 Okay. So the person who asked the question now, I forgot what you were asking. It was about poison, right? So you want to know if you have a poison table and you want to... Can you repeat your question again? So I can write the numbers. Yes, I was asking when or in each situation, do you decide to use a poison where Px is greater than or equals to 2? You say it's equals to 1 minus P where x is less than or equals to 1. Okay. So let's assume here, I'm going to use one of these values. Let's assume here we have lambda of 0,2. So therefore it means we're going to be at this point. And the question is asking you to find the probability that x is greater than or equals to 2. So that is saying you need to be finding those probabilities and adding them together, which then it means you need to add the probability that x is equals to 2 plus the probability that x is equals to 3 plus the probability that x is equals to 4 plus that, plus that, plus that, right? In this instance, because there are 1, 2, 3 values that you're going to add and you can add them together. You can add all these values like 0, 0, 0, 1, 6, 4, plus 0, 0, 0, 0, 1, plus 0, 0, 0, 0, 0, 1. Plus you can see that those other ones are just 0, 0, 0, 0. They will not make any difference whether you add them onto that, right? When you add all of these values, they will give you 6, 7, 1, 0. So that will give you the probability that x is equals, x is greater than or equals to 2 will be 0, 176. Alternatively, you can say the probability that x is greater than or equals to 2 is the same as 1 minus the probability of x less than 2 because less than 2 will be the complement of greater than 2. And in this instance it will be 1 minus the probability of x is equals to 0 plus the probability that x is equals to 1 because we're only going to be adding those two numbers. And that will be equals to 1 minus 0, 8, 1, 8, 7 plus 0, 1, 6, 3, 7 which will be equals to 1 minus. This is the answer. If we add, if anyone calculated, 8, 1, 8, 7 plus 1, 6, 3, 7 which is equals to 0, 1. 0, 7, 5, 4. 9. 9, 8, 2, 4. 9, 8, 2, 4 which will be equals to 1 minus 0.9824 will be equals to 0, 1, 7. I'm not doing it right. 0.755. 0, 0, 1, 7, 4. Which is the same as what we got there. Those are the scenarios that you can use the 1 minus. But you need to know that. Sorry. The complement, sorry. The answer is 0, 0, 1, 7, 6. I don't know if I've got it right on my side. 1, 7, 6. Yes, you got it right. I wrote 4 instead of 6. 1 minus 0.9824 is 0, 0, 1, 7, 6. 0.0179. Check if you did write the values right. You should have 0, 8, 1, 8, 7 plus 0, 1, 6, 3, 7 which should give you 0, 9, 8, 2, 4. 7 plus 7, unless you are using a table with 5 decimals. It will also depend on the table that you have in front of you. If you're using the same table as ours with 4 decimals, then you should get the same answer. Maybe it's my calculator that is rounding off. What kind of a calculator are you using? A SHOP. Okay. Is it a financial calculator or a normal calculator? A scientific calculator. Just check your values right there. Okay, so you need to just know that a compliment of a greater than or equal is a less than. A compliment of a less than is greater than or equal. A compliment of less than or equal is greater than. Because the values need to add up to 1. The probabilities. Okay. So then let's look at how we do answer some of the questions, unless if you do have other questions that you have for me. Lizzie, I have a question. There was an exercise that I watched one of your recordings. And then there was okay. It was given that X was equals to zero and lambda was equals to 0.2. So it definitely is a multiple. Sorry. Yeah. So the answers, the three answers were wrong. The fourth answer was e to the power of negative 0.2. And then the fourth, sorry, the fifth answer was none of the above. So as I watched, you ended up saying the answer is none of the above. But when I was using the formula substituting everything, maybe I might be wrong. But I saw that all that like the lambda to the exponent zero and then and the X. Yeah, like the answer was actually e to the exponent is 0.2. So I was confused between the two. I don't know if you understand. I'm going to go to that question. It was on the recordings, right? Yes. So this is the session. Wait, sorry. I think I would have over the session of today. I did. From, from a power point of fact, because I overall the presentation session. So I know which one you're referring to. Yeah, yeah, that was a major. But if you look at this, none of the above, right? And there is no negative number here, right? Can you see that? But that one. Negative number. I think that one did have. Okay, maybe I'm confusing the two, but that one did have a negative exponent and then. I can look at other negative X. Then the other ones that had a formula that would be the only question that had a formula in. But, but, and also it's fine if it's going to be there. I should find the answer for it. Yeah, I remember this one is this one. I did find one as well. So that is one. Yes, I remember. I think I would have also said. And probably they would have been a none of the above answer somewhere. Oh, okay. Okay. Because this should have been a negative. Yeah, yeah. Yeah. Okay. All right. No, thank you. No worries. Okay. So if there are no other questions, I do have other additional questions that we can go through. In the last whatever minutes that are left. Are there any other questions? Yes. One of the questions on the on assignment to feature requires us to have a poison table where we have a 10.5. But the two poison tables I was able to find do not have that value. The poison table