 OK, so let's look at the properties of a discrete probability. So a discrete probability comes from or we calculate it by using the discrete variable. And you remember what the discrete variable is. Are those variables that you can count? And it's part of your numerical values. So instead of just visualizing them, we can calculate the probability of those variables. Usually they are mutually explicit because we have to list all the possible events that can happen. And remember what we did last week when we looked at the basic probabilities. Also we need to apply those rules here. We know that the sum of all probabilities should always be equals to one. And we also know that the probabilities lies between zero and one. They can never be negative or they can never be greater than one. So for example with discrete probabilities, let's say we have the number of interruptions per day on a computer network. This will be where we have zero interruptions. So there was no interruptions and we can calculate the probability of that. So if this is Monday to Saturday, so we can also just record the number of days that if we never had any interruptions, where we had one interruption per day, where we had two, where we had three, where we had four, and where we had five. And with this we will calculate the frequencies. Once we have the frequencies, then we calculate the relative frequency. And these are almost like your relative frequencies, which we can also come on now. So this we call them relative frequencies, or we can also call them probabilities as such. Because if we add all of them, they should be equals to one. So we can also represent them as FX, which are your relative frequencies. But for now we're going to call them probabilities. So for every number of interactions that happens per day, we can calculate its cross-bonding probability. And we're going to use the same table to calculate and find the probabilities at the later stage. But for now let's look at other properties. So using the same table that we had, we are able to calculate what we call the expected field, which is the average of this discrete variable. So remember all those we had in the beginning. If I add all these probabilities, if I add all these probabilities, they should be equals to one. So no. In your module, when you go write the exam, sometimes they might give you a table and they remove one of these probabilities. So let's say they don't have this zero comma two. So to get the value first before you can do anything, you have to add all the other values and subtract them from one so that you can get that value there. But it's not the base for now. So let's learn how to calculate this expected value. To calculate the expected value of this, we use the formula. The mean is equals to the expected value, which is equals to the sum of your observation multiplied by its corresponding probability. So it means we're going to add the observed value multiplied by its corresponding probability. That's what we're going to be doing. So we're going to multiply this with this, that with that, that with that, and at the end, add all of them up. So we'll say zero multiply by zero comma three. That's what we're going to be doing. So the easy way when we work with this table is to create another row on the table so that you can have all the values there and it makes it easy to add all of them. So what I do is to calculate the x times the probability which says zero times zero comma 35 is equals to zero. One times zero comma zero two is equals to that. That times that will be equals to, until I get to five times zero comma zero five which gives me zero comma zero two five, zero comma two five. Then when I'm done because of the summation, I just add zero comma zero plus zero comma two five plus zero comma four plus zero comma three plus zero comma two plus zero comma two. And it will give me what we call the mean because the mean is the sum of your observation multiplied by the corresponding probability. And with that being said, this is your exercise. Quick, quick, quick, calculate the mean of this corresponding table. The first thing you need to do is to find this value which is find the value of x. Then when you're done finding the value of x, you can then calculate your probability. I'm giving you five minutes to do this, five minutes, and somebody will have to help me answer that question. Okay, let's see, have you posted your answers? Anybody who wants to do this with me? You know that you can use the chat link to talk to me if it's abled on your site. Otherwise, who wants to go for it? Come on guys, I cannot be talking to myself. Maybe I can try. Okay, thank you. So I adapted all of the values that added point 25, point 23, that's point 17, and point 25. I had to get x and I got point one. So that will be 0.1. That's what you got. And then I multiplied one times point 25. Yes. Two times point 23. Two times point 14. Wait, am I missing something? Two times 0.6 is 0.33. Two times 0.32, yeah. Yes. Three times point 17. Three times 0 comma. One seven, point 17. You get point 51. That's 0.51. Yes. And then four times point 15. You get point six. And then five times point one. You get point five. Right, so adding those two, those five, point 25. That's point 66. That's point 51. One, that's point six, that's point five. One point five, two. It gives you one point five, two. And that will be your expected value, which will be one point five, two. So how easy it is to calculate the mean. So let's look at calculating the variance or the standard deviation. So calculating the variance, we use the formula. Since the variance is the sigma squared is the sum of your X observation minus the expected mean. So for example, if we use this table, this will be our expected mean. So we will take this value, subtract the expected mean, and we're going to multiply by the corresponding probability. And that is what the formula is. So it's your observation minus the expected mean squared times its corresponding probability. And that gives you the variance. To calculate the standard deviation, we just take the square root of this, which is the square root of your variance, which your EX will be your expected