Discrete and Binomial Probability Distribution

Hey, good morning and welcome to second semester session on the statistical literacies. Uh if it's your first time attending any of our sessions, uh if you have any technical issues or don't know where to find the schedule or the links to the recordings and all that, you can send an email to CTN Tat which is kept tart at unisa.ac.za Z A and they should be able to provide you with all the relevant details outside of this session. If there are content related questions where you are still struggling with some of your units or sections that you want someone to assist you with, do not hesitate to send an email to me. Uh e boy emunisa.ac.za. Every week when you join the session there will be two links in the chat. I will also uh pin them so that they are always on the chat. Uh please remember to complete the register. Uh and also at the end of the session you can complete the evaluation form to just let them know how the session went. Um and for those who don't know me, I'm Elizabeth Boy. I will be the facilitator for this. And because uh this are statistical literacy sessions. Remember it's not one module or pure content that we look at. We look at techniques, skills in terms of how do we unpack and answer question related to your module. So I might not touch every detail that is in your textbooks or in your tutorial uh guide but I will give you tips and tricks in terms of how to tackle questions related to s certain components or content in your module. And for today uh session we are going to be discussing discrete probabilities which also looks at the basic properties of a discrete probability binomial probability distributions as well as poison distributions. And for today's session, you should be able to know and remember and recognize some of the formulas we're going to be using. Uh you need especially for the binomial and poison. Uh you will be required to also rely on the tables. There are two tables. Normally they give them to you or they are at the back of your textbook. It's called the binomial distribution table as well as the poison cumulative distribution table and we're going to be using those two tables to calculate the probabilities. You also still need your calculator because everything we do we use a calculator to calculate. So I hope you all brought your calculators with you. So let's start with this week's session. What I'm hoping for you to gain in this two hours that we're going to be having together or spending time together is for you to be able to identify whether a variable is a discrete and how to construct a discrete probability and check the properties of discrete distribution in terms of also calculating the uh probabilities the expected values which are the mean values and the probabilities of a variability of your discrete uh tables or distribution, discrete probability distribution. Also to be able to calculate and recognize when to calculate a binomial uh distribution using the formulas and also the properties of a binomial as well as poison uh distribution and its properties. Okay. So I'm not going to bore you around what statistical literacies are all about. We already spoke about it because my aim and my purpose is also is to make sure that you are able to read and understand and interpret as well as question the information that is presented in the numbers or table or graphs or reports that you have and be able to make sense out of it. So what is a discrete random variable? A discrete random variable like we learned in chap in study unit one where we were describing the types of data. It is a count variable. It's where you have whole numbers. Something that we can count. For example, if there's nothing it's zero. You can count that as zero. If there's at least one thing or if there is one pen, you can say it's one. If you are enrolled for three modules, you can say there are three modules. That is a discrete random variable. That is a number you can assign randomly to a certain object, right? Or when a certain object behave in a certain way can produce that count and that number way you can count the occurrence of that uh activity. A random variable is discrete. Like I said, it is when possible values can be counted. Like for example, a coffee, a Cape Town coffee shop. Uh the variable would be the number of coffee cappuccinos. They sell between the time of 8:00 and 9:00. uh because you can count that uh there were 100 customers. They came and bought 200 um cappuccinos in in in that uh hour. Uh it is a discrete variable or if you are working at the Korea uh Korea company. So uh if you want to know how many laid parcels were delivered today because you can count how many are still outstanding or how many were not delivered on time. You can count the number of parcels like there were 20 or 10 or five of them. It's discrete. Therefore you were able to count them. So if you work in a supermarket and you want to tell what time or the time at the till because the time you measure it. If you want to know the time you spend at a till you are going to take your clock and measure that time because you are not counting the time you are measuring it then it's not a discrete. So in that instance that will go into a poison because you can calculate the average of it and that will be something that we going to be thinking about later on. Uh so because it's not counted is not a discrete. So the time you spend the detail will not be a random variable. So anything you can count like 0 1 2 3 4 5 100 200 uh that number that you are able to use to count it's discrete. Um and that is why you are able to assign it as a random variable to a discrete distribution. So in order for us to use the discrete distribution and calculate the probabilities. So it means we should be able to list all possible values that that object or that unit can occupy. And from listing those uh instances we can then able to calculate the proportion or what we call the probability of occurrence in that because the probability will be attached to the number of occurrence that each interval or each