expect you to have a 10.5 value. Did they include 10.5? Like they gave you an average of 10.5. It's very difficult to to understand how would have. I would say question would have been stated. If I don't know. Yes. What they wrote it down. Can you read it? Okay. Can you read the question out? Or you can type it on the check. It's always easier to have all the information because then if you just give me one, one number. It might be that they gave you something else and they expect you to use that number to calculate something so that you can get another view. The table ends in this instance and 10 and then they give you the 20s. So there is nothing in between. They wouldn't expect you to use the values that are not part of the table unless in your prescribed book, your table goes beyond 10. Are you still typing? Yes, I'm still typing. I'm using a phone. If you wrote it down on a piece of paper, you can take a screenshot of it and upload it and attach it. There is a thing to attach a picture and then we will have a picture on here. I can have access to it or otherwise you can post it on the WhatsApp and then I will have access to it. Okay, while we're waiting for that, let's look at some of the questions that I have on here. Do you have a question? Yes, ma'am, I also have a question. I'm using a sharp calculator and I'm trying to do the Poisson formula but I don't get the right answer so I don't know if it's the... I don't know how because even if I type in all the questions that we did previously, I'm not able to get the correct answer. Sometimes it's the right error. Alright, so you're using what kind of a sharp calculator? It's a scientific one. Scientific calculator but scientific calculators are also written like different, right? But it doesn't matter. I'll just open one of the sharp calculator depending. Do you have the one where the values are written in green, right? Everywhere? I know some of them they are written in orange. Yeah, but I mean like you do have the one where there is the median written in green on number four, number five. Yes. Okay, because there are different types of sharp calculators so that is why I need to make sure that I understand the type of calculator you have. So you have a calculator that looks like this, right? Yes. Okay, so let's go to our question that we just answered not so long ago. It was on here. You're talking about not knowing how to answer some of these questions. So I'm just going to use the values that we are familiar with, which are these values that we have here. You can see if I want to calculate. Let's assume now. I'm going to remove all this. Let's assume that we want to calculate the probability that X is equals to one and our lambda is zero comma two, right? And you want to use your formula. So your formula will state the probability that X is equals to one is equals to e to the power of lambda. Lambda to the power of X divided by X factorial divided by X factorial. And now we can just substitute our values e to the power of lambda. Our lambda is minus zero point two times zero point two to the power of X is one divided by one factorial. Our answer that we are expecting should be this answer that we have here. So let's go to our calculator. I've got so many other calculators open here. I will open the right one. So we can first calculate e to the power of negative two. So where do you find your e to the power of negative two? It's on the limb button. You will see that there is a button written e to the power of X is written in orange, right? It's like this e to the power of X is written in orange. So therefore it means we're going to have to press the second function. So in order for us to write this e to the power of negative zero comma two, we will have to press the second function. Lynn will give us this on your calculator. It should look like that. And you will need to press the plus or minus button so that you put this negative number on the question, this negative number by pressing the plus or minus. Not the minus of the basic operation, not this minus here. You're going to press the plus or minus button, which will be the negative and then press point two. And if you press equal, it will give you the answer like this. That is not the end because we multiply in with this value there. And then you just say multiply by what are we multiplying zero point two. I will put it in the bracket because I need to raise it. Well, you don't have to put it in the bracket point two and raise it to the power and your power. If you look at your calculator, this one will give you the power and then you just press that button and it will raise it to the power of one, right? That is one. Raise it to the power of one and press equal. I always like to press an equal sign because I need to make sure that it's doing the calculations right as well. And the last part of it of the calculation is to divide by one factorial. And then I say divide by and then you press one. And the factorial is on button number four is written in orange. And then I will press second function four and it will create one factorial and I press equal. And as you can see, the answer that we got is the same as what we expected zero comma one, six, three, seven. You just need to know how to use your calculator and practice with you. Okay, thank you so much. No worries. Any other question? I have posted the question on the chat. You posted the question on the chat. I need to check first to raise it. So consider a poison distribution with the expected number of occurrence per interval equals to 10.5. Calculate the probability that the number of occurrence per interval is equals to 11. Now, because