 I know that I said our break will. Zoom after 10 past, but because of that one exercise studying any at five past. And we will continue and do the basic probabilities from 10 past until half past. And then we do a lot of other exercises for the remainder of the session. OK, any volunteer? Anybody you don't want to talk to me. Anyone? Sam. OK, Sam number one. Number one find the probability of be happening. I said and then on our. The given there we did not. We have the complement of B. Which is 0.5 and then we know that for complement for probability of B. Is equal to the one minus complement of B that is one minus 0.5. Yes, so the answer is 0.5. The probability of B is 0.5. The probability of B is 0.5. Great, then. That made it easier for us to find probability of A or B because now we have probability of B. I then substitute probability of A with 0.4 my pen was beating out. Yes, and then the probability of B is 0.5. Yes, and then we are given probability of A and B, which is 0.1. Then my answer is 0.8. The last one. The last three. The key way there is mutual exclusive, so we know if it is mutual exclusive. Probability of A and B is equals to 0. Therefore, on the formula above. And we'd only have probability of A plus probability of B. It is close to 0.4 plus 0.5. The answer is 0.9. Happy everybody. Any question anyone with a question? No happy. Great. And that's how you're going to find the questions. They're not going to be as straightforward as we. We do when we do when I unpack the concept. But when we do the exercises, I will try and use the examples. As per the questions you get in your exam or the question you get in your assignment so that you become familiar with the steps on how you can tackle those questions. OK, look at conditional probability and this 10 past 1. We can look at conditional probabilities by the end of this session of 30 minutes. We should know how to calculate the conditional probability. We should know how to define what. How we deal with independent events and how we apply the multiplication rule. All of them they relate to conditional probabilities. OK. What are conditional probability? A conditional probability is a probability that I give. It's when you calculate the probability of A given that probably another event has happened before. So let's say if I want to calculate the probability of me ordering coffee after I had a dessert, I will have to use the conditional probability because it will tell me that what will be the probability that I will order dessert given that I will order coffee given that I had a dessert. And that's what conditional probabilities are all about. It's a probability given that another event has already happened. So let's say we want to calculate the probability of A given B and that will give us the probability of a joint event of A and B divide by the probability of B. So we're always going to divide by the given probability. So it will be the joint probability of A and B divide by the given probability which is the probability of, or the given event which is the probability of B. And that will calculate your conditional probability of A given B. And we can also find the probability of B given A which is the joint probability of A and B divide by the probability of A which is also state that it's the conditional probability of B given that A has already occurred. You know that the joint probability A and B is your joint probability. The probability of A is your marginal probability or your simple event the same as the probability of A is your simple event or your joint or your marginal event probability. So how do we then use this? Let's say we have a statement that reads as follows. Of the people who were hired 27% of them have been promoted in the past two years. 80% of them are main and 24% of them are both main and they are promoted. So in the statement, we are given three values which are all probabilities. What is the difference between what I am given now and the previous ones? The previous examples that we use, for example, if I go back, I know that I'm going way, way, way back. Let's say from this. This are events. Different events are whole numbers. The 300 and the 200, this are events. If they have given us the events, we would have divided them by the grand total. How many in total they were? People who are hired in that company, if they were 1200, we will divide by 1200. In order for us to calculate those events. So this are the probabilities as we have been calculating them. So the probability, the simple event probabilities and the joint event probability. So if we read the statement, 27% have been promoted. Therefore it means this is the probability of promoted which is a simple event. And 80% are men. So this is probability of men, which is a simple event. And 24% are both men and promoted. This is a joint event of men and promoted. And since we know these probabilities, then we can draw a contingency table. It is easy. So if you draw a contingency table, you will say on your contingency table, if this is a promoted promotion and not promotion and this are men and this will be women as we had it before. And this will be women. To get the probability of promoted, I know that that is the probability of promoted. It will be zero comma 27. We can divide the percentage by 100. It will take us to decimal places. The probability of men will be zero comma eight zero. I know that the sum of all probability which is the probability of a sample space will always be equals to one because the sum of all probabilities should be equals to one. And remember, this is where you calculate your total. That's why I'm able to know where to place those simple probabilities. The last one is the joint probability, probability of men and promoted. And that is men and promoted, which is zero comma 24. If you look at this table, it's half complete. It is easy to complete the whole table because the whole table, I have all the other values. So for example, zero comma 27 is a compliment of women is men. And to get that, I can just say zero comma 27 minus