 Today we'll only be doing the activities, but before that I want to run through and recap on what we did on Wednesday because the same concept we learned on Wednesday, we just need to know them today so that we can be able to answer the questions as we go along. And because also on Wednesday, we didn't finish everything. So we will start with the activities that we didn't do on Wednesday, and then we move on into the activities for today. So without wasting any time, on Wednesday we learned the basic concepts of the probabilities, we learned about the probability rules, what are the additional rules, your multiplication rule, and also we did the conditional probabilities. So today we're going to continue with conditional probabilities and do more activities from there and also look at the independence rule. So what we learned, we said probabilities, or a probability is a chance that some event will happen or not happen. And it's always between zero and one. And we said, if an event, we are certain that that event will okay, then it will have a probability of one. Like the sun will come up, it will always be 100% because the sun always comes up and it goes down. The impossible event will be the probability that that event will never happen. That probability of that event not happening, it's equals to zero. And we said also within an event, they are possible outcomes that happens and those possible outcomes, they create what we call an event and also eventually all possible outcome, they come from a sample space, which we will define also a little bit later. And we said with events, we can have a simple event, which means only one thing happening at the time. And we can also have a joint event when two things happen at the same time. Reading while eating, that's another. Watching TV while studying, that's another event because watching TV is that event that you're creating to watch and eating is another event that you are creating by eating. A compliment event is the other event that is not part of the original one. So it means a compliment of A will be any other event that happens and that does not include A. And a sample space is a collection of all possible outcomes or possible events that can happen. And we use those two example of a die because a die has six faces and also the cuts has 52 deck of cuts. And we also spoke about how we visualize the events and we said, organizing and visualizing the events, we can use a decision tree or a VIN diagram or started with the VIN diagram. And we said, within a VIN diagram, you are able to see your sample space. You are able to see your simple event and also your joint event and your compliment events because the compliment events will be those events that are not in those simple events that you would have already identified. Then within a decision tree, it's where you can make decisions as you creating events along the way. Then we also spoke about the contingency table, which is just a cross tab that we can use to visualize and be able to calculate the probabilities. And we said, within the event, you are able to calculate your simple event by using the total and also we can call the marginal probability or way you can calculate your marginal probabilities. And 1,200 will be your sample space and all the events within the table you call those the joint events because there are two things happening. Main and promoted, they create a joint event but whether you are promoted, whether you are a male or a female is just a simple event for promotion. Then we said a simple probability we calculated from a simple probability we calculated from the simple event. And we said to calculate it in terms of a formula we say event satisfying that event or it's a number of outcomes that satisfy that event divided by the grand total which is your X divided by N and in terms of the example we use to calculate the promoted we said it will be 324 divided by 1,200 that will give us the probability of a simple event for promote it. Then we also discussed the joint event and we said joint event we calculated using the joint sorry, joint probability we calculated using the joint events. And also it's event satisfying or the number of outcomes satisfying that event which is the joint event divided by the grand total which is your sample space. And we looked at calculating the probability of a joint event between main and blue promoted. So we wanted to know the probability of main who are promoted which were 288 divided by 1,200 that gives us 0.24 those are your joint probability for main and promoted. Then we also spoke about marginal probabilities and we said with marginal probabilities are the same as your simple event but in this instance you will use the joint events to calculate the marginal probability. So for example, calculating the probability of A will use the joint event of A and B1 plus the probability of event A and B2 plus the probability of event A and B until BK. And that is how you calculate your marginal probability when you are given the joint event. Otherwise, if you're calculating it using the totals you will be calculating your simple event easier because then simple event is just the sum of all the joint events which is the total. So we looked at an example of calculating that if we want to calculate the probability of being promoted then it means if we do not have the total then we just add probability of main and promoted which is 288 plus women and women promoted which is 36. So it will be 288 plus 36 divided by 1,200 which will give us 0.37 which are the marginal probability. And if we have the total we could have just used 324 divided by 1,200 will give us the same answer. So that's how marginal probabilities work. Then we also spoke about other rules that you might have when you work with probabilities like mutually exclusive events. And we said mutually exclusive events are events that can happen at the same time or simultaneously. And we looked at events, also mutually exclusive events are events that cannot happen at the same time. If events cannot happen at the same time then they are mutually exclusive events. And we said for mutually exclusive events the probability of those event. So if we're looking at A and B the probability of these two events will be the probability of A and B will be equals to zero. And remember I said we can use the symbol N intersect or we can write it as the probability of A and B is equals to zero. They mean one and the same thing, intersect or N. Then we also spoke about, I just want to check if the videos are recording. And you are recording, okay. So these are a collect. So we also spoke about the collectively exhaustive events. We said the collectively exhaustive events are events that mix up all this event that are in the sample space. And they need to cover the entire sample space. And with that we said if we're looking at this example where we have A, B, C and D events we said all of them combined they are collectively exhaustive but they are not mutually exclusive because A can also occur in C. So not mutually exclusive or B can occur in C or we can also occur in D they are not mutually exclusive. But A and B on their own they are collectively exhaustive events on their own A and B because