mean, and your XI will be your expected, your outcome. And your PX is your corresponding probability. And once you have that, you can calculate your, you would have calculated your standard deviation when you take the square root of the variance. So let's look at an example. So going back to the same exercise that we did earlier. Remember we calculated it and we found that our expected mean, so if I go back to our exercise, which is this, we find that our expected mean was one comma four zero. So now we're going to use this expected mean to calculate the standard deviation. So our expected mean of one point four zero, we're going to use that to say zero minus one point four squared times the corresponding probability. So let's go for it. So what I do as well to make things easy is to also extend the table. So you can also do it when it looks horizontal. It's just that I run out of space there. So I prefer it in this manner. So what I do, I start first by the square, then I can expand it outside. So what I do, I start with the observation minus the expected value squared. So it will say zero minus the number, our expected value, our expected value that we got was one point four zero. So we use zero minus one point four squared and we get the answer. That minus one point four squared, two minus one point four squared, until we get to five minus one point four squared. And once we have all these values, then it means we did the first part of this of the equation. So we have done the first part of the equation. Now we need to do the second part of the equation. We don't have to go back and do zero minus that because we already have the answer. What we do is we take the probability and multiply with the expected value. So it will be one comma nine six multiplied by the corresponding probability. So it will be zero comma one six multiplied by zero comma two five. And it will give us the answer and we will do for all of the values. And because we have the summation there, what we will do with the summation is is to add all these answers together. And when we add all the answers before we even take the square root, we would have calculated what we call the variance. And when we take the square root of the variance by just taking the answer and we apply the square root on top, then we will be calculating the standard deviation. And that's how easy it is to calculate this. Any question before we do your exercise? If there are no questions, it seems like my PC is stuck, stuck on me. Okay. And continue with your exercise. So remember we calculated this. We found that the mean, remember, that's you continuing from where you left off. Your mean was one comma five two. That was your mean. So now remember to disturb that answer is but incorrect. Which answer? For the expected value, I believe it's two point five two if you add it up. Okay. We go back, back, back, back. You mean yeah? Yes. So if you add all the values, remember I didn't do the calculations. So let's quickly double check again as well so that everybody is a piece. Zero comma two five plus point six six plus point five one plus point six plus point five equals two point five two you are right. So we need to fix our answer here. Sorry, my bad, it's two point five two. Yes. So it's two, two point five two. If you see a mistake, please raise it quickly before we move on to the next slide as well. So now this is your chance to calculate. So we know that our expected value or expected value that we got from calculating this table was two point five two. So now complete the whole table and calculate the standard deviation. That's what you can do is to first calculate the first spot, which is x minus px squared and get for every answer. And then when you are done, you can do your x minus px squared times not px ex ex and this side you can multiply by that. So you can do for all of them. And remember here was zero comma one. And then when you're done, you can also add some of them and take this quiz of the answer. So I'm going to give you 10 minutes for this one. I will also do it on my side so that we can confirm all the answers are correct. If you are done, can I have an indication as well when someone is done so that I can gauge the time as well? Anyone who is done, are you guys still busy? I am done. Okay, thank you. Other people? Other people? Are you still busy? I'm not going to ask you to answer the question. I just want to know if you are done or are you still busy? You can be quiet. One more minute. Thank you. Remember we calculated the standard deviation? So when... Okay, so others? Are you all done? I'm assuming because only one person said I must give them a minute. Before I continue, I just want to raise one thing. Coming to the session and listening to me will not help you understand the content and it's not going to do any justice on you. And also coming to the session and listening to one person always giving the answers and you not doing anything, asking questions or participating in giving answers will not help you because you will leave the session with uncertainties I was very confused and you won't be able to even do the exercises or your assignment on your own and you will rely on other people. The reason why we do this online is for you to engage with me. If not, if the engagement is not there, then I can just record the sessions on my own and post them on my UNISA and you can go through them. And I can also show you where all these recordings are because it's not my first session today. I had session last semester with my face-to-face classes. We have the recordings of the same thing that we're going through with some other exercises that we go through. So I hope going forward, we will all participate so that we can help one another for you to go quiet on me when I ask you to give me an answer. It's not doing me, you are not doing me a favor. This is about your learning. So you need to take charge of it and participate. Even if you don't know the answer, try so that we can see where you're going wrong so that we can help you. If you come and you don't say anything, it's not helping you. It's doing even more damage. You need to go through your learning process because you would have spent two hours or one hour, 30 minutes listening to us, but it didn't add any value. So please, with that, and I hope I'm not going to repeat this in any other sessions coming forward, we will have the engagement. We will participate. We will try and solve the questions together. We will work together. So anybody who wants to try, I don't want to know how you calculated it. We can just populate the table with the final answers to save time because I expect everybody to have calculated them in front of them. So we will have the answers all of us. And if the person giving the answers, if they have one number wrong, you can just correct that person and then we move forward. So anybody who wants to go to give the answers? I'll do it with you. Okay, so the first one. 2.31. I'm going down. If the second one is 0.27, 0.23, 0.23, 2.19, 2.19, do you all agree? 