 and welcome to your sixth session, which will be your second last session for this semester, because I think the majority of people are writing exams and UNICEF is worried that the non-attendance also might impact why the exams might be the reasons why people are not attending. So we might put a halt on the sessions unless the number of people increases in terms of participation, but I doubt looking at today's participation as well, it shows out of the more than 150 students that we anticipated for this semester to join the sessions only very few, handful of people attending the session. So anyway, that's not the purpose of today's session. Today we're going to learn about the basic skills on how to solve basic concepts of discrete probabilities. Since this is our second last session for May, actually, so next week we will look at binomial and Poisson probabilities, which will be the last session for May until further notice. Yeah, so we're going to start this week by introducing some of these concepts by learning how to answer question relating to discrete probabilities and so forth. Before we continue with today's session, do you have any comments or question or query or anything you want Larry to speak now? Nothing. You are all good. There's absence of comments or questions or queries. Let's look at what we're going to be learning today. By the end of the session, you should be able to learn what are the basic principles of basic concepts relating to discrete probabilities. You should learn how to calculate the mean, the standard deviation and the variance of a discrete probability. And most importantly, you should be able to answer any question relating to how to calculate probabilities of a discrete distribution or a discrete variable. Those who are doing 15 or one, like I said, I do not cover the marginal probabilities. If you have any challenges with how do you calculate marginal probabilities? We can have an offline discussion about it in your STA 1501 WhatsApp group as well. Okay. A discrete probability is or a distribution or a variable. We know what a discrete variable is. We learned that in study unit one of your modules. In the beginning, when you start with your modules, you learn about what types of variables you have. So a discrete variable is that variable that comes from a counting process, right? So it means if we need to calculate or find a distribution that relates to that discrete variable, then that distribution will be from events and we learned about events from the basic probabilities, you remember? When we create an event from a counting process, the outcomes that comes from that event will either have to be or not either. The events have to be mutually exclusive. Therefore, it means one event cannot affect the other event and they cannot happen at the same time as well. And they need to be independent as well. So with a probability distribution for a discrete variable, we're looking at mutually exclusive list of all possible outcomes that can happen from that variable and from there we can calculate the probability that is associated with that outcome. For example, if I work at this company and our IT department wants to monitor and look at the interruptions that happens per day from our computer network. So one day there is zero interruption, the other day there is one interruption, the other day there is two interruptions and three interruptions, four interruptions and so forth. And I can calculate what are the probabilities or what are the chances that those interruptions happen if I'm looking at it in a month. Let's say for example, the probability that there would be no interruptions in a day, it's 0.35. The probability that there at least there will be, oh, not at least, there will be one interruption, it's 0.25. Probabilities that there will be two interruptions, there will be 0.2 and so forth. And I can use this information to make any decision in terms of when do we schedule certain things and what are the risks. So by looking at this information, I can see that usually we don't have more than three or four or five interruptions a day. So the risk is too low, but we still have a minimal risk because there is at least one or there is one or two interruptions and they've got about 45% or 45 probability that those things can happen. So with the discrete probabilities, let's go back to the table before I go there. With discrete probabilities, how would you identify that this is a discrete probability question in your exam or in your assignment? It's because you will be given a table that has the x-values and the probability values. Sometimes they try to make it tricky and not give you the probabilities, but they give you the events. You know how to calculate events and event will just be a count. Let's say for example, this was 50, this is maybe 45, this is maybe 40, and this is maybe 30 and so forth and so forth count and you can add all those frequencies, the total to calculate the probability here. Remember now in terms of probabilities, are in the same format. So they are also referred to as relative frequencies as well. So you will take that 50 and divide by the total and that will give you that probability. So if they didn't give you the probabilities but they gave you the x-observations, you can count or calculate the probabilities. So but usually more or less, especially in the exam, they would not try and make things more difficult. So they will give you a table with an x and the probabilities. So you just need to make sure that you understand when you see a table like this and it can be horizontal or it can be vertical. They can make it vertical as well. So from this table, you can calculate the measures of central location, which is the mean, or you can calculate the measures of variation, which is the variance and the standard deviation from that table. So if we want to calculate the expected value, which is also called or referred to as the mean, we need to multiply our x-observation with the probability that will give and we add them. So the formula will be the sum of your x-observations or your x-outcome, multiplied by its cross-pointing probabilities and adding them all up will give you your expected mean. So remember our table, head zero and 0.35, you multiply zero and 0.35 plus one times 0.25 times, now plus two times 0.20, plus three times 