zero comma 24 should give us zero comma zero three because zero comma 24 plus zero comma zero three should give us zero comma 27. And since I know that the probability of the sample space is equals to one, therefore one minus the compliment of a simple event, probability promoted will be one minus zero comma 27, which is zero comma seven three. And you can do that for the whole table. So also here, I can complete this whole table, zero comma eight point eight minus point minus point two four will give us zero comma five six. And zero comma seven three minus zero comma five six. And I can get zero comma one seven. And one minus zero comma eight zero is the same as zero comma two zero. And there is my contingency table and I can answer any questions that they ask. And which is the same as what I have there, where zero comma 27, our simple event promoted, zero comma eight zero, our simple event men, zero comma 24, which is the joint event and the rest of the other values I can quickly calculate them. Because if I have those three values, I can complete the whole contingency table. What if the question is calculate the probability of men and promoted? We know that the joint probability formula is the probability of A given B is given by the joint event of A and B divided by the probability of B. So if I want to calculate the probability of men, given that they are promoted, then it will be the joint event of men and promoted divided by the probability of promoted because we always divide by the given probability. The joint event promoted is zero comma 24 divided by the probability of promoted, which is that probability of promoted, which is zero comma 27, which gives us the probability or the conditional probability of men given that they are promoted is zero comma eight nine percent. Or it's zero comma eight nine or 89 percent. The probability that your exercise, what is the probability of women given that not promoted? And this is your not promoted and that is your women. What is the probability of women given that not promoted? You know that that is given by the joint probability of women and not promoted divided by the joint, the simple probability of not promoted. The joint event of women not promoted, women not promoted is zero comma 17 divided by the probability of not promoted will be zero comma 75, 73, zero comma 73. And that will give us point one seven divide by point seven three is equals to zero comma two three. We look at the conditional probability formula. So remember that the probability of A given B is given by the joint probability of A and B divided by the probability of B. We apply what we call the multiplication rule. It is because they would have given us the probability of a given probability or the conditional probability of A given B, they would have given us that value. They would have given us the simple probability which is the probability of B, but they will want us to find the probability of A and B. In order for us to find the probability of A and B, we can just cross multiply the probability of B, which then tells me that if I cross multiply, if I cross multiply, then I will end up having the probability of A will be the conditional probability times the probability of B. Please note the following. If and only if they tell you that event A and B are independent, then the conditional probability of A and B will be the same as the probability of A because the probability of B has no bearing on what happens or does not influence what happens to the probability of A. Remember, the probability of A given B is given by the probability of A and B divided by the probability of B. So if A and B are independent, therefore that, oh, sorry. If A and B are independent, then the probability of B has no bearing on the probability of, or the probability of given B has no bearing on what happens to the probability of A. Therefore, this statement doesn't stand. It's the same as the probability of A. That is, if and only if they tell you that event A and event B are independent, then it would mean that if we'd go back to our conditional probability and we want to find the probability of a joint probability of A and B, if we want to find that probability of A and B, given that they are giving us the probability of A given B, because A and B are independent, therefore the probability of A and B will just be the probability of A multiplied by the probability of B. This only applies for independent events. Only for independent events. Let's say, for example, they give you event A and B, but they are not independent. They ask you to calculate the probability of A. Remember, they ask you to calculate the probability of A and B. Remember the following. If they ask you to calculate the probability of A and B, it is the joint probability. If it's not independent, it's the joint probability of A and B will be crossed. So the number satisfying the joint event divided by N. You must not get confused between the two. This is when not independent. If it's not independent, the joint event of A and B is calculated by using the event satisfying A and B divided by the sample space. But if A and B are independent, then the joint event of A and B is given by the probability of A times the probability of B. There is your exercise. And this will conclude our conditional probability. Same as what we did before we went to break. In the exam, they will give you a statement that reads, calculate the probability O. If the probability of B is equals to 0.2, the probability of A complement is 0.7, and the probability of A given B is 0.9, which is the conditional probability. And you need to find the probability of B given A. So they will give you multiple statements. One of the statements will say which one is the incorrect one and they will give you the answer for the probability of B given A. From those statements, you need to know how to find the probability of B given A. The first step that you need to do to answer this question is to find the probability of A, because they haven't given you the probability of A because you