6.15, 0.15, do you all agree with the numbers? And I'm going to the last row. The first one, 0.578, 0.578, second one, 0.089, 0.089, and then 0.039. 0.039, 0.329, 0.615, 0.615, and then when I add them, they give me 1.321. Okay, 1.321, the square root of that is 1.149. 321, so therefore it means the square root of 1.321 is 1.149. Do we all agree? Did we all get the same answer? 1.149. 1.141, sorry, my bad, let's rewrite that, 1.149. Happy? Are we all happy? So now let's look at how we calculate the probabilities. Do you have any questions? Any questions? If there are no questions, then let's look at how we... Maybe one for me? Yes? Right, if you're looking and I don't know if this is something that you could be able to ask, if you're looking at the figure, is there any way that you can see that you are incorrectly calculating the outcome? Ah, not necessarily, you will have to redo them. So you will have to go back and double check if you have done them correctly. Yeah, that's the only way. Yes? Okay, it's always good to go back and double check your answers as well. Okay, so let's look at probabilities. So with discrete distribution, you need to understand the following. So for example, you know that you will always get a table that looks like this and you know that this is a discrete probability question. But then when they ask you questions about calculating the probabilities, you will say, oh, but then I already have the probabilities. No, we're talking about cumulative probabilities or the equal probabilities and that's where you need to know the symbols. Sometimes the question might be in a wet phrase to say calculate at least, what is the probability that at least there are two interruptions per day. So then it means you need to know how to interpret that question, that they are looking for more than two, more than or equals to two. So in your study guides, I know that some of the study guides, I'm not sure whether it's for 1610 or 1510, it's somewhere in there. You do have this written out. Some study guides, I think it's 1610. The signs are messed up. They are, I don't know what they did there. So please make sure that you correct them with the right signs. So for example, where they say equal, it means exactly that. So they might say exactly. What is the probability that exactly the two interruptions happen per day? So you will know that that probability will be 0,02. And in terms of probabilities, we will represent it as the probability will be, what is the probability that x is equals to two, and that will just be 0,20. That's straightforward. It's the probability of exactly. So if they ask you, what is the probability that less than two interruptions happens a day? It means fewer than below. It means lower than two. It does not include two. So therefore, it means does not include two, does not include two, but it includes one and zero. And how we represent that in a formula, we will say the probability that x is less than two, it will be represented by the probability that x is equals to zero, plus the probability that x is equals to one. So we have to add both of them for the less than. And that will give us 0,35 plus 0,25. And that will be the probability that x is less than two will be 0,25 plus 0,35, which is equals to 0,6. 0,6, and that will be the probability of a less than. To calculate the probability, let's say they say, what is there? Probability that it is greater than or it's more than. So they will use this with interchangeably. They can say it's greater than or it's more than. So you will need to know that it means it does not include that value. So if they say what is the probability that x is greater than four, therefore that probability will not include four, but it's bigger than four. And the only thing that is bigger than four, it means x is equals to five. I don't know how to write five. I don't know how to erase five. Therefore, it means that probability will be just equals to 0,05. And that is the probability of more than. And what if they say they want the probability of at most? Where they say at most it means more than. It means it's less than or. At most it means no more than. It means less than or equal. So if they say what is the probability that what is the probability that is less than or equals to three, therefore they say they might say what is the probability that at most three interruptions happen a day. They refer into the probability of less than or equal, which then tells me that they are looking for including three. It has to include three because it says equal to three. Therefore, they are looking for the probability of 0,35. I'm just going to write the probabilities only 0,35 plus the probability 0,25 plus the probability 0,2, 0 plus 0,10. So it will include all of these probabilities. And then you just take all of them up, which is 0.35 plus 0.25 plus 0.2 plus 0.1, which gives us 0.4. And therefore the probability of at most three will be equal to 0.4. And that's how you calculate the probabilities. An at least will mean it is greater than or equal to, similar to with the at most. The at least will mean it's bigger than and also include that value. So the probability was at most three. Therefore this would have been greater than or equals to three. And if that is the case for at least, which means bigger than or equals to three or greater than or