0.1, plus four times 0.5, plus five times 0.05 and want to add all of them up, that will give you the mean and that is the mean of a discrete variable. Any questions? Is that clear? Okay, no questions. To calculate, sometimes in the question or in the exam, they might give you a table like, yeah, we've got to suppose the distribution of the number of people per household is as follows. And this is a discrete distribution. We have our x-outcomes, one, two, three, four, five. Sometimes your table can start with zero, sometimes can start with, it depends on the counting process that they used. Sometimes it might be two, four, six, eight, 10, sometimes it's one, two, three, four, five and so forth. So it doesn't necessarily have to start with zero, it will depend on the process that you are following. Okay, and they gave you also the probabilities, but now you need to also realize that at the end of this table, there is an x value. And sometimes in the exam, all these options that they will give you, one of them might be for you to find the value of x because you need to complete this whole table before you can answer the question. What have we learned previously from the basic probabilities is still applicable for today. What we learned in the basic probabilities is that the sum of all probabilities should always be equals to one. So if we know that concept, and if we know that the sum of all probabilities should be equals to one, therefore it means in order for me to find the value of five, all these values of these probabilities from where x is one up to five, if I add all of them up, they should give me one. So find the value of x, to find the value of x, we are going to say x will be given by one minus the sum of all these values, 0.25 plus 0.33 plus 0.17 plus 0.15. And that will give you the value of x. What is the value of x? 0.10. The value of x is 0.10. 0.10. So once you have the value of x, it's easy to calculate the expected value. Now let's calculate the expected value. Remember to calculate the expected value, we say we can also extend from this table and say let's calculate each and every one of them and then at the end we're going to add them together. So we're going to create a total. So the total here when I add all of them, it should give me one. The total here when I calculate and add them, it should give me my expected value. So the expected value, you can either write it out as a formula and say one multiplied by, or you don't even have to use multiplications. You can just put them into brackets. One multiplied by 0.25 plus two multiplied by 0.33 plus three multiplied by 0.17 plus four multiplied by 0.17 plus four multiplied by 0.15 plus five multiplied by 0.10 and that should give you your answer. We can go to the top and say one multiplied by 0.25 is 0.25 and two multiplied by 0.33, it's 0.66. 0.66 and three multiplied by 0.17 is 0.51 and four multiplied by 0.15 is 0.60 and five multiplied by 0.1 is 0.5. So you can just add them up because then I have found that this is 0.25 plus 0.66 plus 0.51 plus 0.60 plus 0.5 and what is the answer? 2.52. 2.52. Keep that value in mind and the later stage we're going to use that for all the examples that we're going to follow that are going to follow. I'm going to use the same table. So just remember that we've calculated the mean and the mean is our expected value is 0 comma is 2.52. Remember that. And also our table five for five corresponds with 0.10 just always remember that later on when we use the same table to just substitute the values. Any question? Is it right? Straight forward. We can also calculate the measures of variation and we can calculate the values of the value of the value of the variance or the standard deviation. And always remember that the variance is the square of your standard deviation and the standard deviation is the square root of your variance. And your standard deviation is calculated by using the square root of the sum of your X observation minus the mean or the expected values squared multiplying it by the corresponding probabilities. So what we're saying is we're going to take this observation X outcome minus the expected mean multiplying it by these probabilities. So it will say one minus 2.52 and then we square the answer and then we multiply by 0.25. That's what the standard deviation formula is and the variance is the same thing but the variance is the square root of is the square of your standard deviation whereas the standard deviation is the square root of your variance. So let's do get an example. Remember our previous example from the original data that we used. So we had our table and I just transposed my table and now I've got my X values and my P values. So now because I'm calculating this complex formula what I did on this I am breaking it down into bits and pieces. So the first bit that I'm doing is that bit. So what I do is remember on this example we calculated the mean and our mean our expected mean from previous exercise that we did was 1.4. Remember that? So I'm taking this expected mean and I'm subtracting the X observation. I'm subtracting the expected mean from this all X observation. So I say zero minus 1.4 and I square the answer. So I'm just doing this bit and I do for all of them I do not add the total because I still need to multiply with the probability. So the next step I take the answer and I multiply by the corresponding probability and I add all of them because it says the sum. So it means I must add all of them and find my variance will be that which is everything underneath the square root is 2.04. Then I take the square root then I get my standard deviation of 1.4. And that's how we calculate the standard deviation and the variance of a discrete probabilities. Let's go back to our exercise that we just did. Remember? We did calculate this. We said here it is 0.1 and we also went and calculated the expected value and we found that the expected value is 2.52. Now let's finalize the whole table. I can't hear your voice anymore. I don't know if it's on my side only or is it happening to everyone? Thank you. Others, can you hear my voice? Yeah, I can hear you. Please. Okay, it's happening on your side. Let me give you some time to go off and come back in. Leave the session and come back in and log in again. That person who can hear my voice, follow the steps. Let's give them a little bit of time. While we're giving them a little bit of time to log in, you can go ahead and draw up this table somewhere and we can do the first bit, x minus the expected value. So we can say x minus the expected value and then square the answer. We can do that. So it will mean we say 1 minus 2.52 squared and then you do the same, continue and continue. In the meantime, I will write the answers here. Okay, I hope he is back on. So let's look at the first one. So you all need to calculate and let me know if I am doing it all the wrong as well. So we say 1 minus 2.52 equals minus 1.52 and we take the square of that answer and that gives us 2.31. I hope you all get the same. And I can also do this while I'm still on this side. I can just multiply that answer by the probability, which is 0.25 and I get 0. And I must keep all my decimals, 0.5776. Go to the next one. 2 minus 2.52 equals, take the square of the answer equal and I get 0.2704. 