my tables don't have 10.5, I need to go and find the prescribed book to see if in the prescribed book there is a table with 10.5. Just give me a second. Thank you for your patience. I am trying to find the table in the prescribed book. Even in the prescribed book they don't have the poison. How come? You see, a 10-bit B part of the tables. Then I wouldn't know why your lecturer will give you a 10.5. Probably they would expect you to use the online calculators. They should give you values corresponding to the tables. Otherwise, they expect you to calculate it manually. So this is when you use your calculators and the formula as well. So I will write the values here. So the question is calculate where lambda is 10.5. And what did they say? Exactly 11. So exactly 11 is equals to. So we need to find the probability that x is exactly which will mean is equals to 11. So you will just use your formula e to the power negative lambda, lambda to the power of x, divide by x. So you just substitute the values e to the power of lambda will be minus 10.5. 10.5 to the power of your x is 11, divide by 11 factorial and you will find your answer. I cannot give you the answer because the question is from your example for your assignment. So that's how you would answer the question. And that is why it's very important to know how to use your formulas because some of the questions might not require you to use the table. You might need to calculate things manually. Okay. So you should be able to answer that question if you get a question like that where the table does not have all the information. Right. Is there other questions? Are there any other questions? Yes, Lizzie. I'm referring to your notes that you've been giving us on study unit 4. You open that. I need help with exercise 7. Study unit 4. Is it on the notes? Like summary notes. Did you say summary notes? Yes. Okay. Yeah, so it's in session 3. Yeah. Sometimes sharing like this way. Let me stop sharing and then navigate directly to where the things are. Otherwise, you guys are seeing my computer. Summary notes. And it is question number. 7. Question 7. Yes. On the summary notes. Did the title say summary notes? No. Study unit 4. Session 3. Basic probability. Okay. So then it's not the summary notes. It is the one that we used in the session on a Sunday. Okay. Okay. Because then they are different. Okay. And you say it is question 7. Question 7. Okay. So let's look at question 7. I hope that is the one that you are referring to. Is it the one? That's the one. Okay. So on question 7. There are multiple things that they give you the answer. If the probability of A is 0.4, the probability of B is 0.3, the probability of C is 0. The probability of A and B is 0, 1, 2. And the probability of B and C is equals to 0. Then A, B and C are independent. B, A and B are mutually exclusive. C, A and C are impossible events. D, A and B are dependent. E, A and B are independent. Which one of the following statements are correct? Now, you need to take each and every one of them and evaluate the statement and see if they are correct. So let's start with the easy ones. The easy one is B. A and B are mutually exclusive. Are they mutually exclusive? Because if they are mutually exclusive, the joint probability of A and B should be equals to 0, right? So is A and B mutually exclusive? No. So we can cross that one out. The others, I cannot test them unless if I do some calculations. So statement number one, A says B and C are independent. When do two events are independent? We can test it by saying the probability of B given C would be the same as the probability of B. Or we can say the probability of C given B is the same as the probability of B. So how do we test that? What is the probability of B? The probability of B is 0,3. 0,3. What is the probability of B given C? So let's test that. The probability of B given C can be given by the probability of B and C divided by the probability of C. What is the probability of B and C? 0. 0. Divide by the probability of C. Which is 0,5. Which therefore is equals to? 0. Therefore this side is 0,0. Are they equal? No, they are not. So is event B and C independent? No. They are not. Number D, I'm going to skip number C because C says A and C are impossible event. Actually, let's not skip them. Are they impossible event? Yes, they are. Is A and C impossible event? If A is an impossible event, then the probability of A would be equals to 0. If C is an impossible event, then the probability of it will be 0. Are they impossible events? No, they are not impossible events because the probability of A is 0.4 and the probability of C is 0.5. So therefore A and C are not impossible events. They didn't give us that they are joined somewhere, but we know that they are not impossible events because they are equals to that. So they are not equals to 0. They are not equals to 0. So therefore they are possible events. So not impossible. They are possible events. I'm going to skip A and B and A and B independent and dependent. We're going to do them together because if one is not dependent, therefore it means it's independent. So let's look at E says that A and B are independent. We know that if they are independent, the probability of A given B should be the same as the probability of A or we can use the probability of B given A should be the same as the probability of A of B. So what is the probability of A? It's easy to find that one. 0.4 Is 0.4. What is the probability of A given B? They haven't given it to us so we can calculate that. So let's calculate it and see. The probability of A given B is given by the probability of A and B divided by the probability of B. What is the probability of A and B? 0.12 What is the probability of B? 0.3 And what is the answer? 