unit uh occupies. And in order for us to calculate the probability, we use a scientific notation. P open bracket X is equals to small X close bracket where your uh P will represent the probability of and in the bracket you are describing which unit and because your random variable X unit you assign it with your uh small X's because you don't know which number of those X values in the discrete A probability distribution there are three properties that you always need to remember. So every probability must be between the value of 0 and one. We did the basic probabilities the last time you remember that a probability can only be between the value of zero and one. It occupies it can also be a decimal number. Right? So it means for every random variable if you want to calculate that probability it has to lie between the values of zero and y. It cannot be bigger or it cannot be less than zero. The the probabilities of all possible outcomes must also sum up or add up to one. So it means the sum of all possible probabilities should be equals to one. And the listed values or your listed x values must be mutually exclusive. So one value must not belong to the other components or the other unit or or the other groups as well. It must be mutually exclusive but it must also be collectively exhaustive. Collectively exhaustive it means it all the values all your random values needs to be accounted for. So one outcome occurs at a time and all possible outcomes must always be included and that's what mutually exclusive and collectively exhaustive means. So mutually exclusive it means one element should not belongs to two groups. It must belong to only one group and collectively exhaustive it means all elements needs to be included in that subset of that subset. So a probability distribution gives uh a possible value and we need to know how to calculate the probability of it. So for example, if a small bakery tracks the number of loaves left unsold at the closing time based on recent weekday, the manager estimates the following distribution occur. So it says um let's let's assume that uh they've got uh they operate on Monday to Friday. So uh so let's assume that on day one uh there were no bread left on Monday, right? Day one. Uh so zero uh how many times uh zero bread were left? How many times one braid were left unsold? How many times two braids were left unsold? How many times three braids were left unsold or four braids were left unsold. So then if once we counted how many laala then we are able to calculate the probability of that. So it says on uh if there were zero bread left so the probability would be 0a 2 0. If there were one bread, the probability is 0a 35. And if the probability two braids were left is 0 comma 25 and if three braids were left the probability becomes 0a 1 5 and when four braids were left the probability is uh 0 comma 05. So for a valid discret distribution every probability should lie between 0 and one. As you can see all our probabilities are between 0 and one. And if and the next pro property set all probabilities must add up to zero. So it means I must add 0.20 2 0 0.35 will be 0.5. Uh 55 + 0a 25 will be 0 8 0 + 015 will be 0a 95 and plus 05 will be equals to 1. Uh so then the property number two uh four then the last property says the list of outcomes should be mutually exclusive. So I know that there will not be a day where there is amper zero uh bread left and one left there will only on that day there's one bread that is left or there is no bread that is left. So they cannot be in between. There's no in between. So the mutually exclusive event here uh statement is accounted for and uh it must be collectively exhaustive. I know that I've got all collectively exhaustive and because all the sum of all probabilities there equals to one. So it means everything is accounted for. Right? So now this is your discrete probability table and from there you can then calculate your expected value. What is the average? what is the the distribution in terms of your variability which is your standard uh distribution or you can then calculate uh what is the probability of more than three loaves uh are left or unsold in the shop in the bakery and you can calculate that and so on so and that's what we're going to be doing just now. So let's see in terms of the probabilities we need to also understand that even though we say these are the probabilities of uh these are the probabilities um assigned to the number of units or the number of unsold lows right or the occurrence of unsold lows when they happen. But there are certain times where you don't just want to know the probability of the exact number but you want to know the probability of more than or less than. And that's where the phrases matter. So for example, you need to also know how to interpret your probabilities by using words or by using mathematical sign. So what we mean by the mathematical sign the equal sign, the less than or equal, greater than or equal or uh greater than where there is no uh a less than or equal, right? So what does that mean? So if it's exactly which means it's equal, it means at that point not more than nor less than at that point. So for example with this where it says exactly two or where what is the probability that the number of rows that are left unsolved is equals to two. Therefore it means on this table we'll just go and say where it is equals to two it means it's 0a 25. So that will be the result that we will put. If they say at most it means when it is less than fewer than right and it must also include that number. So from that number less than. So in mathematical time we say what is the probability that x will be less than or equal to two. Therefore it means from two going down. So it will include those that the days where it was only one bread left and the days where there is no bread left. So it means we're going to add all of those three probabilities and that will give you 0.20 plus 0.35 + 0.25 825 because it says at most and that will give us 0 comma 80. At least means greater than or equal. So it means including that number. So at