they make up all the days within 2014. And they are also mutually exclusive because a day on a weekday cannot be a day on a weekend. Then we defined the contingency table. I'm not going to draw too much on this because we know that the year we calculate this on the total we calculate simple probability within the table we calculate the joint probability from those simple. And we know that the sum of all the values or the probability should be equals to one. And this was just to recap on what we spoke about. We said all probabilities needs to be between zero and one if you get a value outside of zero which is minus or you get a value that is bigger than one it means you did something wrong with your calculation for the probability. Probabilities should just be between zero and one. And we said also a probability we can represent it in terms of the decimal but we can represent it in terms of the proportion which is a percentage format which then we take that relative frequency and multiply it by a hundred to get it to a percentage then be one and the same. So zero will be zero percent and one will be hundred percent otherwise this is zero and this is one or if it's 50% then it will be zero comma five. And we said also the sum of all probabilities is equals to one and if that is the case then a compliment event if we need to find the probability of an A or a probability of a compliment event it will be one minus the probability of an other event since the sum of all probabilities are equals to one. And we looked at this example for the conditional probability because yeah they wanted to find the probability of not drain and we know that that will be one minus the probability of ray and we said that is one minus zero comma seven and that is zero comma three and that's how you calculate your compliment event. Then we also moved on and looked at the addition which is the probability of something or another happening. So we looked at the probability of A or B happening and that will be given by the probability of A plus the probability of B minus the probability of A and B minus the probability of a joint probability and remember this A and B it does not mean you're going to take A and B and then add the probability of B it does not mean that. That is a joint probability which is not the same as A which is not the same as B is the joint probability of A and B. And we said also if and only if an A and B are mutually expressive then the probability of A and B which is that probability will be equals to zero and then when that is equals to zero then the probability of A or B will be equals to the probability of A plus B and this is only if and only if event A and B are mutually exclusive. If those two events are mutually exclusive then the probability of A or B will be equals to the probability of A plus the probability of B. We also went on and looked at this activity I'm just gonna skip through it and then we looked at conditional probability and we said a conditional probability is the probability of one event happening given that another event has already occurred or has already happened. And that is given by the probability of A given B it's given by the probability of A and B happening divide by the probability of B we always divide by the probability of the given or alternatively we can swap them around and find the probability of B given A already occurred and that will be the probability of the joint event between A and B divide by the probability of A and that is conditional probabilities. We looked at this and we also looked at an example and we said sometimes if they give you the probabilities themselves because previously we were working with events and remember that events are whole numbers like five, 10, 15, 20, 100 when you add them they will not give you a decimal but they give you a, sorry when you add them all of them together they should be equals to one whereas with events they are like one, two, three, four, five, six when you add them they give you a bigger number it's just events. So probabilities or proportions like in this instance they give us the proportions of 27% have been promoted 80% have are main so it means they already calculated those probabilities all you just need to know is how to place them in order for you to answer some of the questions that you are given. So for example when they say it's 27% have been promoted then it means regardless of whether they are main or women so we will put it under the total because it is all of them who were promoted with 27% of those and 80% are main so it regardless of whether those people are promoted or not as long as they are main 80% of them were main we put it under main and 24% are both main and promoted so then it means that there's a joint probabilities of main and promoted therefore we put it under and remembering about probabilities knowing that the sum of all probabilities is equals to one therefore it means the grand total or your sample space will be equals to the probability of a sample space will be equals to one and then I can complete the entire table I can find the value for not promoted I can find the value for women I can find the value of not promoted women promoted and main not promoted because to get this value I just subtract 0.27 minus 0.24 will give me main women and one minus 0.27 will give me not promoted so just do those sub directions of the values and then the whole table is complete and then you are able to answer all the other questions that follows. So if we wanted to calculate the probability of A given B but in this instance we want to calculate the probability of main given that they have been promoted so we need to calculate the probability of main given that they have been promoted therefore we're going to use the probability of A and B which is the joint probability so we're going to use the joint probability of main promoted and dividing by the probability of a given, which is the probability of B. So we're going to find the probability of B in this instance is promoted. Therefore, we're going to find the probability of promoted. So we're going to take the probability of main and promoted, which is zero comma two four divided by the probability of promoted, which is zero comma two seven and divide each other. And then we get zero comma eight nine. And that is conditional probabilities. And you can calculate the other conditional probabilities. Now, we also spoke about the multiplication rule and beside the multiplication rule called main all comes from the conditional probability. And this is if you are given the probability of A given B and the probability of B. And they're asking you to find the probability of A and B, which is the joint probability. Then it means you need to apply the conditional probability because we know that the probability of A given B, if they give you that, and they ask you to find the probability of A and B. And this is what you are not given, not given. So you need to calculate that. But they give you the probability of B. Therefore, we can find the probability of A and B by just saying the probability of A given B times, oh, sorry, times the probability of B by removing the probability of B this side when we bring it to the right hand or to the left