0.2704 and I multiply that with 0.33 and I get 0.0832. Please keep all the decimals, right? Or at least keep at least four decimals if you want to get the right answer in your options. The next one, 3 minus 2.52 equals take the square of the answer equal 0.2304. 0.2304 Let me know if my answers are not the same as what you are getting. Multiply that with 0.17, which is the probability and we get 0.039168. Go to the next one. 4 minus 2.52 equals, take the square equal 2.1904. Multiply that with the probability of 0.15 equals 0.328560. Then the last one, it's 5 minus 2.52 equals take the square of the answer and I get 6.1504 and I need to multiply that answer with the probability of 0.1, which gives me 0.615040. Now, once we are done, we can take all these values and add them all up. So I can put this way with them and say this is the same as not even close. I'm not going to do all of them. I'm just going to put the square root of the answer which is the sum of all these values. So let's add all of them 0.5776 plus 0.089232 plus 0.039168 plus 0.328560 plus 0.615040 equals and I get 1.649600 and I take the square root of the answer and I get my answer is 1.28. Happy? Are we all good? I'm sorry ma'am. I got everything correct My mind was a new text. So I don't know where did you get the value of X? Can you please just explain for me? The value of? X. Which one? 0.1. The 0.1, this. You mean here? No, that one. The last one. Okay, so there is a row that is being written X and you wrote is 0.1 with rate. Yes, that one. Yes, yes. X will be one minus the sum of all these probabilities. So we go into say one minus 0.25 plus. You can put it in the bracket 0.33 plus 0.17 plus 0.15. That should give you the value of X. Okay, thank you. Thank you. Okay, and that's how you will calculate the standard deviation or the variance. Remember if they ask you for the variance which is sigma squared, the answer will be 1, if it's 2 decimals it will be 1,65. Because it's this value underneath the square root. Please make sure that you complete the register as well. I'm going to repaste it on the chart. Repost for those of you who joined late. There is always a register that we complete. Make sure that you complete that. Let's continue now and I'm going to give you some time to do more exercises later on. Now let's move on and look at how do we calculate the probabilities of a discrete variable. So in order for you to calculate the probabilities, remember the probabilities. If we have this table, we're going to use the same. I'll use this table to do some examples of each and every one of them. So if I need to calculate the probabilities which are these values, I know that these are probabilities, but if I want to calculate the probabilities of equal, that's what we have been doing. We say probability of that there will be no interruption. It will be 0.35. It's easy to calculate that probability, but sometimes we want to know if there will be more than that, if there will be less than that, if there will be between. What will be the probability that things will be between? So you need to be able to know from reading the question, understanding what the question is asking you to do, and unpacking it in a mathematical way. What I mean. Most of the time, they will give you questions, not with symbol, but with weights. So if they say what will be the probability that exactly zero interruption will happen, or will not happen, or something like that, don't affect. You need to know that exactly means it's equal to, or in a symbol format, it is equal. Or if they gave you an equal, say, what is the probability that x is equal to 1 or 0, you need to know that that means exactly, which means that x. So what is the probability, let's say, what is the probability that x will be equals to 4? That is exactly. And that will be equals to, you just come here and you look at x of 4. The probability that corresponds with that is 0, 05. And that will be your answer. What is the probability that at least three interruptions, or not at least, let's say, what is the probability that exactly three interruptions will happen in a day? What is the probability that at least not, why do I like at least so much? What is the probability that exactly three interruptions will happen in a day? That's your question. 0.1. That will be 0.1. You just come here and get to 0.1. So other weights will be like fewer than or below, which refers to less than. So if they mention weights like fewer than or less than, then you know that it is a less than. What does a less than mean? Less than means that value, but anything below that and does not include that value. So let's say, what is the probability that fewer than four, fewer than four interruptions will happen in a day? Probability that I need to know when they say fewer than or below four, they are referring to less than four. What will be that probability? I must be on four, but I must not include four and there should be all of these values going below four. So therefore it means they are referring to 0.35 plus 0.25 plus 0.20 plus 0.10. So if they say what is the probability of less than four, they are referring to all these other probabilities. So I must just add all of them. And that will be 0.9. So 90% of the time, there will be fewer than four interruptions happening per day. And that's how you use the weights to answer that. Same with greater than works the same because if it says greater than or more than pay very attention to it. More than means greater than. So this you need to always remember more than means greater than fewer means less than. So if it says greater than four, so more than four, that will be X greater than four, that probability will just be 0.5. 