equals to three, it means we only interested in those ones. So it means we're going to only say it is equal to 0,10 plus 0,05 plus 0,05. And that will give us 0.05 plus 0.05 plus 0.1 gives us 0.2. And that's how you calculate the probabilities. And that is not all. Sometimes they will ask you, what is the probability of between? Now, here is where it gets very interesting. They talk about between. Sometimes they don't say whether it is it is inclusive or it's exclusive. You need to be very careful when you answer that question. Most of the time when they don't mention the weight exclusive, then when they talk about between, it also refers to inclusive probabilities as well. So if they say what is the probability that Z is between two values, and they don't mention that inclusive or they don't mention exclusive, you must know that it will be inclusive. But if they mention the weight exclusive, then you will need to know what they mean inclusive of the first or inclusive of the second one. So when they say the probability of between, we can represent it as less than or equal for the between inclusive. It will always have the equal sign because it must include those values. So let's say, for example, they asking what is the probability that the interruptions happen between one, we're going to call this X, one and three. So what is the probability that the interruption happens between one and three? There are three interruptions or one interruptions. So since because it says between and it is inclusive, therefore, it means they are looking at one. Therefore, it says X is one is less one is bigger than X. Gosh, what am I saying? One is less than X. So therefore, it means the value should not be. So X should be more than X. I am scrambling my words today. X should be bigger than one. So therefore, it means it should be bigger than one and it should be less than three. So it should be less than or equals to three. And that's what it says. So X is between those two. It can never be bigger than or less than one, but it can be bigger than one, but it cannot be more than three. So therefore, it cannot also be bigger than three. So it has to be between those two. And with that, it means the probabilities will be X is equals to one plus the probability X is equals to two plus the probability X is equals to three. And we just add one of them. Zero comma two five plus zero comma two plus zero comma two plus zero comma one. And that will give you zero comma five five. And that is the between inclusive. And if they ask you, what is the probability that only one side is inclusive? So let's say, for example, they might say one is X is bigger than one, but it does not include one and it's less than three. So therefore, it means it doesn't start right here. It's not inclusive of that, but it's somewhere between those two. So therefore, it means it will be the probability of X is equals to two plus the probability that X is equals to three because it includes three, but it does not include one. So if it does not include one, but it's bigger than one, so it will only be two and three because it cannot be bigger than three. So it'll only be that. The same as when it is exclusive both sides. Therefore, if they say it is between zero and one and it's exclusive, therefore, it means it does not exist. The probability should be equals to zero, but if they say it's between zero and three and it's exclusive like that. So let's say now they ask you the probability that it's between. What is the probability that X lies between X lies between zero and three? So it's bigger than X is bigger than zero and it's less than three. So it's bigger than zero, but it does not include zero. So it's those ones and it does not include three. So it's only those two. So it will be the probability that X is equals to one first. The probability that X is equals to one plus the probability that X is equals to two. And that's how you calculate the probabilities, which will give us zero comma four, four, five. And that's how you calculate the probabilities. And with that, you have an exercise to do. Ten minutes, exercise one, two, three. Actually, I want to skip this exercise. I want to go to this one so that then to these exercises. Let's skip that one. Let's use this one. Which of the following statements are incorrect? The member at least means greater than or equal. And if they have at most, it will mean less than or equal. Otherwise, the rest would mean as they are. If one person is done just to gauge when you are done, just let me know that you are done so that I can hear how many people are done. And we can recap. And remember the expected number. It's the same as the expected mean, which is the sum of your X multiplied by its corresponding probability. I'm done on my side. Thank you. I'm done on my side. Thank you. So it's done. Okay. It seems like the majority, most of people are done. Anybody who still needs a little bit of time? Doesn't seem so. So anyone who wants to try? Anybody? I can try. Okay. Thank you. For the first one, the probability that the speech therapist will consult with at least one child is one. Wait, wait, let's do it so that everybody can understand how we get it. So the probability that at least one child we will write it as the probability that it's greater than or equals to one. Okay. So what does it mean? So the probability is 0.1. That's one child. So at least one child would get one. The probability is 0.1. Okay. So it means you add all of them because it says it's more than or equals to one. Therefore it's equals to one. So this is the probability. This is the right answer. So we look here for the incorrect answer. Remember that. So the next one, number two. So the probability that the speech therapist will consult with one child on any given day. So if I look at it's one, that's what it's supposed to be. So that is the probability of exactly. So it will say the probability that x is equals to one and that probability will be? 0.1. 