hand side, we're going to multiply and this side you will be left with the probability of A and B. And that is the multiplication rule. So, and that gives you the probability of A and B is equals to the probability of A given B divided by, or multiplied by the probability of B. And if I take the probability of B underneath the probability or divided by the probability of A and B, you will see that we end up with the conditional probability. However, if in the question they say, even A and B are independent, only if A and B are independent, then the probability of A given B will be given by the probability of A. If and only if even A and B are independent, then the conditional probability of A given B will be given by the probability of A because then it means B does not have any bearing on what's happening or what happens to A. So then those will be equal. And if so, then the probability of A and B will be given by the probability of A because then the given gets replaced by the probability of A multiplied by the probability of B. This only occurs if and only if A and B are independent. So if in the question they're asking you, calculate the probability of A and B and they gave you the contingency table and they never said anything about A and B being independent, you must remember that this is X divided by N. Number satisfying the joint event divided by N. In case they say A and B are independent, then the joint probability of A and B will be given by the probability of A multiplied by the probability of B, only for independent event. Let's look at this example. Probability of B is equals to 0,2 and the probability of A complement is 0,7 and the probability of A given B is 0,9. Find the probability of B given A. At some point, some people might say, oh, it's easy. This is the probability of B given A and this is the probability of A given B. Maybe we can swap this around and say it's the same thing. It's not. Finding the probability of B given A is given by the probability of A and B divided by the probability of A. Now, if those are the things that we need to answer this question, we need to go back to our statement and ask ourselves questions. What is it that we are given here? We are given the probability of A and B? No, we are not. We do not have that. Are we given the probability of A? No, we do not have that. But we are given that probability. So it's easy for us to find the probability of A because the probability of A is one minus the probability of A complement. And since that is the probability of A complement is 0,7, therefore, this will be 0,3. So I have that answer. But I do not have the probability of A and B and I cannot use, they never told me here that A and B are independent. So it means probability of A and B should be equals to X divided by A. That's what I need to use. But I'm not given any X. However, I am given this probability. So I'm given the probability of A given B. I know that that is equals to 0.9, but it does not help me at this point because if I'm given the probability of A given B, I can write the formula as said, this 0.9, they found it by using A and B divided by the probability of the given, which is B. So since they found it using this formula, I can then come back and say, do I have the probability of B? Yes, I do have. Do I have the probability of A given B? Yes, I have. So I can substitute the values. Substituting 0.9 is equals to the probability of A and B divided by, I have the probability of B, which is 0.2, 0.2. Then I can use my multiplication rule because this is 0.2 multiplied by 0.9 multiplied by 0.2 is equals to the probability of A and B. And therefore my probability of A and B will be equals to 0.9 multiplied by 0.2, which is 0.18, 0.180. So I have now my probability of A and B. So I can go back to my formula. I know that I needed to find the probability of B given A, which is equals to the probability of A and B divided by the probability of B. Finding this is easy. Now, I have my probability of A and B, which is 0.18 divided by my probability of B, which is 0.2. And that gives me 0.8 divided by 0.2, which gives me 0.9. And that's how you find the probability of A given B. Oh, B given A. Sorry, my question. Just quickly ask. Is it not divided by the probability of A? Oh, divided by the probability of A. Yes, sorry, you are right. I divided the wrong day. Sorry. It's divided by the probability of A and we did find the probability of A is 0.3. Thank you very much for picking that one up. So it's 0.18 divided by 0.3. Yes, you're right. It's 0.6. Thank you. And that's how you will find that, oh, sorry, because I had a couple of questions. And that will be how you answer that question. So remember all these venues that you have. So we'll go to the next section and do the independence. So two events are independent, if and only if. And when do we use this? Sometimes in the question, they will give you statements and ask you to verify whether the statements are correct or incorrect and they say even A and B are independent. You need to remember these formulas. You need to remember this to say, how do I test that even A and even B are independent? You can only test them by applying this. The probability of a conditional probability of A given B is equals to the probability of A. So if the conditional probability of A given B is equals to the probability of A, then it means your two events are independent. Or if the probability of B given A is equals to the probability of B, then those two events are independent. So you will use this formula to test whether they are independent or not. You will use this formula if and only if when you calculate the conditional probability and you are using the multiplication rule and they told you that even A and B are independent, then you use the conditional probability. You will say the probability of A and B is equals to the probability of A times the probability of B because of event A and B being independent. But you can also use these two equations to test the independence of event A and B. Okay, so let's look at this example. We calculated all the values. So we know that the probability of A is 0.3. We know that the probability of A and B is 0.18 and we know that the probability of B given A is equals to 0.6, that we know now. Question number one, it says are event A and B independent? So you need to go back and think about that how do I answer this question? We can answer it two ways or one way or once. We can say, if we look at the probability of B given A, is it equals to the probability of A? Or you can say probability of A given B, is it equals to the probability of B? If they are equal, oh, sorry, I'm doing it all wrong. It must be the other way because for independence, irregardless of whether A was given, has no bearing on B. I had it all right the whole time and then I changed it. So this is B and this is A, so sorry, my bad. So for independence, the probability of B given A should be equals to the probability of B or the probability of A given B should be equals to the probability of A. So we already have the answers here. So we can just check. We can check the first one. What is the probability of B given A? The probability of B given A is 0,6. Is it equals to the probability of B? What is my probability of B is 0,2. Are they equal? They are not. And then that is your answer. Even A and B are not independent because the probability of B given A is not equals to the probability of B. You can do the same with the probability of A given B because you don't have, in the exam you don't have to do two of them. You can choose one of them. So, but you will get the same answer. So let's say with this one we say the probability of A given B, so we know that there is 0,9. It needs to be equals to the probability of A and our probability of A is 0,3. 