0.4 0.4, therefore it's 0.4. So is A and B independent? They are independent. Are they dependent? No, they are not because if they were not independent then they would be dependent. Which one of the following statement is correct? So it's going to be 5. Only E. And that's how easy it is to answer the probability questions. Are there any other questions? Or can I use the questions that I have? Today it's Q&A. Based on the things that you don't understand let us help you understand them better. Nothing. I'll do spot checks in terms of questions that I'm going to pick on as we go along. So remember this, right? You must always, always remember when you do discrete probabilities, binomial probabilities, and Poisson distribution probabilities. The weight phrases corresponding to the symbols. Meaning, if you interpret the symbol what do they mean? You need to always remember that. So let's look at this one question. So this question says, consider two events that a primary care giver had to go through during lockdown. Waking from home and secure environment. Waking from home is represented by H and secure environment is represented by S. So they could have just said... Hi Lizzie, I can't see the question. Oh yeah, I didn't share my screen. Are you sure? I could see the previous question now. I don't see anything. And now? Yeah. Okay. So yeah, that could have just said consider the two events that primary care giver had to go through during lockdown. Waking from home and secure environment. They could have said A and B if they wanted to. But let's say they use the weights themselves. So H and S are our two event. They give us the probability that H is 0,53 and the probability of S which is secure environment is 0,56. And they also tell us that the probability of H and S is 0.38. To answer questions like this, if they give you the probabilities and they don't give you in the table you can create a contingency table so that you are able to answer the question with ease. Like I said, they could have just given to us as A and B because if it's A and that will be a compliment. If it's B, that will be B compliment. You can write them like that. So because they gave us the letters to use we can use our letters. So they said it's home and home compliment and we have secure and secure compliment. But they're also telling us some information because we know that secure and home and secure is 0,38. We can put it there and the probability of home regardless of whether it's secure or not secure is 0,53. 0,53. And the probability of secure regardless of whether it is and this is where you put your totals, right? And this is where you put your totals and that will be 0,56. 0,56. You can take the information and create your own contingency table and now I can calculate all the other measures that are left on there because these are probabilities. This should be equals to one. Therefore here will be 0,47. And here will be 0,44. And here will be 0,81. And here will be 0, this one is the tricky one. It's 0,15. Oh, there we go. And here 0,29. As you can see I've completed my table I can answer any question that I need to answer from here. So the first question, we're looking for the correct answer. 0,18. 0,18? 0,18. Wait. For H complement. H complement and what? This, I already have 0,18 for H complement and secure. Are you referring to the same? Yes. Okay, so now we can answer the question. So the probability that H or S you need to go back to the formula. Remember the formula? The probability of A or B is given by the probability of A plus the probability of B minus the probability of A and B, right? So we can write it in the same manner using H and S. So here we need to find the probability of H plus the probability of S minus the probability of H and S. Our probability of H they gave us 0,530. Plus the probability of S is 0,560, minus the probability of the joint probability of H and S is 0. Let's calculate. What is the answer? 0,1. 0,7. Therefore, this is not correct. We're looking for the correct one. If event H and S are independent so they are telling us if they are independent what do we know when events are independent? If event H and S are independent then the probability of H and S should be the probability of 0,6. So you know that the probability of H and S should be the same as the probability of H. What is the probability of H? 0,53. Therefore, they are not independent because this should be the same as that and they telling us it's 0,56. That is not correct. That's how you're going to check because they told you that they are independent. If they are independent then the probability of H and S should be the same as the probability of H. The probability of H is 0,53 then it's not correct. It should be 0,53 there. If S compliment or if S with a C-gap is the compliment of S then the probability of S compliment is equal to 0,47. So we've calculated all of them. So what is the probability of S compliment? We find it here at the end. 0,44. It's 0,44. This should be, it's not correct. It should be 0,44. So that is incorrect. If H compliment or if H with a C-gap is a compliment of not S this is my mistake here we should say of H the probability of H compliment H compliment is 0,47. So this should be I can't change it. My bad. This is 0,47. So therefore that is the correct answer that they are looking for. Event H and S are mutually exclusive. H and S and they gave it to us there. Are they mutually exclusive? No. So they are not mutually exclusive because they are equals to 0,38. And the only answer they would