 It right it's Dean beat. Oh Dean, but it's D. Okay, so you Dean beat, okay All right Beat, okay beat. Oh, I see Thank you We learn every time Okay, so Let's get to this week's session welcome To your fourth session where we work into what submitting our assignment to when it opens which will be Very soon. I think this week as well so Would study unit five we're going to split it into three parts so this week we're going to concentrate mainly on the discrete probability distribution and And The next two weeks we're going to look at the other two parts of the discrete Probability, which is poison and binomial so we'll split them into those three sessions two sessions so let's dig in today because Today I realized that the video I've shared with you didn't work on my UNISA and I think I've shared the Wrong other video as well on the what's up So we're going to do things different this week because then you didn't have enough time To watch the video that I wanted you to to watch and I'm not going to upload it anymore Because then everything will be covered in this video okay, so By the end of the the session today you will learn What is a probability distribution of a random discrete variable? How to calculate the mean of a discrete variable the variance and the standard deviation and also how to calculate the probabilities of a discrete Variable before we start that let's go back to the beginning and Start with unpacking the definition of what a random variable is a random variable represent a possible outcome Especially of from a numerical value from an uncertain event so for example When you you create an event of tossing a coin then that Event that you are creating Will have some outcomes coming out of that but that Event that you are creating should be random one and should be You should be able to count The number of times certain things happen within that random randomness that you are counting For example, if I'm tossing five coins and I want to count how many times the coin will land on a head and that the number of times that when I toss it Lends on it on one or those five point lands on a head probably all of them land doesn't land on a head they all land on a tail so the Outcome there is zero because that will be the outcome that the coin never landed on The hate how many times when I toss all those five coins they all of them Oh, not all of them one of them lands on a head and Four of them don't land on a head. They land on a tail. I can count how many times those Times okay, and I can go on and on and on and on and on and on until I have got all my Outcome based on the five coins that I am tossing and that is creating a random Event then from the that random event creates a variable that I can use to count How many times certain things are happening? So we know Also that a random event can either be in both cases It can either be because we're talking about a numerical values. Yeah, it can either be a discrete variable or it can be a Continuous variable for example when a continuous variable will be when I take a thermometer And I check the temperatures of people whether the people have high temperature or a Temperature of zero or one or two or three things like that and then I can use that They they the continuous variable to check The discrete variable will be when the outcome can be countered Because it will come from a counting process and we did discuss the type of variables from study Unit one remember that so in study unit five We discussing things like discrete random variable in study unit six. We're going to discuss everything relating to continuous Variable so today study unit five. We only going to concentrate on discrete random variables so a Discrete random variable can only assume a countable values like for example, we use that if I thought Oh, I roll it die. Let's say I roll it die twice Let X be the number of times four. Okay, so I have a die It can either I can roll it and no four appears that will be zero I can roll it and it appears once or twice or three times or four times Depending on the number of times that you are rolling that die Um, oh if I toss a coin five times Let X be the number of times it lands on a head and it can be zero one two three four or five and so on so when we talk about a probability distribution for a discrete variable usually we talking about Events that are going to be mutually exclusive. Remember those events cannot happen at the same time. So it means they only they cannot influence one another as well, so they should be independent as well and It must also be collectively Exhausted because then it must include all possible outcomes of a sample space of that what happens with Your outcomes they should have also Some probability that corresponds to that outcome that is happening because for the For the fact that we are able to count how many times a coin will land On a on a head and we say zero time So if I toss it five times and five times of it Three three times it lands on a tape Therefore, it means zero time or if one time it lands on a on a head Therefore the frequency will count as one if I toss it and three times of three times it lands on On a on a head therefore I will Count the frequency of three times as three if I toss it and it lands on it tape Then it will go into the zero time because it didn't land on a head So you just need to count the frequencies and after counting those frequencies then you can calculate Your probabilities as well. So for example, these probabilities