0.1. And it means this one is also correct. Then we move to the next one. The probability that the speech therapist will consult with no children on any given day is zero. So therefore it's what they are looking for is the probability that x is equals to zero. And since it does not exist on the table, it means the probability also should be equals to zero. Yes. Okay. Therefore it means this is also correct. So we go to number four. So the probability that the speech therapist will consult with more than five children on any given day is zero. So I go to the fifth block. So it will be more than is greater than? More than the five, greater than five. More than five and since there is nothing more than five, therefore the probability should be equals to zero. And therefore it means that is correct. Then we left with one last question. The expected number that the number of children consult with a therapist on any given day is four. That would be the calculation of the expected probability or the expected number. What I did was I took each value and multiplied it by the probability. Yeah. So one, one times point one, I got point one. Mm-hmm. Three times point one, five. I got point three. Mm-hmm. I'm going to point three. Point three. Point, point three. Yeah. Yes. Three times point three, I got zero point six. Mm-hmm. Four times zero point three, I got one point two. And five times point zero point two, five. I got two point four, five. Three point four, five. You got two point two, five. Oh, no, one point five. Three point two, five. Three point four, five. Three, four, five. Three, four, five is the sum of all of them. This is one point four. One point two, five. Sorry, one point two, five. One point two, five. So now we need to do the sum of all the probabilities. So you add all of them. And the answer you get? Three point four, five. Three point four, five. And therefore it means that is the incorrect one. And while we still add the exercises, there is your next other exercise. And when you are done, I just want to hear who says I'm done first so that I can gauge how many people are done with the question. I am done. Thank you. I am done. Thank you. Those who are done, you can continue and answer the next one which just asks you to calculate the expected number. Same question, just calculate the expected number. I'll go back and wait there. And when you are done, those who have responded to the first exercise now, are you done to this exercise? Others, are you still busy? I'm done, mom. Thank you. You can also calculate the expected, the expected number. Done, you get the second one. Thank you. Done, expected value. Thank you. Done with the second one? Thank you. Who wants to do this one? The first one. Which one is the incorrect? Anybody? I'll try. Okay. Must I tell you which one is incorrect to go through each of them? You can go through each one of them. Okay, so for the first one I said is correct because if you add all of them together they should come to one. And what can you do get? For number four, I've got a 0.3. So that is correct if you add all of them up. Okay, then I said number two is incorrect. Why is it incorrect? Because I said four would actually equal 0.3. So this is the incorrect one, but we can go through the rest of them. Okay, then I said three is correct. Because if you add it's actually equal to or greater than four. And that equals to one. Because you add all the values, the same as equal to one. Yes, that would be the correct one. And greater than or equals to four. Yes, and then I said number four, it's less than or equal to four. So I added zero, one, two, and three. No, because it says greater than or equals to four. Because if it's greater than or equal to three. So because there's nothing after four. So it's only going to be the same as the probability of x equals to four. Which is zero comma three, which is equal. And we can just disregard that one statement. Okay, so the expected values. I said it was number three, 2.5. It's 2.5 because you multiply all of them. So you will see the sum of x times px, which gives us zero comma one. Times zero, which is the same as zero. Then I said one times zero point one five is zero point one five. You can continue. Oh, sorry, then I said two times zero point two is zero point four. Three times zero point two five is zero point seven five. And then four times zero point three zero is one point two. When you add all of them, they will give you two point five. And that's how easy it is to do stats. And actually there are no more exercises. That will be the end of today's session. Just to recap on what we did, we looked at how to calculate the probability distribution of a discrete variable. And by doing so, we looked at how to calculate the mean, the variance, the standard deviation of a discrete variable. Then we also looked at how to calculate the actual probabilities by using the sign. You need to remember always that the greater than, especially the words at least and at most, they like to use those. And we're going to continue using these terminologies and these words, even from today going forward. Because when we start looking at other probabilities as well, when we look at the sampling distribution, when we look at hypothesis testing, we are going to use the terminologies such as the greater than or equal to the at least. At most less than, you will need to know how to interpret them in terms of the sign when they give you in a wet phrase. And that concluded all the questions that we had for today, or the content that we had for today. Then I will see you tomorrow when we do binomial distribution and Poisson. Now, those two content, the sections that we're going to cover tomorrow, make sure that you have a past exam paper somewhere. If it's printed, have it ready. At the back of the past exam papers, they always have tables. And we're going to use those tables. You will be allowed to use them even in the exam, whether you go sit for the exam and write, or when you are writing online, those tables, they will be given to you. If you have them as a tutorial letter, because I know some tutorial letters they do, gives you tutorial letter 103 or 104 that has tables. Download it, have it ready somewhere. We're going to use that as well, because we're going to use the tables. Instead of you using the formulas to calculate, it's best to use the tables. It makes it easy and quicker to calculate using those.