have been number four. And that's how you will answer some of this question. Sometimes they will give it to you in a in a question like this as you can see from here I've got a table with home and away and they told me that H is home S is secure environment so the compliment is insecure the compliment of home is away and you can answer the question relating to this. And here they are asking you to find out whether secure and insecure are compliments of one another. Events that home and secure are independent therefore it means you will need to prove that the probability of home given secure is the same as the probability of home and if not then they are not independent. The probability that home and away are mutually exclusive you need to go and test whether the probability of home and away is away. I'm going to use away as a I'll use the first letter to represent since here they use the first letter to represent. Home and away is H are they equals to zero if they are equals to zero then they are mutually exclusive remember you're looking for the incorrect answer right. The event home and insecure are not independent you're going to find whether the probability home given insecure should be the same as the probability of home or should not should not be the same as the probability of home therefore they will not be independent. So let's check let's test that let's test all that you need to complete the entire table if we want to use some of this values so yeah it will be longer writing are you aware that it's no longer writing is it no longer writing or no longer showing on your side. No longer showing on our side and sorry Lizzy I'll take you back I didn't I asked about the 0.18 but I didn't get there you say that you were given so now I see it on question two but no no no you were not given remember I put all the things that you were given it was the 0.38 0.53 and 0.56 we completed the entire table this was didn't get how you calculated the 0.18 sorry oh we take 0.56 subtract 0.38 it will give you 0.18 okay thank you no worries so yeah we add I'm gonna use the same 53.47 0.53 0.47 and yeah I would have been 0. 0.56 and 0.44 yeah I'm just VISA VISA from the table that I was using so how do we answer this question easy compliment event it means it's another event that is part of the sample space and is not included in that so secure and insecure are these two events so it means secure is a compliment of insecure insecure is a compliment of secure we know that so if this was s this would be as compliment or if this was I if this was I this will be I compliment something like that because one they are compliment of one another so this one is right correct but we looking for the incorrect one the event home and secure are independent so we need to prove this what is the probability of home probability of home is 0.35 so we know that probability of home is 0. probability of home is 0.53 what is the probability of home given secure that we need to calculate all given secure we calculated by using the joint probability of home always remember that the joint probability of home and secure divide by the probability of home so the probability of home and secure home and secure is 38 0.38 divide by the probability of home home is 53 0.53 what is that probability 0.72 0.72 so therefore this is the incorrect one because they are not the same the probability of home and away are mutually exclusive those two are compliment event so the probability of compliment event they cannot happen at the same time so somebody cannot be home and also be away at the same time so that will be correct now the last one is asking because then the probability of home and away will be equals to 0 there with the number C the event home and insecure are not independent so here we need to solve for something we need to prove that they are not equal so let's prove that so probability of home we know that it was 0.53 let's see if the probability of home given insecure which will be the probability of a joint home and insecure divide by the probability of the given which is the insecure the probability of home and insecure is 0.15 the probability of insecure regardless of whether they are home or not is 0.44 0.34 0.34 0.34 yes 0.34 0.34 they are not independent because they are not equal that one was 0.53 this one is 0.43 so that is correct so the only question that is incorrect is B I hope you do understand how you're going to answer the questions so sometimes you will not be given only the table with 2 you might be given the table with 3 categories or 4 categories maybe home, away and on holiday something like that you still need to be able to calculate the probabilities based on that okay and sometimes the questions might look like this an experiment can result in 1 or 4 likely events that can happen and these are the events that can happen and they say let A be event where event 1 and 3 happens and we know how you can calculate the probability of A because there are 2 outcomes out of 1, 2, 3, 4, 5, 6, 7 is 2 out of 7 you can calculate the probability of that but they are not 7 sorry they are 4 my bad, I counted these events you should count these events there are 4 of them there are only 2, 2 divided by 4 gives you 0, 5 3 divided by 4 gives you 0, 75 2 divided by 4 will give you 0, 2 for event C as well so based on those events they ask you which one of the following event their statement is incorrect you need to use the A's and the B's to answer the question so A and B are independent how do you prove that A and B are independent the probability of A given B will be equals to the probability of A now the challenge you will have here