you would have calculated them from the frequencies Therefore this probability are related to your relative Frequencies right remember when you calculate your relative frequencies you take your frequencies and you divide it by the total Sometimes it will not be as straightforward as this because the table that they will give you for the discrete probabilities might look like this It might be That they give you the outcomes and they say in this family They are five There are zero people living in this household There is one person. There is two people. There are three people living in this household And we conducted this survey and we found that only four people are living in this family There are zero people in that household or maybe let's say this household are people in primary school Let's call it like that. How many children in this household are in primary school So it's four of the families that we interviewed Do not have children in primary school two of them have children in primary school Ten of them have ten children two children that are in primary school Thirteen of them have children in primary school. So I can call this my frequency or my count So this is my frequency and I can calculate my frequency my relative Frequencies right and my relative frequencies. I will calculate it based on this frequency count, which is four plus two Is six plus ten is sixteen plus thirteen Is 29 So that will be 29 and therefore four divided by 29 Will give me I must just open my calculator quickly Otherwise, if you have your calculator quicker, you can give me the count. So four divided by 29 one one three seven point One three seven one one three eight So I can keep two decibels. So I will say point one one four right point one four And two divided by 29 only three decibels Two divided by 29 Not point not seven Not point not seven not point not seven and ten divided by 29 zero point three four zero point three four Thirteen divided by 29 Not point four five Not point four five not point Four five. What do we know about probabilities as well? The sum of all of them should give me one. So let's add all of them They should all give me one. So let's see point one four plus point zero seven plus point three four Plus point four five Should equals to one as my total right Yes, it's concrete That is total. So you can see that I can create my relative frequencies based on this Random free random variable that I have which are the number of children that are in primary school And this will be my probability of x And which he makes up a table like this and I can answer any questions relating to this probability table Or my Probability distribution table. This is what we call a distribution table because it Distribute all my random variables And it includes the probabilities that are related to those random variables that I have. These are my wins from my random variable Interruptions per day and this is my random variable of number Of children in primary school so Okay, so based on this probability distribution, we can answer some questions like for example, we can calculate the expected value remember now with a Reset only Study unit in study unit three. We said we can only calculate The mean the standard deviation and all that for only numerical values but here we're talking about categorical values, but We are talking about categorical values in terms of a discrete Probability distribution and we are able to calculate the expected Or the mean of that But the mean of that will not be the sum of the values divided by how many they are And that is the reason why we cannot use The measures of central tendencies as we use it to calculate the mean of a continuous variable And as we calculate the the the mean of a Discrete variable if we want as well here. We're talking about calculating the expected mean of A probability distribution and the formula that we use is Mean is the sum of your expected value or your outcome multiplied by your probability the corresponding probability what we mean Remember our table had our x outcome and its corresponding probabilities to calculate the mean we say The outcome multiplied by its corresponding probability plus The outcome multiplied by its corresponding probability plus and we continue adding all of them because Of this summation summation means adding up. So we're going to say 0.0 Multiply by 0.35 gives us zero one multiplied by 0.25 gives us 0.25 2 multiplied by 0.2 gives us 0.40 3 multiplied by 0.1 gives us 0.3 4 multiplied by 0.05 gives us 0.2 0 5 multiplied by 0.05 gives us 0.25 and we just add all of these values together 0.00 plus 0.25 plus 0.40 plus 0.30 plus 0.20 Plus 0.25 gives us the expected mean of 1.40 And that's how you calculate the expected mean of a discrete variable Here is your exercise now Based on what we just learned. We know that the sum of all probabilities So the sum of all probabilities Equals to one on this table that we have it has an equal An equal mark A question mark Therefore, it means we don't know what the probability that corresponds with Outcome four is so this we have africa checks knows that election time The number of daily fake news Post about politicians follows the discrete probability distribution Let x be the number of daily fake news posts And here we are given the outcome in terms of daily fake news posts And they can be zero which is no daily fake news at least one day Not at least one daily Exactly one daily fake news two daily fake news three daily fake news or four daily