0.05 because it's just that one probability after the table, which is 0.05. Later on, I can also show you some of these other alternatives that you can use instead of going through the whole entire table. Okay, so that is for greater than. What about less than or equal or at most? So you also need to pay attention at most. If they say at most, these are the weights that they tend to use most of the time, then we're going to learn about this from now on until you go write the exam. When we do hypothesis testing, when we do normal distribution, when we do sampling distribution, when we do a binomial distribution, this weights you need to always remember at most will mean less than or equal. So it means at most for the probability that at most for it means X is less than or equals to four. They must also include four. So it will be 0.90 plus 0.05. I'm not repeating all these values again because we did do them. So the answer will be 0.95 because it's from four. Now we are including four. We are including four going that way. Or alternatively, instead of saying adding all these values because there are so many, we can say the probability that X is at most for looking at this because there are just there's just five at the end. We can say this is the same as one minus the probability of X is equals to five, which then it's one minus 0.05, which is equals to 0.95. And you see that both of them give you the same, but you need to practice if you're going to use alternatives like this. Less than or equals to five, meaning instead of me calculating and adding all the probabilities that there might be too many of them. I can just find a complement of that event. Because a complement of less than five will just be the probability of one minus the probability of where it's equals to five. And that is what we just did. So they've been one and the same thing, those two. Right, so we're done with at most, which is no more than. Also, that is the other way that they like to use no more than. It means at most it means less than or equal. So you need to know this weights. Oh, sorry. I want to clear my screen. Other way that they like to use it's at least an at least refers to greater than or equal. Or we can also refer to it as no less than no less than or at least refers to greater than or equals to that little. So if I need to find the probability that X is greater than or equals to three greater than or equals to three. Therefore, it means it starts from here and also include all the other values that are above three. So it means it's zero comma one zero plus zero comma zero five plus zero comma zero five, which is equals to zero comma two zero. And that is at least. Then we also have the between. So with the between, very tricky with the between, but you need to know and remember you need to read the statements correctly. So the statement will say it's between or it lies between. However, some key words such as inclusive or exclusive, they mean something. You will need to read it carefully. So if it says inclusive, therefore it means it's got an equal sign to it. If it says exclusive, then it's got a less than sign only no equality sign. This are two scenarios that can happen. However, sometimes it's not easy to say inclusive and exclusive when you have two values and you want to say one is inclusive and the other one is exclusive. However, they can give it to you in a symbol format and say because I'm not going to do those two because it's easy to do those two or less. Let's do them. Let's do them. So let's say for the first one, they say what is the probability that X lies between. So this will be X. X lies between two values and this we're doing inclusive. So it's inclusive and it lies between one and five. If it lies between one and five, therefore it means starting from here, ending here and everything in between. So that will be 0,25 plus 0,20 plus 0,10 plus 0,05 plus 0,05. Or I could have just said because this will be equal 0,65. Am I right? Yes. It will be 0,05. Or I could have just said this is the same as one minus the probability of X equals to zero, which is the same as one minus 0,35, which will be 0,65. You can do it that way as well. So that is the between inclusive. What about between exclusive? I'm going to use the same question. I'm lazy to retype everything, but I've got strength to cancel everything. So if it says it's exclusive, then it means it's less than. So it meant it is any value from one to five, but does not include one and five. So it's only those ones. So it will be only 0,20 plus 0,10 plus 0,05, which is equals to 0,35. That is exclusive because it's excluding one and five. However, what about this one? What is the probability that X lies between two values, one and five, but it is less than or equals to one and it's less than five. It's got inclusive and exclusive because then X is less than or greater than one. But it's also less than five. So it's between those two. So what does that mean? It says, if I start from here and I go this way, that is my value of X less than one because I know that it's going there. It's less than X is a value greater than one. But also I know that X is less than five, but does not include five. So it means it's going to start from here and it must go that way. So therefore it means this is 0.25 plus 0.20 plus 0.10 plus 0.05, which is equals to 0.5, 0.6, 0.60. And it can be vice versa as well. So it can be inclusive of five, but not inclusive of one. So you also need to do the same. So when it's exclusive of one, so it means it doesn't start at one, but we know that it's any value bigger than one. So it's any value there. And it's any value less than or equals to five. So it means it's all of these values, which is 0.20 plus 0.10 plus 0.05 plus 0.05, which is equals to 0.40. Any question? Because now we're moving into doing more activities or exercises. Anything that you don't understand that you still need clarity on your needs. Are we happy? Are we good or are we cool? So there are no questions. Let's do some exercises. So I'll do this one with you. And