will need it to answer this question. Why? You need to do that. Remember, this is the probability of B given A. So to find the probability of B given A, I'm going to change the color of my pen. To find the probability of B given A, it's the probability of a joint event of A and B divided by the given probability, which is the probability of A. In order to find that probability of A is not given on the statement, you need to first find that probability of A. What else are you not given? You are not given the probability of A and B. Therefore, you need to go find the probability of A and B. How do you find the probability of A and B? You are given the conditional probability of A given B. How did they find that answer? They found this answer by using the probability of, the probability of A and B divided by the probability of B. Therefore, it means 0,9 will be equals to that, because that is your conditional probability. So your probability of A given B. A given B is the probability of A and B divided by the probability of, and then you need to go find the probability of A. Therefore, it means you must use the multiplication rule to find the probability of A and B. Once you have the A and B and you have the probability of A, then you can only find the probability of B given A. And remember, this is one option of many options in your answer. Option A might give you the probability of A, option B might give you the probability of A, option C might give you the probability of B. Remember, you're looking for the correct or incorrect answer in your assignment or in your exam. So you will need to know how to answer each statement. But to find only that one option, correct answer, you need to know how to do step number one, step number two in order for you to get to the answer you are looking for. That is your exercise. I'm gonna give you some time to do it. And then we will come back and do it together. So at 25 to, I will ask you to work it out together. Always remember to keep your mics muted all the time, only unmute when you ask a question or you answer a question or you need to comment. Okay. Anybody who wants to volunteer to assist the rest of the class, you can just unmute yourself and let's wake it out together. Sim. Okay, so it seems like you and I will be coordinating these classes going forward. Okay. Okay, I started with number one, find the probability of A. Since we are not even probability of A, then we have probability of compliment of A. I said it's one minus probability of compliment of A, one minus 0.7. And I got 0.3. Number two, I used the formula that you gave us there for probability of A given B, because you already have 0.9 there. And then we have 0.2 for probability of B. So I used probability of A given B equals to probability of A plus A and B over probability of B. Sorry? The probability or you are here? Yes, I'm there. I'm there for what? Sorry. So it's fine. So I have 0.9. Okay. It equals to probability of A and B over 0.2. Then I cross multiply 0.9 by 0.2 to get the probability of A and B. The answer is 0.18. Then it was easier for me to find probability of B given A, because I have all the figures now. Over probability of A is equals to 0.18 divided by 0.3. The answer is 0.6. And that's how you find the probability of B given A if you are given the probability of A given B. Any questions? If there are no questions, then we move to the independence. Two events are independent. If and only if we already covered some of this, the probability of A given B is equals to the probability of A or the probability of B given A is the same as the probability of B. These two statements, you can use them to verify if two events are independent or dependent on each other. Event A and B are independent when the probability of one event does not affect the other event from happening. And that's independence. For example, using the same values that we had previously, we can determine whether event A and B are independent and we can also determine if they are mutually exclusive. For example, if event A and B are independent, we need to find that the probability of A given B should be the same as the probability of A. The probability of A given B is 0,9. The probability of A, we did find the probability of A, previously it was 0,3, which is the complement of that, which is 0,3, therefore they are not independent because they are not the same. Our event A and B mutually exclusive, therefore it's the event A and B equals to zero. We did find that information because we did calculate it, we said the probability of A and B is 0,18. So the probability of A and B is equals to 0,18, therefore not mutually exclusive. Exclusive. Yes, you can ask. Actually from the previous slide, I think you are moving a little bit fast. Hello? Yes, the slide. No, not this one, the previous one again. Yes, here, I did not really get how did you work out the 0.8? Remember the probability. Sorry, the 0.18. Yes, remember the probability of A given B, which is 0,9, how did they find it? They use the formula, the probability of A and B divided by the probability of A, because if we know, this is the conditional probability formula. Is? Let me use a pointer. So this is your formula. The probability of A given B is given by the probability of A and B divided by the probability of B. Our probability of given, we know that is 0,9. We substitute that into the formula. The probability of A and B, we are not given anyway, so we need to find it. The probability of B, we are given the probability of B, 0,2. We substitute into the formula. Then we cross multiply because we need to find or make the probability A and B subject of the formula. So 0,9 multiplied by 0,2. Which makes 0,9 multiplied by 0,2 is 0,18. Okay, thank you so much. I got it. Thank you very much. Any question? We were on exercise too. If there are no questions, I had several exercises. I'm going to skip exercise one, but you can do exercise one on your own time. You can come back to this video and pause this and do exercise one and also exercise two. They come from the same