fake news And they are corresponding probabilities, but we don't have The probability for four so we can't calculate the probability for four. How can we calculate it? anyone One minus all the other values One minus the sum of all the other values and therefore you can calculate it for us So the question mark is one minus The sum of all the values zero point one plus zero point one five Plus zero point two Plus zero point Two five and what is the zero for four? No point three Zero comma three Not three point three. Oh you guys are on the Okay, so we know that this is zero point three now. Let's calculate the expected Value so to calculate the expected value We know that the formula is the sum of your outcome multiplied by each corresponding probabilities So all you do is multiply zero with zero point five I can write here p P times x and you can calculate them there and then add them together So zero times zero point one will give us zero one times zero point one five will give us Zero point one five two times zero point four zero point two will give us zero point four three times zero point two five zero point seven five zero point Seven five seven five zero point seven five so therefore we have zero plus zero point one five plus zero point four Plus zero point seven five What is our expected? We haven't done four We didn't do four. Yes four times zero point four One point two It's one point two One point two so we add one point two. Oh you are awake Thank you for being awake on a Sunday Okay, so I'm it shows that I'm still in a weekend mode. Okay, then we add all of them What do we get as an answer? 2.5 Is that 2.5? 2.5 2.5 and that is Our expected mean so We are also able to calculate The discrete variables as well We are able to calculate the variance of a discrete variable And the variance is the sum of your outcome minus the expected value, which is the mean that we calculated Squat you squared the Your outcome minus the expected mean squared Multiply the answer with the corresponding probabilities and because it's a sum you will do it for every individual outcome Otherwise we can calculate also The standard deviation, which is just the square root of your variance. So how do we do that? Remember our table the original one with the interruptions per day. So yeah, our x-values. I've just transposed the table Originally we were waking with it in this and in that way now. I just transpose it And so you can also in the exam find it in this manner, but usually most of the time it is Transpose the way we started with Okay, so To calculate the standard deviation because the variance is everything that is underneath the square root We can calculate that and also calculate the standard deviation So i'm gonna do both at the same time Because you can't get the standard deviation without calculating the variance So we're going to split the calculation into two parts. We're going to first do The first bracket, which is contains the square. So we're going to say Zero minus the expected value remember our expected value that we calculated previously Was 1.4 right our expected value From our original calculations was 1.4 so now Going to say zero minus 1.4 and square the answer we get 1.96 1 minus 1.4 Squat we get the answer and we get up to 5 minus 1.4 and we get the answer Then once we have the answer we can move to taking the answer multiplying it with the probability Which is the second part that i'm doing here. So taking 1.96 multiplying by 0.35 I get the answer of 0.686 And I do everything all of them up until I get to 12.96 multiplied by 0.5 which is 0 comma 4 8 I add all these values together When I add all these values together, they give me everything underneath the square root, which is my variance So they give me my variance My variance of 2 comma 0 4 When I take the square root, which is the square root of 2 comma 0 4 I get my standard deviation Which is 1 comma 4 to 8 there And that's how you calculate the Variants and the standard deviation Of your discrete variable. So using the same exercise that we did calculate The standard deviation of this question So remember you said the expected mean we calculated it. You said was 2.5, right? That's what we got So now We need to calculate the whole thing. So you can do everything together You don't you don't have to do it the way I did it. So what you can do is zero minus 2.5 If you want to save time because you're working on your calculators. So zero minus 2.5 And you take the square root of that Or the square. So we not the square root, but the square of the answer And that gives you 6.25 and you multiply that with point one And the answer you will get there will be 0 comma 6 to 5. So what I'm doing here is x minus the expected value squared Multiply by the cross-pointing probability and that gives me those answers that I'm gonna write there So you also can do that. So you can give me the answers for the rest of the questions as well So 1 minus 2.5 Equals square the answer which is 2.25 multiply that With point one five. I just did the The two You can do the rest keep at least four decimals in order for you to get almost at least the correct answer when you complete the question 0.3375 So I'll wait for you to finish the whole question You will tell me the other values 2 minus 2.5 Square 2 minus 2.5 equal and then square the answer and multiply that by 0.2. What do you get? It's zero comma zero five Zero comma zero five. So let's go to the next one um three minus 2.5 equals square the answer and multiply that with point two five Equals what you get? Nobody Zero comma zero six two five zero comma zero six two five Let's go to the next last one It is four minus two point five Equals Square the answer You get two point two five, right? Multiply that With point three. What do you get? 0.675 0.675 Now add all these values together And give me the answer So you just go back and add all these values together What do you get when you add all these values together? 