then the next few exercises, you will do them on your own. And this is the same as what we have been working with. So the same table. So let's assume that with the two questions we answered previously, those were option number one and option number two. And this is option number three, option number four, and option number five on your multiple choice question paper. You can see that, right? So it means you need to evaluate all the statement to find the correct or incorrect answer as per the question that they were asking you. Now let's evaluate these questions. So we know, we calculated this and we know that this is equals to 0.1 so that we have a complete table. And now we can answer the question. What is the probability of more than four people pay household? More than? Do you still remember what more than is? Yeah. It will be the probability that X is greater than four and what will be that probability that it's greater than four? It's 0.1. It will be 0.1 because it is only this probability that it's bigger than one, but bigger than four. Is it right? Only thing that is very important is that. What is the probability of between two and four inclusive? Between and inclusive. Do you still remember how to write that? Probability that X lies between and it says inclusive so therefore it's greater than or equal or less than or equal. Two and four. Always write the smaller value first and then the larger value last because it's between the two. What is the probability that it is between two and four inclusive? It will be 0.33, right? Because it's between that and this inclusive. You know that it's more than two, but it's less than four. So it's 0.33 plus 0.17 plus 0.15. And the answer is 0.65. Inclusive. What is the probability of less than three? The probability of X. Less than three. Does it include three? 0.33 plus 0.25. And that is equals to 0. 0. 0.58. 0.58. Because it says it's less than three does not include three so it's only those two. What is the probability that at least two? Number of there are two number of people pay household. What is the probability that at least there are two people pay household? 0.33. At least, remember, at least it means you need to find the probability that X is greater than or equals to two, right? Pay attention to the weights. At least two, you can do it two ways. You can add all of these values or you can say it's the same as one minus the probability of X equals to one. And that will be one minus 0.25, which will be equals to 0.75. You could say it is the probability that X is greater than two, which will be 0.33 plus 0.17 plus 0.15 plus 0.1. And that should be 0. Are we good? Are we good? Are we good? Are we great? I'm going to give you five minutes to do extra exercises. Are there any questions before we move to the next exercise? No questions. So it means you guys understand this. Always remember, if you haven't written this down, write it down some way. I'm going to give you time to write this or take a picture of it with your cell phone if you are watching this on your laptop. Or if you are watching on a laptop, do a screen grab of this presentation. Some people, I think one person asked how do you do a screen grab of your laptop? Some laptops have the capability to do an entire screenshot, which is the screen script. It's written there and you just need to press the function button in the screenshot. Sometimes it's on button F, some number, F9 or something. You just press the function button on the windows button and that screenshot. Or you press the alt and the screenshot. Some laptops have the Microsoft screenshot or screen grab functionality. If you click on your start button, on your windows start button and you type snippet like SNIP PPI. Let's call it what you call it. Snipping, if you type snipping on your start button. So you just press the start button and say snipping and the snipping tool will come up. It's like a magnifying glass sort of. So you just say SNIP and then it will say snipping. It will have a scissor next to it and then you just open that tool. And that tool will help you do a screen grab and you just press on the new and it will do a screen grab. Let's do it this way because I know our sessions are meant to help you with skills. So let's go into that for some few minutes. I'm going to share my entire screen right now. So we're only not going to learn stats only today. So let's learn about computers as well. So on your PC you go to the start function there. You click on the start and you type SNIP and you will have two. If your Microsoft Office has the SNIP and sketch, it will come up there. You can see there, it's the SNIP and sketch. You just open that SNIP and sketch and it will come up there and you press new. And you can see here it says for shortcut, you need to press the logo, the window logo, press the shift button and then press the S. You press them all at the same time. If I do that, what they told me, it will snip the entire screen. Can you see that? And then you just have this type of things that you want. So you can SNIP to shape and end. And you can snip the entire full screen. So but if I don't want the entire full screen, I can use the rectangle and it will snip whatever I want to snip. So let's say I want to snip that, it will snip it. And if I want, I can just use the snipping tool, so new. It will also open the same and then I can snip the presentation. As I'm presenting, you can always create your screenshots like that. And then you will see there is a picture snipped there. And you can save your picture. There is a save button. You just save your picture in where you want to save it, save S. And you can call it width to symbol. And you can always refer to that. You will always have it some way. So that is the SNIP and sketch. But there is not only the SNIP and sketch, there is also snipping depending on your laptop who installed some of the applications. Some of them they come with your laptop, with your windows. So a snipping tool, it also works the same. It's a snipping tool. You can also save how you want to snip or you just do new and you snip whatever you want to snip. And there it is. And then you can save. You will say save to snipping and then you say width to symbol. And you save. There we go. Then you will have it and you can open it any way you want to open it from. And that is snipping. Okay. Back