least three will mean all the numbers bigger than three including also three. So in this table we only have three and four. So we're going to add the probabilities of three and the probabilities where it's equals to 4. So that will be 0a 1 15 plus 0 comma 05. You add only those two because it says greater than or equal. If they ask you for more than one. So here it says more than. More than means it does not include that number. So more than it means bigger than that number. So bigger than one would be where x is greater than one. So where x is greater than one. So when there were more than one soul uh uh uh one bread sold it means they it include when the days where there were two breads are left left unsold, three braids left unsold and four braids left unsold. So it means we're going to be adding all these three uh properties. So it means we're going to say 0.25 25 + 0a 1 5 + 05 which will give us 0a 4 five sorry so in terms of interpretation uh I don't think you are going to be asked uh to interpret read the results when you see them. So for example, if we were calculating the probability of x= to 2 which is 0, 25, you will say it is 25% of the chance that at least two brands remain unsold at the closing time because you just multiply 0a 25 with 100 and that gives you the proportion or the probability in a percentage format. Sorry. Are there any questions? No questions, no comments. Don't hesitate to ask if there are any questions or you don't get uh because I don't see the chat when I I present. So please feel free if there is something that you want me to explain further do not hesitate to ask. So now let's look at the exercise since uh it's your chance to now do some work and respond. So we'll do one by one and then you need to post on the chat. I need to see your answers on the chat. Please don't be shy. So we'll do A and then you write your answer. Then we do B then you write your answer C like that. so that then we can engage uh with every question at every point. So question number exercise one a Johannesburg mobile repair uh mobile repair business records the number of same day uh screen repairs or which are denoted by x completed by a technician. The probability distribution is given by this table below uh where they record the number of uh same day repair screens. So some days they've got zero screen that they need to repair. Some days they've got one screen, some days they've got two, some days they've got three, some days they got four uh with their corresponding probability. And where they rep uh there is an unknown number. And question number one says find the value of K. It means you need to go back to the properties of a discrete probability where it says the sum of all these values should be equals to one. So because you don't know what K is. So it means if I add all these other number and subtract that those numbers from one I should get K. So what is that number? What is K? So you can write in the chat or you can unmute if you have the answer. Someone already posted in the chat the value of K. I don't know what the value of K is substance. Remember if you also agree with the other numbers you can also uh the like there are emojis you can use on Okay, I see already we've got two answers in the chat. What about the others? Uh and Ramcha Ramana. Okay. So what we also know I don't know if anyone wants to answer that. So you can say k will be equals to 1us the value of 0a 1 0 + 0a 28 + 0a 2 + 0a 1 2 and that will give you 0 comma 28 and that is the answer for A. What is the answer for B? What is the probability that X will be less than or equals to 2? Do we have the answer? You know, you can also unmute and talk to me. I don't bite. So the probability that X is less than or equals to 2 it means we're going to add 0 1 0 + 0A 28 + 0 28 which is equ= to 0 6. So it is the same as the probability of x = to 0. If we have to write it in a formula format, the probability that X = 0 plus the probability that X = 1 plus the probability that X = 2 and that will be those probabilities. What is the probability that X= 3? Anyone can unmute and say it out loud. >> 0.22. >> That will be 0.22. And what is the probability that X will be uh greater than 1? 0.62 62. The probability that X = 2 plus the probability that X = 3 plus the probability that X = 4 which is 0 comma 28 + 0 comma 2 2 + 0 comma 16 which is equals to 0a 66 or in a way uh the other way that you can also answer the same question um where it says the probability that x is equals or greater than one because there you have more numbers you can also is the same as one minus the probability of those numbers but I don't want to go that route anyway So, so we can also say the probability of X is greater than 1 will be the same as 1 minus the probability of X is less than or equals to 1 which in a way it's 1 minus the probability of X is equals to 0 plus the probability that X is = 1 which is 1us the probability of 0 comma it doesn't give 66 or did I calculate incorrect my thing up 0.124 for number four. Not 0.16. >> It's 0 >> 1. >> Oh, one two. >> Oh, yeah. So, this we got it all wrong here. So, it should be eight uh >> 0.62. >> So, this will be also uh 0a 62. Sorry, my bad. You can write it in that way. And later on, I'm going to demonstrate this method more uh especially when we work with probabilities uh binomial probability or discrete u poison probability and so on because sometimes adding all the other numbers might be longer than shortening it up. So you can use either method and you will see how it applies. Okay. So now we also can get questions where they ask you to calculate the expected value or the mean. So let's assume that our discrete probability for this table where it also includes up to five. They give us the the values of our x which is from uh days where they zero up to when there are five units sold or uh left and sold or whatever they the the type that they are tracking here and with their corresponding probability. So to calculate the expected mean we use the formula. So in a short version it's the same as saying the expected mean or the expected value is the sum of your value times its corresponding probability. So what does that mean? It means we're going to take this x unit multiply it with the probability and do it for all of them and then add them because the sum means summation. So this is a summation. It means adding up. So in a way you can then just say 0 0 * 0a 35 is equals to 0 0 because any number that we multiply with 0 it's 0. 