is that given you A and B but you can calculate A and B the A and B on this event how many events falls within A and falls in B so if I have a circle here and a circle here and this is A and this is B in A we have E1 and E3 in B we have E1 E2 and E3 can you see that? I'm able to create A and B A and B probability will be given by 2 divided by by 4 which is 0.5 so I do have my probability of A and B because it is divided by now let's find out if A and B are independent so we know that the probability of A is 0.5 the probability of A and B let's go find that the probability of A given B is given by the probability of A and B divided by the probability of A and B we did find it because we were able to calculate it is 0.5 divided by the probability of B which is 0.75 what is the probability of A given B? 0.67 0.67 0.67 so they are not equal therefore they are not independent we are looking for the correct answer the probability of A or B is equals to 1 so let's go find out because here we can find the probability of A plus the probability of B minus the probability of A and B Liz before you continue sorry can I take you back? Yes so without doing those circles that you did up there to get the probability of A and B how else can we get it? because you can look at the values that they both are in both you can just do this there are two values that are in both A and B so that's the only way you can find that okay so the probability of A is 0.5 which was given plus the probability of B 0.75 minus the joint probability of 0.5 which is equals to 0. 0.75 because 0.5 and 0.5 will cancel out you will be left with 0.75 so that will be will not be the correct answer the probability of A and B is equals to 0.25 B and C B and C so we now move to B and C let's not use the VIN diagram so how many values are in B and are also in C there's only one right? so 1 divide 1 divide by 4 0.25 0.25 so this will be 1 divide by 4 which is 0.25 which is the value that they are looking for the probability of A given B we did calculate that we did find it the probability of A given B we did it 0.67 so this will be incorrect A and B are mutually exclusive we did find that they are 0.25 so that will be correct and that's how you will answer the questions okay so we spent more time on the probabilities and B's and basic probabilities so you can also find more exercises to do this is yeah when you go and answer questions relating to binomial because they are all asked in the same question paper binomial, Poisson all those things when you read the question and you are given a percentage and you are also given a number then you need to know that you are doing a binomial if they didn't mention that this is a binomial probability you need to know and remember that that you are given an N and you are given the probability of success right and to calculate the average you still remember that the expected mean of a binomial is N times pi so yeah will be 20 times 0.35 which is equals to 7 sorry which is equals to 7 we looking for the incorrect statement right the probability that at most you always need to remember that at most means the probability of X less than or equals to 1 therefore you will need to find the probability of X equals to 0 plus the probability that X equals to 1 and you need to go to the table I need to share my entire screen right in order for you to be able to see the table have you gotten the number the answers so this is binomial so you need to go to the binomial table and on my binomial tables my binomial tables are I need to rotate them this one I need to rotate it the other way around so we know that we looking for 0.35 and N of 20 okay so how many columns 1, 2, 3, 4 4 columns this is 10 N of 20 is this one rotate and rotate it until clockwise so 0.35 N of 20 X of 2 or 3 so we looking for those 2 values is 0.002 plus 0.002 0 0.002 0.002 0 which is equals 2 0.0022 which means that is correct that is correct we still looking for the incorrect one the probability that only one branch which means we looking for X is equals to 1 what is the probability of X is equals to 1 we still on 0.35 the probability that X is equals to 1 is 0.002 0 0.002 0 which means that is correct the average number of branches closed so we still looking for the average we calculated it previously and yeah it says it is 19 so that we know that it is 7 it cannot be 19 to be 7 and the probability that at least 1 the probability that X is greater than or equals to 1 you can find this by saying 1 minus the probability of X less than 1 that is not include 1 right so that will be 1 minus the probability that X is equals to 0 which is 1 minus the probability of 0.002 which is 0.99998 and that is how you will answer some of these questions we can go on until quadratum I just want to go to the Poisson and the discrete list remember with the discrete the table can be like this or they can put it like this Py so this is Y right they can put it like this it's 1 and the same thing so it can look like the way it is horizontally or they can put it vertically where it will still the table will still have the same values whether it's like horizontal or vertical and so on and so on right that will mean it means 1 and the same thing so you need to take the table as you see it and answer the question as they ask you but before you do if the question has a question mark so you can see that yeah we have a question mark you need to first calculate this question mark by saying 1 minus all this other the sum of all the other views how much is it what is number 2 probability of number 2 will be 1 minus you can just minus all the values 0.009 minus 0.376 minus 0.167 minus 0.6 