to the presentation. Yes, you can also do the print screen. And print screen, it captures your entire screen as well. Not the only thing that you want to concentrate on. But I don't have time to do the print screen right now. And you will need that when you are writing your exams because I think I assume that with model it says you need to take a picture of your screen at a certain point and send it. You need to know how to use all these things. But that is one of those things that I wanted to share with you today. Not stats related. I hope it is helpful. Okay. Here is your exercise after I have taken your mind off stats for a bit. You have five minutes to answer the following question. Africa check knows that around election time, the number of daily fake news posts about politicians follows the following discrete probabilities. There can be zero fake news, one fake news, two fake news, three fake news, four fake news. And they gave you the probabilities of those outcome. And there is a question mark at the end. Let X be the number of daily fake news post. Which one of the following statement is incorrect? What the question is asking you is to find the incorrect statement. The table has a question mark. Therefore, it means something needs to happen on this table. So you need to complete this before you go on. Once you have done that, you can just answer all these questions. Emma. With multiple choice questions, we evaluate each statement. I'm going to give you five minutes to go through each and every one of those statements and then choose the incorrect answer. And you can post your answer in the chat after you have evaluated all of them. Then we'll come back and do some feedback on it. Five minutes starts right now. Remember, if you have any question in between, you can ask anyone can also respond. Sorry, I have a question. Hello, can you hear me? Hello. Hello, can you hear me? We can hear you. Oh, okay. I guess my question is more related to how you answer because the way this is worded, like for number five, it says none of the above. As in none of the above are incorrect or is it a case of, because it's asking which of the following statements is incorrect. And it's incorrect to say none of the above is incorrect, if you know what I'm saying. Firstly, you must establish the probability for four and then cancel out the correct ones so that you can get the incorrect one. Yes, yes. And that's, no, that's exactly what I did. And I did find one of the incorrect ones. I was just more concerned about the fifth one as a statement that's incorrect because that statement is incorrect. I think that statement number five translates to all of the above, that is one, two, three and four are correct. And none of them are incorrect. Okay. That's how you'd interpret that. Okay, I see. Okay, so if all of them were correct, then that statement would more translate to being the answer. Yeah. Okay, I see. But yeah, it's the way it was worded so. Okay, thank you. Cool. I should go out so often. I enjoyed listening to all of you. Okay, are we, are we still busy? Yeah, still busy. Yeah. Are we done? Yes. Okay, so. Yes. What is number four's probability? 0.3. 0.3 because it's one minus. Yeah, it's one minus all these other values. Okay. Now let's answer every statement and then we'll come back to the question. Number one says the sum of all probabilities equals to one. Is that correct? Yes. Yes. That is correct because if we add all of these probabilities, we should get a total of one. Number two, it says the probability of X is equals to four. It's equals to zero. What is the probability of X is equals to four? It's 0.3. It's 0.3. So therefore this is the incorrect. So number three says the probability of X less than or equals to four is equals to one. That means from here going there is equals to one. Is that correct? Yes. That is correct because if we add all these probabilities, they will be equals to one. The probability that X is greater than or equals to four. So there is nothing on the other side. So therefore starting from here going above, which means the probability of X is equals to four. And we found that it was 0.3. That's that. Statement number five, which is one of those statements that most of the time they do include this kind of statement, which how you read them, they might confuse you. But someone already explained what this means because it says none of the above. Remember, if we go back to our question, it says which of the statement is incorrect. So it means this statement says if I look at all these statements above, they are all incorrect. They are all correct, but we know that this statement is incorrect. So it makes this statement correct because it says none of the above. So it means not all of them. It's very confusing. Can you see that? But what I always suggest is that this statement, you always come back to it. If none of the statement doesn't correspond to the correct question you are asked. Sometimes it can say all of the above, none of the above, any of the above, things like that. But you always revert back to it if your answers are not one of those that are above as well. So that one is also correct because we know that at least one of those statements is incorrect. Your next exercise, calculate the expected number. What is an expected number? That is the mean. I'm going to give you the formula just now. Which is the expected mean is the sum of your X observation times its cross-bonding probabilities. Are we done? Yes. That is not a convincing yes. Are they still people calculating? No, also I'm also done. Okay. So let's do that because I think most of you are done with that question. So we just need to multiply each one of them. X multiplied by, so we say X multiplied by the probability, I can write the answers on here. And we know that this one is 0.30. So 0 times 0.1 is 0, right? 1 times 0.15 is 0.15. 2 times 0.2 is 0.40. 3 times 0.25 is 0.75. 4 times 0.30 is 1.2. Is it 1.2? Yes, 1.2. So we can just say 0 plus 0.15 plus 0.40 plus 0.75. Plus 1.2 and the answer will be 2.5. 