1.75 1.75 Which is Our variance So 1.75 take the square root of that Take the square root of the variance 1.3229 What should I put to two decimal places? Let's leave it to two decimal places 1.32 1.332 And that's how you calculate The standard deviation of a discrete variable easy, right? If you are lost, let me know right now before we move on to the next To the next question Unfortunately, I am I don't always look at the chat if some if people post on the chat, especially with When we doing some explaining When I'm doing some explanations of hey, so If there are no questions, we move on now. Let's look at how we calculate the probabilities as well So remember I've raised a hand I just want Okay on on this exercise what I would like to maybe You to emphasize uh The calculation it is actually Let's say if I'm on Number three says three minus 2.5 squared Multiplied by 2.5. Am I correct? Yeah, so When you are working on your calculator You see this thing is inside the bracket I expect you to also put it in the bracket if you don't put it in the bracket. Therefore, it means you're going to say three minus 2.5 You must say equal because you didn't put it in the bracket, right? So you will say equal and then once you have the answer once you put the equal and you have your answer Then press the x squared button and then press equal and then press What what do we need multiply by and then you will see Multiplied by 0.25 And probably you will need to also press again the equal sign to get the answer That is what you need to do on your calculator if you are using the bracket and you're following what it does Yeah, so you will use bracket and you say Three minus 2.5 and you close the bracket and then you press the x squared button and you press equal And or you don't have to press equal. You can press multiply by but I prefer to use the equal side So often and once you have the equal sign and then press the multiplication And 0.25 and press equal That will still give you the the answer that you have there All right. Thank you so much. Well said No problem. Thank you Okay, so now let's look at how we calculate The probabilities of the same distribution table I'm just gonna explain Um several things and then we'll go and do some more exercises. We almost done with The basic things. Okay, so in terms of your discrete probabilities and not only in terms of the discrete probabilities from now on going forward This is more relevant Going forward as well Some of the questions you will be given to them They will be given to you in a weird phrase type of like questions You need to use those weights to convert it into a mathematical expression in order for you to answer The question for example When they say what will be the probability that exactly you need to know that exactly is the same as equal Or equal to Uh, or they might say fewer than less than um Below so you need to know how to take these weight phrases And assign the side. It's very very important. For example Let's use this, um Our example that we came with from the beginning the interruptions per day of a computer network So we're going to use this To answer questions relating to this so i'm gonna pick any of the number look i'm gonna stick to two All the way so i'm going to use two as my My question for answering any of these questions So if I need to calculate the probability that it is exactly I need to know that they are asking me to calculate the probability that equals to two exactly two So they are asking me to calculate the probability that it is equal to two So therefore because the probabilities are calculated. Yeah, I just need to go and take The probability which is zero point To zero so what is the probability that at least uh four? That's your question Your question. What is the probability that at least four? anyone It is 0.05 yes, because it's that probability day So you need to be able to read this the quicker someone answers The quicker we move to the next one Don't make me ask you again and again and again and again time Time time time. Let's keep time time time. Let's work together. Okay, so if they are Yes, is it going to be 0.5 of uh four not including of five because I see they say um greater than We said exactly Oh, sorry. Sorry. I thought you We still on exactly exactly now You move into the next one or the next the next next next next the next next Okay, so What if now we that we done with exactly so everyone understand what exactly means, right? What about when they ask you when it is fewer than or they say it is below two When they say what is the probability that x is below two it means x is any value that is not two, but it any value that is less than two So those values that are less than two are zero and one right It's not two, but it's any value that is less than two So it means they are asking you what is the probability that x is equals to zero Plus what is the probability that x is equals to one? That is less than two all of those so it means you're going to say it is zero comma three five Plus zero comma