0.661 minus 0.016 and the answer will be 0.371 are we on the same page yes right cool and then you are able to answer any of these questions so the probability that y is equals to 2 it means you were calculating that value they say it's equals to 0 and that's what you are looking for but that is incorrect the probability that it's between this is inclusive so when it says inclusive remember to include 0 and 5 so it's all of them what do we know the sum of all probability should be equals to 1 so this should be correct the probability that y is greater than 1 so it does not include 1 so it's all the values from here going there so you just add from 0.00 so you can add all of them or you can say 1 minus those two values or you need to add all four of these values 1 minus 0.00 9 minus 0.37 6 is 0.615 or you can say 0.371 plus 0.167 plus 0.061 plus 0.016 it should give you the same 6 as 0.615 which is correct the probability that y is less than 2 less than 2 or equals to 2 so it includes 2 as long as there is an equality sign to it therefore it means it includes 2 so you count from here to there so you just add 0.615 plus 0.371 and it should give you 0.756 and it should give you 0.75 because 0.09 and 0.37 6 gives us 0.615 plus 0.7 0.756 and the other one says between but does not include 0 and 5 so it means you only interested in 1 up until 4 so you just add those 2 or you can say 1 minus 0 and 5 and that will give you the answer you are looking for or you can add on 4 of them 1 minus 0.009 short cut minus 0.016 but you need to know what you are doing when you take short cut and the number is 0.975 so short cut it means you take that 1 minus those 2 values outside and that is your discrete probabilities you will find questions that looks almost like that otherwise then they will give you questions where they ask you find the expected value what you need to remember regardless of what they are asking you to calculate or find what is the expected value the expected value is is the mean parameter is the average whether it is if it was for Poisson then it will also be referred to the variance so that is the expected value it is the expected value or we can also call it the expected the expected value or we can call it the mean we can call it the average it is the mean value the average they mean one and the same thing so if in the exam or your assignment they ask you what is the expected value you know the answer to the question it is as easy as that you can't even do any calculation except if it is a Poisson you need to know that it is also the same as the variance and if they ask you for the expected value of a Poisson that is easy to calculate or to find because you don't even need to calculate anything remember also you need to be able to know how to use your formulas so for example this autism South Africa have found out that one out of two people with autism struggles from social interaction assume we randomly select six people living with autism what is the probability that only three people with and now here all of a sudden you are not given the percentages of something like that they give you all the information you need you can calculate your pi by using one out of two it means one divided by two which will give you 0.5 which is your probability of success of that people living with autism and then what else are you giving you giving n and you are giving your x of only three people which is three all they want you to do is to calculate the probability but if you look at the answers this is not something that you can go and calculate it's something that you need to use in the formula and calculate certain things and without certain things so what do we know we know that the probability of x is equals to your I'm going to use ncr in this instance ncr and pi to the power of x 1- pi to the power of n-x or you can use the formula so depending on which one you feel comfortable with x is equals to how do we do this is it n in fact I'm not going to use that other formula now I need to go and memorize what that formula looks like the formula I'm just going to show you it's do we have formulas on this nope we don't have formulas but you know what formulas are there to use anyway so I'm going to use ncr because then my n is 6 my r will be 3 because here I'm calculating the probability that x is equals to 3 so therefore my pi I did find it 0,5 to the power x, x is 3 1- 0,5 n is 6 minus x, x is 3 and you just calculate what is ncr remember you use your calculator so let's say those who are using your calculator your ncr is it oh it's written in orange there it is so you just press 6 second function ncr and then 3 and press equal and that is equals to 20 and that is my answer and looking at this so the second part of the equation I do not have to do anything just say multiply because my bracket means the same thing as multiply 0,5 to the power of 3 and multiply by 1 minus 0,5 is 0,5 6 minus 3 is 3 and I look for the right answer and the right answer is that one and that's how easy and simple it is to just answer the questions and I think we're going to go over that time because now I want to go to the poison but we did poison so so much so we are right on time in terms of Godat and this one gives you 41% so we know that this is the pi, 6 students is our n and sorry not 6 is our n I'm giving you wrong things, 6 is our n and yeah they say at least half of the 6 so you need to also think outside of the box what is half of 6? half of 6 is 3 so therefore our x of at least means greater than or equals to 3 so we need to find the probability that x is greater than or equals to 6 and in this instance I will say go and use