2.5, which is option 3. If they ask you to calculate the standard deviation, you can use your expected value to calculate that. Okay, next question. You also have five minutes to answer this question. A speech therapist knows that the number of children that consult with her on any given day is given by the following discrete probability. Let X be the number of children consulting with a speech therapist on any given day and those were the numbers. Which one of the following statement is incorrect? You need to convert the words into symbols to answer this. I'm just going to go in and highlight the key words that you need to pay attention to. At least one child, children. Let's do this. One children. At least one. No children. More than. And since I cannot highlight below, maybe I can highlight below. I have something blocking my view on this one. Those are the things that you need to be aware of. So evaluate all five statements and choose which one is incorrect afterwards. Remember also to ask if you don't understand. Are we winning? Almost there. Yes, yes. How many are there anyone or is there anyone who is doing STA 15 or one the descriptive probability statistics? Is it called descriptive? Yes, I think so. Yes, I am. Remember, we will not cover the marginal probabilities where you have your X and Y. We only work with one discrete variable. So if you want to learn how to do the marginal probabilities, let's have that discussion on the WhatsApp group. Maybe we can set up a session for that sometime during the week. It can also be about 30 minutes of 45 minutes. It doesn't have to be long. That would be great. Thank you. No problem. Are we done? Yes, yes. So let's answer the question. So which one of the following statement is incorrect? That's what we are looking for. But we need to evaluate each and every statement. The first statement says the probability that the speech therapist will consult with at least one child, which means the probability that X is greater than or equals to one. What is that probability? It equals to one is all the probabilities added together. So therefore it means that statement is correct. The probability that the speech therapist will consult with one child, it means we're looking for the probability that X will be equals to one. And that is only that probability where X is equals to one will be zero comma one, then that statement is correct. The probability that a therapist will consult with no children, which means they are asking what is the probability that X is equals to zero? Is there such probability that we can calculate? Nope, no. That will be equals to zero. That is an uncertain event. And that probability is zero because it does not exist on the table. What is the probability that the therapist will consult with more than five children? Probability that X is greater than five. Is there anything above five? No, there's nothing. So that will also be an uncertain event that will be equals to zero. And that will be correct. Number five it says find the expected value, which we know that is the sum of your X observation times the cross-bonding probability. So we can go and calculate our X times the probability. Let me know if I am having them all wrong because I'm counting them in my head. 1 times 0.1 is 0.12 times 0.15. It should be 0.30. 3 times 0.2 should state 0.6. And 1, 2. 5 times 5 is 5. 5 times 2 is 10. 1.25. Yes. Right. So therefore, yeah, we can say it is 0.1 plus 0.3 plus 0.6 plus 1.2 plus 1.25, which is equals to 3.45. 3.45, which is not equals to 4. And that is our incorrect statement. So therefore option 5 is our incorrect statement. Next question. Usually this type of questions, they like follow one after the other. You might find in the exam, usually discrete probabilities, you get three or four questions. Or they might ask you, one of these probabilities might also be a discrete probability. In the assignment, you might have three questions from discrete probabilities. So you just need to know how to answer all of them. So let's continue with the last question I think of the day, which would take us to the end of the session. Using the same information, we have already calculated the expected mean. And we know that that was 3.45, right? The next question says, what is the standard deviation of this number of children consulting with the therapist? Remember, the standard deviation formula, if I can just give you the formula, is the square root of the sum of your observed minus the expected squared times its cross-bonding probability. So you can come here and do x minus the expected squared. And then also do the same and multiply that with the cross-bonding probability. Or you can do it all at once times its cross-bonding probability. You can do that all at once. Remember also to keep all your decimals so that you don't lose any integrity in terms of your numbers. Remember, if you're struggling or you don't know what is happening, ask those who joined late. Please remember to complete the register. Are we winning? Almost there. Manisa, did you take the square root of the answer or did you keep all the decimals as well? Because if you didn't keep all the decimals, you might not get the answer. If you didn't get the square root, take the square root of the answer, you might also not get the answer. So just double-check those things. We left with nine minutes. Are we there yet? Remember you take 1 minus 3,45 and then you square the answer of that and then multiply by 0,1. And you do the same for all of them. I think I know what my problem is. I calculated the wrong table. Okay. Can we do the answer? Or are you still busy? No response. So let's do the answer. Okay. So we know that my table is very small there at the top. I'm going to write all the answers at the bottom on my equation. We say 1 minus 3,45 squared. Multiply by 0,01. And that gives us... You let me know if I've got the wrong answers as well. 0,6,0,0,2,5,0. Plus, because we have to add because of the summation. Because the question says the sum, the summation says we are adding. 2 minus 3,45 equal squared. Multiply that answer by 0,15. I get 0,3,1,5,3,75. Plus, 3 minus 3,45 squared. Square the answer times 0,2. I get 0,0,4,0,500. I'm not going to write it double zero. I'm just going to stop right there. And then we get 4 minus 3,45. And then get the answer, square that answer. Multiply by 0,3. I get 0,09,0,7,5,0. Plus the last one. 5 minus 3,45 equals... And then I get the answer, multiply that... Oh, square the answer. I press the square. Multiply the answer by 0,25. I get the answer of 0,600,625. I'm keeping enough numbers. There might be some numbers that are very long. I'm just keeping at least 6 decimals or 4 decimals so that I don't lose the integrity of the answer as I get my decimals. So adding all the values... 