0,3, they are not equal. Therefore it means the two events are not independent. Are event A and B independent? No, that is the answer. No, they are not independent. Are event A and B mutually exclusive? How do I know that they are mutually exclusive? Probability of A and B should be equals to 0. Is the probability of A and B equals to 0? The probability of A and from my answers, the probability of A and B is not equals to 0 because it is equals to, oh, sorry. You can say it is equals to 0,18. Therefore it is not equals to 0. Therefore, no, it's not mutually exclusive. And that is how you test whether statements that you are given are mutually exclusive or they are independent by using the formulas that you know. Okay, and that concludes what we were supposed to cover on Saturday and with that we can move on into the activities for today. Any question? Any questions? Speak now. When we do the activities, I will give you all the formulas because I will understand that at this point you are not familiar with all the formulas. So I will give you the formulas, but you need to work things out as well. And sometimes I will not give you all the formulas. So you will need to derive some of the formulas from the one, like the multiplication rule. So I'll just give you the conditional probability formula and you will need to do the other. For example, the additional rule where we look at the probability of A or B, if events are mutually exclusive, I'm not gonna give you the mutually exclusive event probability formula, I will only give you one. So you need to make up your mind and know all your probability concepts so that you know that for mutually exclusive you have to drop the A probability of A or B. Okay, so if there are no questions with that regard, then let us go into activities. I just want to share my screen, not the... So are we ready? We're going to start. Okay, exercise number one. Autism South Africa collected the following information on specialist consulting with children living with autism, ASD. The table below shows the number of boys and girls consulting different specialists. What is the probability that at a random chosen or that a randomly chosen child who consulted a neuropsychologist is a boy? Which means calculate the probability of neuropsychologist and a boy. So are we given a table? Before we even start answering the question, let's first complete the entire table. So first calculate all the values that are given. So I'm only going to do this one activity with you once and then I expect you to do... There will be questions relating to the same table going forward. So we're going to use the same table again and again and again and again so that we get used to the activities. So 90 plus 30, it's 120. So you can do the calculations with me and answer the questions. 45 plus 15. Oh, sorry. It's 60. Yes. So you need to write this table somewhere because you're going to use it in all the activities and it looks like this. It's not completed for all the questions as well. So you need to quickly write this table somewhere with all the values completed because when I move to the next slide, the values that we are putting here will disappear. And then now I can calculate this value here as well which is 55, by saying 55 minus those two values minus 30, minus 15. That's 10. Because if I add all of them, it should give me 55 because that is the total. And that's 10 and then 30 plus 10 is 40. I can also add all the values. Since they didn't tell us how many they are so it's 120 plus 60 plus 40 equals to 220. And this also, we can add them or I can just say 220 minus 55 or I could say 90 plus 45 plus 30. It should give me the same answer which is 165. Even if I say 220 minus 55, I should get 165. So do you have the table? Remember you didn't? Not yet. Okay. Not yet. Yes. Now, we can go and answer this question. Remember the question we were asked was calculate the probability that a boy and neuropsychologist or neuropsychologist and a boy. So there is neuropsychologist and there is the boy. So we know that the probability of A, the formula that we know is the probability of A and B is equals to X divided by N. Number satisfying the joint event divided by N. Are we done? Should be quick. If you can. Okay. So what will be the probability? 0,5. And boy, number satisfying the event. Neuropsychologist and a boy is 45 divided by the total which is 220. Then your answer is 0,5. 2,05. 0,5. 2,05. Which means our correct answer is option five. That was easy. Remember we can also use the checks to post. So let me see, I didn't open my check function. I'm sorry about that. That is why I didn't know whether you guys have answered the question or not. Remember, let's use the check function as well while we're busy with the activities. Okay. Next second question, like I said, we're going to be using the same table. So you need to have complete, you have the table in front of you to answer the following question. The probability that NP or a boy, we know that the probability of A or B is equals to the probability of A plus the probability of B minus the probability of A, B, not all. So you can also do the same with your probability of NP or boy. You can complete the equation. Fiso has answered others. Do you agree with what he has? You can like his answer or post a new other answer that is different from his. Are you done? Are you lost? Talk to me. Hello, lost. So let me give you the formula. If you look at this formula that I gave you because you will get the same formulas in your exams as well. And this are the formulas that you need to familiarize yourself. So take the same question and rewrite it the same way as you see this. So we know that A or B and I wrote it here. So if I rewrite it as NP or B, I know that I need to write the probability of A. So which means is the probability of the first one which is NP. So it will be the probability of NP irregardless of whether he's a boy or a girl plus the probability of B because there is all B. So here I also have a B. So it will be plus the probability of a boy minus the joint probability of A and B minus the probability of NP and a boy. And a boy. So I know this is a joint probability. This is simple probability, simple probability. So because here we are given events. Always remember events, the probability of A will always be given by X divided by N which is the observation satisfying that event divided by how many there are. For a joint event probability of A and B always is going to be given by the event satisfying a joint event divided by the total sample space. So what is the probability of neuropsychologist? How many events satisfy neuropsychologist? 