Two five and that is zero comma six zero Right Am I counting right? I think and I hope so that my counting is right. What is the probability? Yeah, what is the probability that it is fewer than four That is your question. What is the probability that it is fewer than four? Okay, I think we're gonna add all from zero one two and three Yes, because they're less than four Yes, you're going to add all this together Which is which is equals to zero point nine zero point nine. All right So you're getting the grip of everything, right? So we get in there we get in there we get in there you understand So i'm just gonna erase this because i'm gonna run out of space Now we move on to the great tata Which they can ask the question above they can say What is the probability that They are three or above three above no above two. I remember I said two is my my my number above two So what they are asking is what is the probability that x is greater than? Two because it's above two So therefore it means they are talking about any of those values that are above two Not including two but anything that is bigger than two which is greater than two And that is zero point one zero plus zero point zero five plus zero point Zero five because it's three four and five probability that it's three four and five That's zero point two Which is Right now let's move on to At most so we done with I'm not going to ask you to do that So if they ask you above or more than so you need to remember all that More than above they mean is greater than What about when they ask you at most and they like asking at most at least so now we are at most at most means It is at most two will mean if x is less than or equal to two And therefore it means it must include two so it will be the probability that x is equals to zero Plus the probability that x is equals to one Plus the probability that x is equals to two It's all of them. So now with at most It means it is Everything below two and including two because it's equal less than or equal Which means it includes that number So therefore zero comma eight zero comma eight Because it's zero point two plus zero point two five plus zero point three Okay What is the probability that it is no more than Five But is the probability that it is no more than five It's zero point zero five Ah No more than five What is that probability You can even find the probability without even doing any calculations because we know that the sum of all probabilities should just Be one Yes, you getting it you getting it you getting it Okay, so we're done with at most at least is the opposite of at most as well Because at least says it is greater than or equal So if I need to find the probability that x It's greater than or equals to two therefore it means it includes Two and any value above two Any outcome above two will be included So it will be the probability that x is equals to two the probability that x is equals to three The probability that x is equals to four And the probability that x is equals to five which will be adding all these probabilities and that is equals to zero point four zero point four So you understand that I'm not even going to ask you with no less than Because now you guys understand everything And now we get to the between Between they need to tell you whether the between if they tell you in words Is it inclusive or is it exclusive because inclusive will mean they in it includes the values therefore it needs to have an equality sign to it and the Exclusive it means they should not be any equal sign next to it. So for example What is the probability that x is between a and b inclusive? So Let's use i'm just going to remove all the the paint all at once So if I need to find the probability that x lies between and it's inclusive it lies between two And five Because it says inclusive So it says x is greater than two, but it's also less than five. So I can start from two And I must end at five. So it says it is less than but it also greater Sorry, it's greater than two, but it also less than two. So it will be between So I've just given you the answer. Therefore, it means it is x is equals to two plus x is equals to Three plus the probability that x is equals to four Plus the probability that x is equals to five Because it's inclusive Of all the values less than greater than five, but sorry greater than two less than five It's those values in between And that will be requires to zero point Four again, right? When fall Yeah, so if they ask you what is the probability that x is between two and five, but it includes Exclusive So therefore they're asking you what is the probability that x is not equals x lies between two and five But exclusive So it means it doesn't include Two and five and that will be I look at this question. It says it will be any value Not including two but less than two and not including five Less than five So that will be only the probability of x is equals to three Plus the probability that x is equals to four only So which will be that will be zero point one five And that is inclusive sometimes they can mix up the exclusives So it can be the probability that x is great Less than greater than or equals to two But it is less than five You need to know how to get to that. So let's read that If they have one equality And the other doesn't have so it says It includes two and it's any value greater than two And it says but it Does not include five and it's any value below five So that will be from there today Because it doesn't include