1.28,3. Yeah, but adding all these other values, I get 1,64,7,5 and plus other numbers or like zeros and some numbers. I'm just going to keep 4 decimals. And if I take the square root of this number, I get 1,28. The value here underneath, this is what we call the variant side, which is sigma squared, which if it was, it would have been number 4. And the answer is number 5. You just need to practice, practice, practice, practice, practice. Okay, so that concludes... We are right at the end of the session. You have learned so far by now the properties of a discrete probabilities, how to calculate the mean, the variance and the standard deviation of a discrete variable, and also how to calculate the probabilities of a discrete variable. Remember, if you're given a discrete probability or a discrete variable table, you will recognize it by the X value and the probabilities. Because the previous time, you would have done the basic probabilities where you also use some table, the cross-tabulation table. You will see the difference that a cross-tabulation plus this one, they look exactly, they look different. So with that discrete probabilities, you will get a table either in a horizontal format or in a vertical format, and then it will consist of the observations or the outcomes and the corresponding probabilities. And you can use that to calculate the expected mean, which is the sum of your X observations or your X outcome multiplied by its corresponding probabilities to calculate the standard deviation or the variance, which the standard deviation is the square root of the variance. You will use the square root of the sum of your outcome minus the expected value squared multiplied by the corresponding probability. So which is the sum of your outcome minus the expected value multiplied by the corresponding probability. In terms of calculating the probabilities, always remember that they can give you questions in a weight format. You need to convert it to a mathematical symbol in order to find the answer. Like weights like, at least, meaning greater than or equal, at most, meaning less than or equal, between, and if it's between, is it inclusive or exclusive, or relating to whichever one, inclusive and exclusive. When it is no more than, what does that mean? When it's more than, what does that mean? You need to practice all those things. You need to know them because they are very, very important to know them. Those who are doing STA 1501, like I'm going to repeat it, we didn't cover the marginal probabilities. If you want information about the marginal probabilities or you want us to go through a session where we look at the marginal probabilities, please get hold of me. I will create a session where we can have a discussion and look at examples of marginal probabilities. The other modules, you do not need to worry about that. I will see you next week as we do binomial and Poisson distribution. Please bring to that session tables. We are going to use the binomial table and the Poisson table to answer some of the questions. Through the session, where can you find the table? Some modules, they do give you tables as part of your tutorial letters. Some modules, tables are part of your study guide. Some modules, since you have a prescribed book, the tables are always at the back of your prescribed textbook. Otherwise, if you got hold of past exam papers or past tutorial letters, they also do contain tables, which you will need. We're going to refer to that. Therefore, it means, as from next week, every time you attend any of the sessions, you need to bring that table with you or the tables with you because we're going to be using them from now on going forward. Other than that, are there any questions, comments or queries before we end of the session? Lizzie, just a question around the end one. The recording of the day session, how soon will that be loaded to the Mayonnaise side? And then a second follow-up on class tomorrow. Will there be classes still continuing on the Sunday? Okay, sorry. You are asking the question on the wrong platform in terms of the class for tomorrow because I don't want to create those confusions. Remember, if you are in my e-tutorials class, if you are part of my e-tutorial, correspondence about the classes already posted on my module, we have classes. And I don't want to include that in here because then it will raise some other questions and other people's expectations. I don't want to do that. I invited you to this session for this paper, for this session, for the facilitation of learning only. So I will answer the first question. The recordings, if they are not available, you need to send an email to ctentact.unisa.ac.za. The recordings usually takes about 72 hours to be uploaded. Right. That is that one question that I can answer for this paper. The other one, I can't answer. I can't answer yet on this platform. Okay, so if you have any other questions relating to e-tutorials, you must post them on the WhatsApp group for the e-tutorials, not on this platform. Because we don't talk about modules, we don't talk about tutorials, we talk about basic statistics. Any other questions before we close off this session? Yes, Lizzie. The formulas and the tables you sent on the WhatsApp group, is it part of those binomial distribution? What tables that I sent on the WhatsApp group? On the 1501 group. Did I send something? There were formulas and tables. Who sent them? Not me. I didn't send any information. It was you. When? Don't confuse the WhatsApp group and the telegram. I must sit and check. I haven't shared any documentation with any of the WhatsApp groups. I think it's not a list, it's not this group. Yeah. It might be in your tutorial WhatsApp group or something. I think Lindua might be right. Maybe it's the other one. Thank you. No problem. Elizabeth. Yes. The slides for last week's class, I see that the video is uploaded, but the slides are not there. Oh, the slides are there. I saw my apologies for that. I've uploaded the slides. You should have access to them now if you go to the folder. Okay, thank you. No problem. Are there any other questions? If there are no questions, I'm going to stop this recording and I'm going to tell you.