45. Nope. 60. It's 60 and we also divide by the total sample space which is 220. Plus we also have a simple event here. How many are boys? 165. It will be 165 divided by 200 and 20. Minus the joint event of neuropsychology and a boy will be 45 over 220. Then for this we can write it as 60 plus 165 minus 45. Divide by 200 in tweet. And when you can't play that, what do you get? 60 plus 165. 165 is 225 minus 45. You get 180 divide by 220. The answer you get is. 04281181818181. Which then it means the answer is those who are lost she got it. She no way she made a mistake. So guys, when I give you an activity and you feel that you don't know how to answer the question, don't wait until I ask. Just ask say, I am lost. Repeat, help me, just ask. Let's look at the next example, or exercise. Same question. What is the probability that a randomly chosen child is a boy given that he has consulted a neuropsychologist? The formula that we know about conditional probability because we are told given. So we know that the probability of a given b is equals to the probability of a and b divided by the probability of the given, which is the probability of b. So knowing this formula, try and rewrite your formula based on this. The probability of a, probability of a boy given b, which is neuropsychologist. And then complete the whole equation. The probability of a boy given, they visited a neuropsychologist, or they consulted with a neuropsychologist. And remember also, the probability of a and b is given by x divided by n. The probability of b is given by event satisfying, the event divided by that. So you need to always remember that, that you always need to divide by your n. And I think for those who never did meet at high school and you've never learned about divisions, because you will get two fractions, one for the probability of a and b because it's s divided by n divided by the probability of b, which is x divided by n. So you will have two fractions. So you need to always remember for a division, we say kcf. Keep the first fraction, change the sign to a multiplication, and flip the second fraction. So if I have 3 divided by 4 divided by 5 divided by 8, those are two fractions. kcf says I must keep the first, which is 3 divided by 4. I must change my division to a multiplication, and then I must flip. So the value that is at the bottom must come to the top. And the value at the top must come to the bottom. So then it means it's 8 divided by 5. And that is what you're going to do when you answer this question. And with multiplication, we know that we can simplify already, or we can multiply what is at the top with what is at the top and multiply the bottom with the bottom, and then find the answer. So you're going to get that. And that is the only hint I can give you right now, especially for those who, for the first time. I'm sorry. Are we happy now? Fiso has posted that, and Lungile has posted that. I'm not sure if Lungile is posting the previous, or is it for this one. But anyway, we have two answers. Let's see, we have two answers. Darabo also posted the same as Fiso. Are we happy? OK, let's answer the question. The probability of B given N is given by, or B given Np will be given by the joint probability of both, which is the probability of boy and neuropsychologist divided by the probability of neuropsychologist. Now, the joint probability for neuro, boy and neuropsychologist is 45 divided by 220. 45, I don't know what my pen is writing right now. 45 divided by 220 divided by, because I need to remember that there is a division. The probability of neuropsychologist irregardless is 60 divided by 220. Applying the KCF methodology, we keep 45 divided by 220. We change the sign from a division to a multiplication. We flip, 220 comes to the top, and 60 comes to the bottom. In terms of math, we can simplify. In terms of math, we can simplify. 220, then cancel out. And then we are left with 45. So because this will be one there and one there. So 45 times 1 will be 45 divided by 1 times 60 is 60. Which then gives us 45 divided by 60 is 0,75. And if I look at that, that will be number 3. Esfiso and Karabukorit. Happiness, are we happy? Miss Liz. Yes. So when you calculate this, do you say, because it says boy given, is that how you write it? Then that's how you will calculate it. If it's chopped around, it means you're looking for the boy instead of the neuropsychologist. So if the question was, is the boy consulted a neuropsychologist given that he's a boy? So a child consulted a neuropsychologist given that he's a boy, then you will start with neuropsychologist on this site and then a boy as a given. So this given tells you when to put the line. So this site is the boy and that site is neuropsychologist. Sometimes they might not ask the question in that manner. They might say, what is the probability that a chosen child, given that he consulted a neuropsychologist, is a boy? So you must read your questions carefully. Given that he consulted a neuropsychologist and is a boy, you will need to know that the given part goes with the neuropsychologist. So you must read carefully. OK, so moving on to more exercise. Same. Now we have different statements that we need to verify if those questions are correct. So we're going to work together now. You must unmute your mics because we need to go statement by statement by statement by statement by statement so that you know how to answer this question. And if you know the formula by now, when we answer the question, I want you to tell me what the formula looks like and then how do we calculate it? Don't take shortcuts. Don't give me, yes, this is correct. That is incorrect. I want to make sure that you practice as much as possible so that you know how to answer the questions, even if we're not looking at the same question. OK, so completing the whole table again, 120, 60, 40, and 220, and 10, and 165. So now, number one, how do we answer this? We're looking for the incorrect statement. Probability of a boy is x divided by n. That is 165 divided by 220. And it's 175. And is it the same? Is it correct? 175. This is 175. Therefore, this is correct. Moving on to the next one, the probability of boy or a girl, I will write the formula, which is the probability of boy plus the probability of girl minus the probability of boy and girl. So how do we find the probability of a boy? This is 165, the total for boys. OK, it will be 165 divided by 220. Probability of a girl? 154 divided by 260. When probability of a girl and a boy? Zero. Which will be zero, because a boy and a girl cannot happen at the same time. Therefore, they are mutually at the same time. Yes. So calculating the probability of a boy and a girl, boy or girl, it's equals to one. Because 165 plus 55 is 220 divided by 220 will give us a one. So therefore, it means that is correct. Calculate the probability of a girl or speech therapist. So yeah, we need to find the probability of a girl plus the probability of speech therapist minus the joint probability, probability of a girl and speech therapist. And that will be given by, let me remove this. I'm going to use the same formula. That is given by girl, speech therapist, and girl, and speech therapist. What is the probability of a girl? How do we calculate this? It's like a life to life. Probability of speech therapist. 