five So any value less than five will mean two three and four So you will get the probability of zero point two Plus zero point one plus zero point zero five And that is zero point three five And that's how you will answer the probability questions Any questions before we move on to the exercises because we are done With me explaining how you will answer questions relating to discrete probabilities Are there any questions? No No questions Then I'm going to give you five minutes to answer that on your own and we will Do some feedback just now Let me give you some question some time to reflect on the question on your own Remember when you have an answer you can post it on the chat And then everybody who agrees with that answer can always Put a smiley face on it or a stand up or a laugh Because you love it and so on and so forth The two is wrong Yeah, number two is wrong Okay, so you guess I guess that everyone is done with the first one Remember we need to evaluate all the questions since we are doing some practice work anyway This is not the exam in the exam The first time you get to the answer you just move on to the next question because it's Exam and it's crunch time with assignments and with activities Practice practice practice So it means go through all the options and evaluate all the options Spend extra time on the question Okay, so let's do that Check Africa. We've been working on check Africa We know that zero point number four is zero point three, right? If I still remember This value here is Is zero point three Okay, so what we didn't cover is what if they ask you questions about The probability that is not on the outcome that is not on the table All right, you remember from basic probability An uncertain event has the probability of zero, right? Which means An uncertain event is an event that cannot occur or does not occur which Also a mutually exclusive event will have a probability of zero But yeah, we're not talking about mutually exclusive events. We're talking about an uncertain event Okay So for example, if they ask you what is the probability that X is X is five Automatically that probability should be equals to zero because there is no five on this outcome So that is an uncertain event that is an event that can never happen All right, so let's answer this question Let X be the number of daily fake news post which one of the following statement is incorrect Number one states the probability of X is equals to one or the sum of all probabilities equals to one Is that correct? That's correct That is correct because if you add all these values, you will get one Number two says the probability of X is equals to four is equals to zero What is the probability of X is equals to four? 0.3 0.3, so that is the incorrect one the probability of X less than four or equals to four It's one It's one because all the values from four and down so it is equals to four that is correct The probability that X is it's greater than or equals to four Is 0.3. Is that correct? Yes It is correct because it says any value from four and up and we know that the table ends at four So therefore is the same as the probability of X is equals to four That's the only value that we have which is 0.3 which is correct And we know that this is another statement that they would like to use to confuse you very much You can even ignore all the none of the above most of the time when you have your answer correct your correct answer But it says none of the above are incorrect, but we know that number two is incorrect. So that's statement is correct Okay, so We came to the end of What I wanted to talk to you about and discuss in terms of the discrete probability But it doesn't mean we came to the end of the session. We ending at half past We've discussed the probability distribution of a discrete variable We looked at how to calculate the mean the variance the standard deviation and how to do the probability of a discrete variable Do you have any question before we go into the exercises in the next 30 minutes Anything that you still unsure of and you still want me to clarify Nothing nada next Okay, so we won't have any problem when we do some exercises There is your exercise number one I'm gonna give you some minutes when you are done. Remember, you can use the check to post your answer and then we'll do the reflection Remember the expected Value It's calculated by the sum of your outcome multiplied by its cross-point improbability So it means you can just take x times px and calculate all the values Are you winning Okay, we'll get to that Somebody gets 0.78 We will get to that Okay, so it seems as if Are the people who still are calculating? Yes All right, we'll give you some time Why the others are still being um, do you know why I can't see the The chat option on my team Uh, probably you are not part of the you did you join us So I did accept the invitation Okay, so yeah, probably because you you might not be Part of the group but I enable the chat For this meeting, I don't know why you would be able to See the chat Check now. Are you still unable to see the chat function? Yeah, I'll drop off the call and join again Okay, are we are we done? Okay silence means you are all done Okay Let's get the answer 0.0 times 0.6 Is zero right one times 0.20 And two times 0.1 So 0.2 0.2 three times 0.5 0 0.5 0.15 Yes, and four times 0.3 0.12 five times 0.1 0.05 0.05 and six times 0.01 will be 0.06 so add them all together So when we add all of them together we get 0.78 which is option