is incorrect the probability that at most what is at most what is at most it's less than or equal so what is the probability that at most 17 people visits before you jump into conclusion go to the table and check how many x values are there right so we go to the to the table just to look at how many they are so it says the probability that 17 there is 17 there so what will be that probability when there is none 17 does not exist so the answer is it remember we're looking for the correct answer right so the answer here will be the probability that at most 17 but 17 doesn't exist it will be less than or equals to 17 so the sum of all of them will be equals to one I'm gonna come back to that one let's validate because it might be correct and it might not be correct that is 17 does not exist on the table but it says at most so what is the standard deviation of the front desk so the standard deviation so what we know is this is the expected value and it is also called the mean and it is also called the variance so if if the lambda is also called the variance then what is the standard deviation the standard deviation will be the square root of the variance right so the square root of the variance is 1.7 is 1.3 so therefore c is incorrect we're looking for the correct answer number d the probability that only one person so this they're asking you to find the probability that x is equals to one only one person visit the front desk at any given point so we go to the table and we look for probability of one and that is zero comma three one zero six and zero comma three one zero six which this one says is zero comma three two three zero which is incorrect the probability that two persons visit we're looking for the probability that x is equals to two so where x is equals to two the answer is zero point two six to erase is zero comma two six four zero zero comma two six four zero so that is incorrect so the catch with the second one which is b is that they say the probability that at most 17 but we know that 17 is not part of this but at most means adding all of them so even if 17 is not there at most 17 it means we will it would have been zero for 17 for some reason like we have zeros on this once so let's assume that from from 10 until 17 it would have been zero zero zero zero because they don't have they don't exist they are mutually exclusive so if we add at most which is less than or equal so zero zero plus zero plus zero plus zero until we get to 10 and then when we get to nine we add the one zero point zero zero one zero point zero zero three all of them the sum of all of them will be equals to one therefore it makes number b the correct answer as the only way i can i can validate that one question and that note that concludes today's session is that all the notes if you still don't have access to the whatsapp groups there is the link to the whatsapp group otherwise thank you very much see you on sunday bye okay sunday we're doing um study unit six and seven remember that so you need to go and read all about chapter six and chapters oh sorry study unit six and study unit seven that's when we do that revision okay it's the unit three you're talking on sunday and uh i might i might also ask for it for you to excuse me on sunday as well because sunday is my bed day i totally forgot about it but um but we can talk on whatsapp anyway uh oh i didn't stop there quite a bit