120. 120. 120. Right. Probability of a girl and speech therapist. 31 to 20. And calculate that. What is the answer? 0.659. I also put this point. Pardon? 0.659. 0.659. Therefore, that is the incorrect one. Calculate the probability of speech therapist. It's 120 divided by 220. And it's the same. It's correct. So that is correct. Calculate the probability of speech therapist or neuropsychologist. Which is given by the probability of speech therapist plus the probability of neuropsychologist minus the probability of speech therapist and neuropsychologist. And decide to do B0 because they cannot happen at the same time. As long as they fall within the same category, they can never happen at the same time. So they are mutually exclusive. The speech therapist, we already calculated it before and we found that it's 0.545. Sorry for the speech therapist and for the neuropsychologist. We calculated it so many times before. It's 0.272. 0.2. It's 45 divided. Sorry, neuropsychologist is 60. So it's 60 divided by 220. Which is 0.273. So it is 0.545 plus 0.273 gives you 0.818181. Is that correct? 0.8181. Next, which one of the following statement is incorrect? So you need to do the probability of speech therapy and a boy because his joint probability is just X over N. Sorry before you continue. Please fill the contingency table. Sorry, mine is on another page now. Sorry. I told you to write this some way. We're going to use it so often. Okay. 10 and 165. Okay. So now let's answer the question. S T and a boy. It's 90 divided by 220. 90 divided by 220. Do we get the same? Yes. Correct. Probability of a girl and a P psychiatrist and a girl. 10 divided by 220. Yes. Do we get the same? Yes. Now we go into the probability of giving. So we need to go and find the probability of S T given boy, which is the probability of S T and a boy divided by the probability of a boy. So you can go and calculate the probability of S T and a boy. We already have that. You said that is correct. You can go and calculate the probability of a boy. So calculate the probability of a boy, which is 160 divided by 220. Isn't it 165? Oh, 165. Yes. 165 divided by 220. 0.75. 0.75. We had this some way. We didn't even have to do that. So since I have the probabilities, I can just use the probabilities. So the probability 0.409 divided by 0.75. And that should give me the probability of S T and a boy. It's 0.545. Which is correct. Now calculate the probability of NP and a girl given NP. Given that is a girl, which is NP and a girl and the probability of a girl. So it means we need to go and find the probability of a girl. It's 15 divided by 220. 155 divided by 220. 0.25. We need probability of NP. So you can go and find the probability of NP and a girl. And that is 15 divided by 220. 0.068. Which is 0.068. Now we can substitute the probabilities onto here. And that will give 0.068 divided by the probability of a girl. 0.25. And the answer is 0.273. 0.273. And that is not the right answer. It gives us the probability of neuropsychologist and a girl. We can do the last one. Now with the last one they're asking us to where is that the probability of a boy and probability of neuropsychologist given that is a boy. Is it the same as the probability of neuropsychologist given that is a girl and is it the same as the probability of neuropsychologist? So we did find the probability of a girl and we said is 0.273. Did we calculate the probability of neuropsychologist? We did calculate that and we found that it was 0.273. 0.273. Did we calculate the probability of neuropsychologist and a boy given that is a boy? We didn't calculate that unless it was calculated previously. Nope, we didn't. So we need to calculate that. The probability of neuropsychologist given that is a boy. We calculated the probability of a boy given neuropsychologist not the probability of neuropsychologist given that is a boy. So that is the probability of neuropsychologist and a boy divided by the probability of a boy. So we can go and calculate the probability of neuropsychologist and a boy which is 45 divided by 0.220 which is 0. 0.205 divided by the probability of a boy we did calculate it earlier which is 165 divided by 220. We found that was 0.75. The answer is 0.273. The answer is 0.273. Therefore, it means they are equal. So it is correct and that's how you determine whether the statement is correct or incorrect. So we know that for us the incorrect one. Happiness, you can take three minutes break. I need to go and get water. Those who wants to continue working, I'm going to go to the next question. You can work it out and see if you are able to make sense out of it and then we'll come back and discuss it just now. Take the three minutes break and then we'll do this. Okay, are you all back? Hello, did I mute myself? Okay, we're back. One of the questions was asked on Wednesday was around how many questions are like calculations and how many are like theory? So with probabilities, there are not so much in terms of theory, you just need to know certain concepts as well and answer them. So like this is one of the example where I might say theory comes into play but you will still also need to do some little bit of calculations because they're asking you to test whether things are independent or are they mutually exclusive? So like we did with the activity when we were finishing off with the recapping, we looked at two questions where we needed to look at that. When it comes to this basic psychology, basic probability as well, sometimes the questions they might ask you in this format, like already in a formula for format with the answers, sometimes they might ask you in a sentence format and you need to make sense out of those sentences in order for you to answer the question. So yeah, using the same information we need to find which one of these statements is correct or is incorrect. Remember for independence, we use the conditional probability. We need to test whether the conditional probability of B remember that we need to test the conditional probability of A given B whether it's the same as the probability of A in this instance. So looking at this three statements, the first one says independent, independent, and this one says dependent. So if they are not equal then we say they are dependent on each other. But if they are equal then we say they are independent. So let's check this three statement. So we already did calculate previously remember the answers some of them are here. We calculated the probability of neuropsychology given that is a boy and calculated neuropsychology is given that is a girl things like that. So these questions as well they will be asking those. So they're asking if neuropsychology and a boy are independent. So therefore they're asking us if the probability it will get less of how you write it because they just want to know if event A and B are independent. So here we can say the probability of neuropsychologist given that is a boy we want to test if this is equals to the probability of neuropsychologist. That's all what we want to do with line number one. So if we go back to the question that we answered previously because we already have the answers we don't have to go back and repeat all of them. So we know that the probability probability of neuropsychologist given that is a boy we say 0. 