three Suppose X simply said Um, why did I get 1.38? You must check your answers because now all the The vapes are here. You probably added the 0.6 On the on the 0.78 Okay, let me check That's what I think We got on each one of them Okay, the answers they Uh, please make sure that they no no, sorry Please switch off the radios when you join the sessions When there are music playing on youtube when we place the videos It will tell us because it's copyrighted music sometimes It might ask us to cut off the section. So we might delete the chunk of Of the sessions as well. So make sure when you when you talk there is no music playing in the background Or people playing in the background and if there are music, please post on the chat because we don't want to Take out any anything from the video so that we can learn from these things Okay, so sorry your question No, no question. Sorry for cutting you but I It was just because of that music but that was playing in the background Uh, suppose x represent the number of students in sta 1610 the probability distribution is as follows And this is the probability distribution If the mean so it means they've calculated the mean you don't have to go and calculate the mean If the mean is 2.52 Then the variance of the best students in sta 1610 is so it means calculate the variance And you know that the variance is equals to the sum of your outcome minus the expected value squared Multiply by its corresponding probability That's what we know. So you can come here and say x minus the expected squared Times px and then do your answers here So the first one I can start it off so that if you are calculating and you're not getting the same answer You know that you are doing some calculation wrong way so Gonna give you time to do the question I will write my answers in silence And I will stop right there so that you can give me I don't have to give you all the answers Are we winning? Yes, we're getting there Are we winning? Are we done? Yes, the answer It's 2.00211 I get 1.6496 Let's let's see how we work it out. So 1 minus 2.52 squared Equal squared Multiply by 0.25. Do you get 0.5776? Yes Right And number is The second one we get 0.89232 2 minus Is that correct? And number three, is that correct? Yes And Yes 0.52856 And number five, is that correct? Yes Correct If we add all of them So you add 0.576776 I'm not gonna write all the values here Saving time Because they are here So you just add them all up Add them Add them Add them When you add all of them What do you get? 1.6496 1.6496 which is option number one So please check your calculations When you're doing the calculations To make sure that you are calculating correctly as well It's very important that you double check your calculations And the answer is number two So let's move on to the next one The probability distribution of a discrete random variable follows the following X represent the number of cars owned by a family X are those probabilities are given Which probability is incorrect? So let's we're gonna do this together I think the couple of questions we're gonna do them together I'm not gonna give you time and then There are about eight questions So we are on question number three So it means the others you can do on your own As practice after this session Which probability is incorrect? The probability that X is greater than one Is equals to 0.35 Yes correct Greater than one Therefore it means we must add all those probabilities That is correct So it's the addition of those two probabilities That is correct The probability that X is less than two Or equals to two It means all of them right So here we're calculating all of them We're adding all these probabilities Yeah that's correct Is correct That could be correct Okay moving on to the next one The probability that it is between One and two Includes because it's got greater than or equal Or less than or greater than or equal Or less than or equal So it is including It's inclusive And they say it's zero point and that is correct Number four The probability that X is less than one It does not include one Because they say it's less than So it means it's this probability here That is true The probability that it is between zero and one But exclusive of zero and inclusive of one So X is greater than zero So it means it's any value greater than zero But it is less than one or equals to one So it's any value less than or equals to one Which is the same as Which is zero for max For zero and that's how you answer the questions Easy right It will be easy right When you get assignment questions Yes Now let's look at it when they give us weird problems So yeah the weird problems The following discrete probability shows the number of workers From Sassol per day which follows a discrete probability distribution Which one of the following statement is incorrect Here Is our probabilities and our outcomes Statement number one The probability that between one and five learners are absent On a given day is zero point nine five Assume one and five are both inclusive So what are they asking? They are inclusive So our X inclusive will mean greater than or less than or equal Right It will include the equality side And they say one and five Always start with the smaller value and then go to the beta value as well So what do they mean? They say any value between that and that One and five Which is zero point nine And we're looking for the incorrect statement right So that will be zero point nine Nine because it's adding all of