data. If you look at exercise two, let's say this is one of the exam questions that they will give you. You must see that they have incomplete at the whole table. There are missing information. Missing, missing, missing, missing. You need to complete the whole table in order for you to calculate the probability of NP or boy, which is neuropsychologist and boy. And using the formula, the probability of neuropsychologist plus the probability of a boy minus the probability of neuropsychologist and a boy formula to calculate this. But in order for you to calculate this, you will have to find the totals and complete the whole table. This is your exercise for now. We unpack this exercise. They use the contingency table. And this is, I think it comes from your assignment question. You can also pause the video later on and work through them. Which one of the following statement is incorrect? So, yeah, they're asking you to find only one incorrect statement. But all questions that they're asking you is probability statement that you need to calculate. Remember, they are giving you events. So you will always have to calculate the probability by taking the event. If it's a simple event, you will take the simple event value and divide it by the grand total. So the first one, they're asking you to calculate the simple event of social media. The second one, they are saying is your complement of social media, is it the same as the probability of traditional media? Which is 0,25. You will have to find that. You can either calculate the probability of traditional media and find the complement of that because the complement of traditional media is social media, which they have stated in there. Is it the same? Number three, they are asking you, what is the probability of a simple event for celebrity? So you also need to calculate that simple event. You can use the marginal event or you can use the total to calculate this. All of them are simple events. And the last one is the probability of So this is simple event, spot and star, which is that color. So all of them, they are asking you to calculate the simple event and validate and find the one that is incorrect. The other type of an example that or exercise that you will have in your assignment questions are like this. They also want only one incorrect answer. They have given you option number one, you need to calculate the joint event. When you calculate joint event of celebrity and sports star, celebrity and sports star, you need to tell us if that is correct or incorrect. The probability of a joint event of celebrity and social media, celebrity and social media. The next one, you need to calculate the probability of a joint event of celebrity and traditional media, which is celebrity and social media. And remember, because this are joint event, this is the event satisfying the joint event divided by the total. So you will have to use that. And number four and number five, these are conditional probabilities. The first one, which is option number four is asking, what is the probability of a social media given that the person is a celebrity? So we're going to calculate that conditional probability, which is the joint event of social media and celebrity divided by the given probability, which is probability of celebrity. Option number five, the probability of social media given sports star. Also, the joint event of social media and sports star divided by given probability of sports star. And you will need to find the incorrect answer from there. And I think, let me leave it on exercise number four so that you can do this one. And we will, what time is it now? I don't have my time with me. I'll give you two minutes to look at the statement. Okay. Have you figured it out? Which one is the incorrect statement? I'm not going to give you the answers because this is your assignment questions. So this first statement, option number one is asking you to find the condition, joint probability of celebrity and sports star. Can you find that? Is it, are you able to calculate it? Are you able to find, can celebrity and sports star happen at the same time? No, they can't. No, they can't. And therefore it means there are mutually exclusive events. So when there are mutually exclusive events, the probability of a mutually exclusive event is equals to? Zero. It's equals to zero. Then the second one, they are asking you to find the probability of celebrity and social media, which means you need to use 1,800 because that is the joint event of celebrity and social media. Is it correct? I don't know because you need to calculate it. So events satisfying that divide by the sample space, which is 4,000. So I'm just gonna write only one time. So that will be 8,000 or 1,800 divide by 4,000. And you should get the answer for that joint probability. The second or the third one, find the probability of celebrity and traditional, which is that one. I'm not going to do that. You will have to sort it out and find whether this is correct or incorrect. The last bit, which are option number four and option number five. I'm only going to do one, which is option number four. Probability of social media, given that the person is a celebrity, it's given by the probability of social media and celebrity divide by the probability of celebrity. Which then we need to find the probability of social media and celebrity, which is 1,800 divide by, because it's still an event. We need to divide that by 4,000. And we need to find the probability of, conditional probability of, conditional probability or sorry, the probability of celebrity, which is celebrity, which is that one day. So the probability of celebrity will be 2,285 divide by 4,000, since they both are still event in meds, because this is a division. So we can still write it as 1,800 over 4,000 multiplied when it's a division, we can change the sign. And when we change the sign, we flip the second fraction. So therefore it means 4,000 comes to the top and 2,285 goes to the bottom. And when it's like that in a mathematical way of doing things, then we say 4,000 and 4,000 cancels out and we are left with, we are left with 1,800 divide by 2,285, which then gives us the value of divide by 2,285, 0,79. 