0. 0.733 and the probability of neuropsychologist given that is a girl is given by 0.273 that's what we did there. Remember this one was wrong. And then we also find that the probability of neuropsychologist we said it is 0.273 we already calculated all these values as well so I'm just going to go to the question and answer it. So the first one neuropsychology and a boy we said it is 0.273 remember that 0.273 probability of neuropsychologist 0.273 they are equal therefore that is the correct statement. The next question is asking event neuropsychology and a girl are independent so we do the same we can do the same with the next one. So we need to find that the probability of neuropsychologist given that is a girl should be giving us the probability of neuropsychologist they should be equal. We did calculate the probability of neuropsychologist given that is a girl we found that is equal to 0.273 and we said the probability of neuropsychologist is to 0.273 and that's what we did in the previous one. We did calculate them here neuropsychologist given that is a girl we said is the same as that. So we are just validating the question here so it's independent do you want to ask a question? Yes I'm lost. Why are you lost? Where are you lost? I'm lost sir. Village of neuropsychologist with me. Let's go back to this question that we did previously remember we did this question. Yes it's not 0.273 we calculated the probability not end you must remember this is not end if they're given. We calculated that we did this so we said it's correct here because we calculated the probability of neuropsychologist given that is a boy we said it's 0.273 I'm fine now. We calculated this neuropsychologist given that is a girl we said it is this if we didn't calculate all these values we could be calculating them again using the formula there we could have been doing this but we already have the answers from the previous question that we answered so we're using those answers to validate some of the statements and that's all what I'm doing right now. So we have that neuropsychologist is the same as neuropsychologist given that is a boy they are equal so therefore that statement is correct and also the second statement is correct because neuropsychologist is the same as the neuropsychologist given that is a girl they give us the same now the third statement it says event psychiatrist and a boy are dependent so therefore they are saying to us in this we need to test that the probability of psychiatrist which is P given that is the same as the problem is not the same as the probability of a psychiatrist so since we didn't calculate any of these views we need to go and calculate them so let me just double check we calculated probability of STST so I'm just going to go back so we never calculated psychiatrist on its own so let's go ahead and do this calculation so the first one I'm going to remove all this because we don't need them anymore now we're done with them so we're going to start with the first site which is the probability of psychiatrist given that is a boy it's given by the probability of psychiatrist and a boy divided by the probability of a boy so we know what the probability of a boy is it's 0.75 but we don't know what the probability of psychiatrist is so this is 0.75 from the previous exercises that we did we did calculate the probability of a boy as 0.75 now let's calculate the probability of a psychiatrist and a boy we did calculate that so I don't have to go no we didn't we did and again so we need to calculate psychiatrist and a boy so which is p and a b which is 30 so the probability of p and a b is 30 divided by 220 which is 30 divided by 220 do you get the same? 0.136 is 0.136 so now divide that divide by 0.75 what do you get? 0.181 do you get that? 0.181 0.181 0.18 I'm getting 182 okay so now I know what this probability of the probability of b given p given b is I need to find the probability of p so the probability of p it's given by x divided by n which the probability of p is 40 so it will be 40 divided by 220 and what do you get? 0.181 do you get that or which is 82 0.182 do you also get the same? yes therefore 0.182 0.182 is equal to 0.182 therefore it means they are not dependent but they are independent so then this is incorrect so you just need to check the statement so if they are equal they should be independent if they are not equal then they will be independent so because they are equal then they are independent not dependent statement number four says ASD is more common in boys than in girls so this is just a general statement so remember this are the number of boys and girls that consulted but they have ASD so this says it's more common in boys than in girls therefore it means boys they should be more boys than girls so if I look at this there are more boys so yes that is correct so we know that the first one was also correct the last one it says the speech therapist and psychiatrist are mutually exclusive speech therapist and psychiatrist if they are in the same category they cannot happen at the same time therefore this is correct because they are in the same category so they are mutually exclusive which is correct and that's how you validate the statements so you will have to go through each and every one of them do some calculations if you have to or use the knowledge that you know in order to make that decision like with the last statement any question then we move to the next exercise exercise seven state let X represent an event that someone lives with ASD it represents someone lives with ASD B represent that someone does not live with ASD so A is a complement of B or B is a complement of A in this instance which of the following statement is correct so number one it says the probability of A and B is the same as the probability of A times the probability of B is that statement correct so why that one who said no why are you saying it's not correct looks like you are adding them together you need to be adding them together but here we came for the joint probability of A and B where they both meet so is A this question you need to ask yourself the following is A and B independent the question is not saying are they equal is A and B equal the question here is A and B independent because if they are independent then this statement will be correct therefore it means the probability of A given B should be the same as the probability of A and because we are not given as many information for us to make that decision we cannot assume statement one is correct at this point because how will I know that A and B are independent if I cannot find the probability that A given B is the same as the probability of A at this point we can leave it as a question mark or we can make a decision based on that based on the probability of A being the same as the probability of A so if we look at number two it says the probability of A given B is equals to the probability of A which is the statement that I just said the question we need to always ask ourselves is A and B independent does event A has any bearing on what happens to event B that's all what you need to be there's a person