the table because the challenge with the table is you don't have 41% let's go there are you able to see the table as I scroll we have we have 40 and then you have 45 can you see that therefore it means in order for you to answer this question you will have to calculate it manually and because there are 16 and we know with Poisson let's go back to Poisson n corresponds to the number of n so it's easy so this will be the probability that x is equals to 3 plus the probability that x is equals to 4 plus the probability that x is equals to 5 plus the probability that x is equals to 6 or remember this is the probability of the probability of x greater than or equals to 3 you can say it's 1 minus the probability of x less than 3 which means you only need to add very few of them the probability that x is equals to 0 plus the probability that x is equals to 1 1 plus the probability that x is equals to 2 you just need to add all 3 of them instead of 84 depending on you how you want to do it so that's how you will answer the question so based on either one of them you will get the same answer I'm not going to do that because I'm just giving you in terms of the Poisson they will give you the average you just need to remember that this is your lambda and they say at least 4 therefore you are going to find the probability of greater than or equals to for now with the lambdas it depends because if I go to the lambda table and the lambda says 1.5 and you can see that it ends online so if the question was the probability that at least 4 so at least 4 it means I must add all this so it means I must add all of them all you can do only those ones it depends on you I will just add this because there are 1, 2, 3, 4 and also here 1, 2, 3, 4 the effort is still the same there are 5 but I mean the values are very very small so the effort will still be the same so you make up your mind as you go along in terms of which one will be easier to calculate whether to add and subtract 1 from 1 or to just add all of them so that's how you will answer the question because that is greater than or equal this one as well is a day or more 2 or more so 2 or more is like the probability of x is equals to 2 2 or more 2 or more I'm not going to go into details in this one at most that that means exactly that means exactly so you just use the equal equal equal everywhere you calculate and this says the estimated value so the estimated value is the same as the average so you just go to the average of lambda of 20 so remember 20 on this one is the last one it will be this table with the 20 but the x values will be different for the 20 as well and then they're asking you about the variance remember this is poison so they also did say it's poison the average the variance is the same as the lambda if my lambda is 20 therefore this is also 20 and what is the mean what is the mean the mean if my lambda my average is 20 therefore it means my mean is also and then the others you need to calculate and those are some of the questions I just put together as well for you to go through but practice practice practice practice it's very important it's very very very important so what happens oh that's the other thing I wanted to say what happens when you have an average of 1.2 let's go to 1.2 1.2 and if you are sitting on 1.2 let's let's assume that we are here on 1.2 and they ask you find the probability that X is less than or equals to 12 find the probability that X is less than or equals to 12 there is no 12 here but because of the question the way it's formulated it says less than or equals to 12 it means it's all these values including the imaginary ones that were not there so we know that the table ends at 9 so it's adding all of them and we know that the sum the sum of all probabilities where lambda is equals to 1 that or 1.2 that will be equals to 1 because if I add all of them they should give me 1 that you should always be aware of but if they ask you what is the probability that it's greater than or equals to 12 greater than or equals to 12 there is no 12 here so that will just be equals to 0 because it doesn't exist it's an impossible event that can okay but if it says less than or equal you can just add all of them even if 12 is not part of the table if they say on this one they say dating is not part of the table but if it's less than or equal so it means adding all of the values up and you just need to remember to do that so I'm looking at this one where it says 10 but we are on 1.4 but I'm just giving you some example as well okay so I will upload this note after I made those small corrections that I needed to do on one of these slides and that concludes today's session I hope you should be able to do that especially those who haven't done at least one submission you are able to do your first submission remember to look at all the all the recordings that we did on a Sunday they should give you at least more or less some ideas in terms of how you can target or answer your questions in your assignment you can see you in July but I'm always available on WhatsApp throughout to support you and assist you with anything you want remember if there is any question that you struggled with and you want to discuss with me during don't take screenshots just remember write down and share so that we we stay in the bounds of what you can do and not do in terms of what is allowed and what not allowed so remember that always right not that I'm going to give you your answers but I will lately steer you in the right direction in terms of how you tackle some questions other than that have a blessed Sunday I will see you in July enjoy thank you thank you thank you