0,79. And why did I have to do that one? I didn't know that will be the answer. And you can do also option number five. We're looking for the incorrect statement. You can stop there or you can validate all of the other options. Any question? We have five minutes. With that five minutes, just want to recap on what we did. And close off the session. So what we have done today, by the end of this session, you should be able to complete your assignment one questions and be ready to submit your assignment. And what we've covered today, we looked at the basic concepts of probabilities such as the sample space, the events, contingency table, which are your type of visualization that you can use, like your contingency tables, your VIN diagram and your decision trees. And we looked at what a simple probability is and a joint probability is. We also covered the basic rules of probabilities, which includes the conditional probabilities, which are the probability of A or B. We also looked at when do events become mutually exclusive and when they are collectively exhaustive. We also looked at conditional probabilities, which included the multiplication rule and the independence. Because with the multiplication rule, you just cross because you are looking for the probability of A and B. And for independent event, if the conditional probability of A and B, if the conditional probability of A given B, then it will be the same as the probability of A, because what happened to B has no bearing on what is happening to the probability of A. And that's all what we've covered today. And enjoy the rest of your session. And if we look at some of the formulas, I must highlight this. Some of the formulas that you will be using for answer the basic probabilities that you always need to remember is for compliment event, which is the probability of A will always be one minus the probability of a compliment. And for additional rule, we use the probability of A or B and remember if and only if the event A and B are mutually exclusive. Therefore then the addition rule or the probability of A or B will just be the probability of A plus the probability of B. What else? And yes, and also for only if and only if an A and B are independent or they tell you that they are independent then also the conditional probability of A given B will just be the probability of A. Or the probability of the conditional probability of B given A will just be the probability of B. Like the joint probability of A and B or conditional probabilities, if and only if the events are independent then the probability of A and B will be the probability of A times the probability of B. Or you can write it as the probability of B times the probability of A. It means one and the same thing. And that's it for today. If you have any questions. Yeah, just one for me. In the example, I don't give you these formulas, right? They do give you these formulas in the exam. So for example, let's say, because now there's going to be a confusion. So since you are writing online exam because of the COVID-19, these formulas are available to you to use because you have them in front of you in your study guides in everywhere. Your exams are not proctored. So nobody is watching what you are doing on your own spare time. So you have all the formulas. But let's say you're going to change, you need to change and say, oh, your exam is written in November and it is a venue-based. You need to go to your NISA center and write your exam. Then you need to know the following. Formulas are only going to be given. That formula and that formula will definitely be given to you. Only those two formulas. The rest, you can derive them from using those formula. If you are given the conditional probability, you should be able to calculate the multiplication rule formula. You should know how to find the independent rule formula. The only other formula that they might give you is just the probability of A. They might say it's X over N because we also, at the later stage, when we do the probabilities as well, especially when we in some later on in hypothesis testing and sampling distributions, we're going to be using some of these concepts again. So they will give you this formula that looks like this. And they might also give you the formula for the joint probability. Remember that formula for the joint probability is just the joint probability for events satisfying the joint event divided by N, which is different to when the events are independent. You need to be very careful as well. So if events are independent, then we use this formula only for independent. Otherwise, the event satisfying event, joint event divided by the sample space. So those are the only formulas you will get in the exam. You will not get the complement formula in the exam. You need to know that there is one minus the other event that you are given. You will not be given the multiplication law or formula. You will need to know that it's the same. You will not be given the probability of B given A. You will only be given one way. It says the probability of A and B. And you will also be given the probability of A or B. But they will not give you for independent event or sort of for mutually exclusive event the way we know that the joint event is zero. They will not give you that. You just need to know that. Okay, any other questions? No questions? If there are no questions, enjoy the rest of the weekend. Please don't be strangers. If you need help with your assignment, there is WhatsApp and there is my UNISA for us to engage and I should be able to help you. And thank you, Yuhan, for posting on my UNISA. Last week I had challenges with logging onto my UNISA. I did see your comments. I will go through them and respond. Thank you very much, guys. Enjoy the rest of your weekend. Don't be strangers. I'm here to help you. And I want to see you succeed and pass this module. Thank you very much. Thank you very much.