who living with disability have any influence on what the person has on the person having ASD person with ASD does that person has an influence on what happens to the person who does not have an ASD person and since that is not independent then also B number two as well cannot be correct it says the probability of B is not equals to the probability of A complement we have the probability of A and we also have the probability of B and we know that the probability of A is a complement of the probability of B therefore it means A can be the complement of we can write it as we can write B as the probability of A complement B B is a complement of A because A and B picks up the whole sample space I think someone is making noise I think someone is making noise now it's time to hear you I'm sure I will move myself with the neighbors okay remember we have A and B they all make up the sample space and I said the probability of A is a complement of B because the sum of all makes the sample space so B is a complement of A so we can also write it as the same S B is the complement of A but B is not equals to A but it's a complement of A complement of A as A to the power of C remember when we were doing the complement I said we can write the complement with a C at the top or we can write the complement with a B and we know that the sum of the probability of A plus the probability of B should be equals to what which is the same as the probability of A the complement which is B which is A complement will be the same as one we can write B as A complement because B is the complement of A and yeah it says it's not so it also means that is not correct so we know that all those three statements are incorrect because A is not independent of B and B is not independent of A and we know that B is the complement of A so this statement says they are not so we cannot assume that I'm going to jump all of them and go to five based on those three statements at the top that I said they are not correct therefore it means even A and B are not independent is even A and B mutually exclusive yes they are mutually exclusive because everyone living with ASD cannot be also living without ASD you can either be one or the other therefore only option four is the correct one you cannot be living with ASD and not be living with ASD so you have to be one or the other your coin cannot be ahead in a tail at the same time it can be one or the other it can only land on one or the other either it lands on a head or it lands on a tail it can never have two on one the same you can never be living with ASD and not be living with ASD so you are A and B in this instance they are mutually exclusive and that's how you answer the questions you're going to have to interrogate the understanding of everything that you've learned in the basic psychology or in the basic probability I know I keep on saying psychology because we're talking about psychiatrist and all that okay let's look at another example we only have seven minutes left so let A represent that someone lives with disability or ASD and B represent that someone does not live with ASD suppose that the probability of A is 0.96 and the probability of B is equals to 0.04 find the probability of A or B remember what we discussed previously there you need to take that into consideration when you answer this question so you have the probability of A or B as your formula and we know that there is the probability of A minus the probability of A and B answer the question the answer is 3 why is the answer 3 the probability of A is 0.96 and the probability of A is 0.04 and A or B is 0 because there are mutually exclusive events and therefore the probability of A plus the probability of B is equals to 1 so if we didn't do exercise 7 before you will still when you come to this question you will still need to ask yourself the same question is A and B mutually exclusive am I able to calculate the probability of A plus the probability of B or the probability of A and B you cannot from the statement that they have given you as well okay let's see if we can in the next 4 minutes fit in another long question which one of the following statement is incorrect with regards to the probabilities if an event has a probability of 0.25 then that event has a 25% chance of occurring is that statement correct a statement is correct because they just take that and convert it into propositions number 2 a probability is a chance of an event not occurring that doesn't sound right that is incorrect that is incorrect when 2 events are mutually no sorry yes that is incorrect when 2 events are mutually exclusive they cannot occur together is that correct correct yes that is correct when 2 events are independent they can occur together but they do not influence each other actually number 2 is correct remember is a chance of an event happening or not happening because probability either an event can happen or an event cannot happen if an event cannot happen if it's an uncertain event which cannot not happen we call that an uncertain event which has a probability of 0 and if an event is happening then that we call it a certain event and it can have a probability of 1 so this question is asking so this one this next one it says when 2 events are independent oh no oh no I am lying to you actually this is not correct because probability is a chance of an event happening whether it's not happening or not happening but it's a chance of an event happening this next question is asking when 2 events are independent they can occur together but they do not influence each other of course 2 events can happen together and not have an influence on the other and we say they are independent so remember when we do the probability of A and B those 2 events are happening at the same time but we know that the probability of the joint event we know that B has bearing on what happens to A because we know that we replace that probability of A with the original conditional probability as we know it it would have been the probability of A and B will be given by the probability oh sorry will be given by the probability of A given B times the probability of B since B does not have any bearing on A does not influence A therefore for independence we say the probability of A times the probability of B but they can still happen at the same time and that is independent event so this statement is correct statement number 5 it says the intersections of 2 A and B is denoted by A and B is that correct yes that's correct and that is correct so likewise you have activities 10 up until I think 20 10 up until 19 you can work through that if you have any questions you can ask on WhatsApp or on my UNISA or even here on MST so there are plenty of activities that you can do to get used to before you start answering your assignment questions and with that it is 2 o'clock and that tells me it's time to recap and check it and close the session thank you for being part of the sessions today if you have any question or comment feel free to ask now so you have from question number 11 up until question number 19 to do on your own and ask if you have any challenges otherwise then I will see you on Wednesday when we do study unit 5 which is discrete probabilities which also we