them Otherwise you could always say also it's the same as It is the same as the probability that One minus Because there are so many to eight You can say it's one minus the probability that X is equals to one It will be the same as the between the statement Because it will be one minus zero point one Which will give us zero point nine You can also do it that way Instead of adding so many numbers You can take the complement of that The probability that eight least What is eight least? Would sorry, ma'am would it be equal to zero One minus probability equal to zero Oh sorry I took it's equals to zero Yes, I took the number from that But I wrote one day in state of zero My bad Thank you for that Eight least what does eight least mean Greater than or equals So yeah we're looking for the probability That X is greater than or equals to They say at least one So greater than or equals to one I think this question has a problem But anyway we're looking for the incorrect statement And then both of them can be correct I think there is a mistake on the question It might be looking for the correct question So not the incorrect We'll circle back to that Because this one says at least one So it's still the same because it's all of them Adding all of them from one and above Which is equals to zero point Zero point nine as well The probability that at most one At most one What does at most mean Less than or equal So the probability that X is less than Or equals to one It's equals to Zero point Three right because it's those two values It's zero it's probability of zero and plus the probability of one The probability that One liner is absent The probability that one liner is absent What do they mean Equal to one Equals to one and that probability is equals to No point two Not point two because it is just this probability here Which means there is a typing error on this question and I think I got it from last year's Tutorial letter or questions Which means the answer is option four Should have said which one of this statement is correct The probability that between one and five are absent on a given day assuming that One and five are both not inclusive So it means exclusive So yeah, they're asking you to find the probability that X Lies between one And five because they are both not inclusive Which means they are exclusive so it means we only looking at Those ones So that probability is zero point two plus zero point three plus zero point one five which is equals to It wasn't a typing error. I mean I think the first two were meant to be Point nine zero instead of point nine five Hmm Yeah, I think so I think one of the only incorrect answer is This would then be three Wait, sorry one and two I think were meant to be Point nine zero Yeah They put the five in error the five in error, right? So let's yeah, so the five is the error So number three is the one that would be incorrect Because this one is correct. This one is correct incorrect. Yeah And this one is also correct. And this one. Yeah, so let's assume that yeah, I'm gonna Rub out On the five and rub out the five here And then that means only option three would be the incorrect one, okay I understand now Thank you for picking that one up So we wait on question number four. So I'm gonna leave you with Exercise number five. You can do them. I've posted this on my unisa. So you have access to the notes With all these additional questions. So a question number five There is the question so you need to calculate the probability Of at least at most the expected value the variance and the between you can see there the between it's inclusive and exclusive and question number six There is a missing value that they say it's probability three You need to calculate it and see if it was zero comma one five and the between And at least at most I told you that they like at least and at most so you need to know those things by heart by now um And the probability that all students you should be able to find that And number seven Also the same a speech therapist will consult four children So this will be equal because they say with With which is the same as exactly With most So which is at most Between inclusive exclusive you can see there there are your hints The probability that the speech therapist will consult at least So probably here they left at at most At most But when you study at unisa you will realize that most of some of these past exam papers and Tutorial letters they've got so many Errors a little little errors that make you scratch your head, but Most of the time they they will give you the points if It's their error if they Found the errors as well or if students are complaining about the errors So when you pick up an error in the assignment send an email to your lecturer, it's very important The more people send the questions the more It will be resolved And let me know as well when you send those messages so I can follow up as well All right, and the last questions I think I said eight right So the other question is just asking you to calculate the expected The expected value and the last one will ask you to calculate the variance um, and That concludes today's session. We are right on the dot or not on the dot two minutes All right, and Are there any questions if there are no questions? Thank you for being here today I really appreciate your participation out of 500 students All of you are very special. I appreciate you For sacrificing your Sunday with me. Thank you Thank you. We thank you. Thank you. Thank you