1 * 0a 25 will give us 0a 25. 2 * 0a 2 will give us 0a 4 0. 3 * 0a 1 it gives us 0a 3 0. 4 * 0a 5 would give us 0a 2 0. And 5 * 0a 05 will give us 0a 25. And then I just add 0 + uh 0a 25 + 0a 2 uh 4 0 + 0a 3 0 + 0a 2 0 + 0a 25 gives us 1 comma 4 0. So you just add all these probability uh the product of those probability times it corresponding probability and add them together and that gives us 1 4 0. So that is our mean. Once we have our mean, we can also calculate our standard deviation. So to calculate the mean, we just multiply the unit times its corresponding probability. to calculate the variance or the standard deviation and mostly the standard deviation because the variance is the square root of your standard deviation. Also similar the standard deviation is the square root of the sum of your x value minus the expected value which is the mean squared times its corresponding probability. So the formula even though here it looks like it's so complicated but it's just as straightforward as that if you don't want to confuse yourself so similar on the same table you can then just create the formulas and then use them. So what I'm going to do is I'm going to first simplify it in a way because it's got the summation which means I must add everything and adding I must add where I multiply where I subtract or the square of where I subtract the x value with the expected mean and multiplying it with its corresponding probability. So you can do all this in one and then add them at the end. So what I do I do the expected uh the observation minus the expected squared and I calculate then I take this value that I calculated then I go back and multiply it with your uh probability and that is the next section that I do there. Remember the expected we calculated it there. So I'm not going to show you again how to calculate it because it uses the same table. So we calculated the expected we know that it's 1.40. So we say 0 - 1.40 squar and gives us 1.96. And then you multiply it with 0a 35. That gives you the answer. And then you do for all of them. Then at the end you add all these values the 0a 68 0 comma 04 0 0a 7 07 72 until you get to 0a 648 and that will give you uh sorry they don't give you that my bet they give you 2 04 so it means we do all of this gives you uh 0 comma uh uh 2 04 then you take the square root and it will give you 1 comma 4 283. So after you get uh 2 so you still take that 2 04 and that will give you 1 comma 4 283. I hope you do get the flow of how I did it. Is that clear? So, and that's how we calculate the standard deviation. Here is your exercise. So, let's enter number number one. Find X. Where this X Where is this Okay, seems like we know what the value of x is. So x will be 1 minus X will be 1 - 0 25 + 0a 33 3 + 0a 1 7 + 0a 15 which will be equals to 0a 1 0. Calculate the expected value. So what you need to do is you can say not the sum but your x * your px and say 1 * 0a 25 is 0a 25 and say 2 * 0a 33 will give you 0a 66 and then continue with the whole 3 * that will be 0 comma 5 1 and 0 comma 6 0 and And then you just add all of them up. Just add all the values up. And what is the answer? Still completing. you. Sorry, I was muted all along. I was speaking to myself. Uh uh fun bale you have the answer of uh 0 comma 2 466 with a question mark. My question is uh is it for any of the questions when we were calculating um the inner side of the equation where we do 1 - 252 um times it corresponding probability of 0a 25 or the wrong formula Uh, okay. >> My answer is incorrect. I I applied the wrong formula. So, I'm going to do it again. >> Okay. So, I realized that my mic was muted. So, I don't know if I explained the whole process. So, I'll start again for the sake of the recording. Um, so in order for us to calculate the standard deviation, we already calculated the expected value because that's the value we're going to be using there. So on the same table, what I do, I'm going to ignore the first line, the red first line, because we used to calculate the expected value. So now I created another line with only the inner side of the equation of this standard deviation. So what I'm saying here is I'm going to take my x value, subtract it from my expected value and then square it and multiply with the corresponding probability. So, and then I did that here to say 1 - 0 25 which is uh sorry 1 - 2a 52 which is our expected value squared and once I get the answer then multiply that answer by 0a 25 and then I get the answer of 0a 5776. To do the second one, it's 2 minus 2 52 square the answer multiply with the corresponding probability which is 0 33 and the answer I get will be a very long digit and I have to write all of them. So only round off when you get to the final answer. While you're still in the problem mode, you write all the decimals. So I write all the decimals that appeared on my calculator. Then I move on to the next one which will be three multip uh sorry 3 subtract 2a 52 squared and then multiply by 0a7 sorry and then I get the answer of 03916 and then go to the next one which is 4 subtract 2a 52 square the answer multiply by uh 0a 1 4 then I get the answer then I do the 5 will be 5 - 2a 52 square the answer multiply by 0a 1 0 because we did calculate it so our X is 0 comma uh 1 1 0 multiply by that right and then we get 0a 61504 once I have all these values because I've got to sum all of them. So I just add 0a 5776 plus 0a 08 and all the digits until I get to 0 comma 61504. The answer after I've sum up all of them gives me 1 comma 6496. And I said this answer is what we call the variant which is also a sigma squared. All right. But the standard deviation is the square root of the variance. So it means I must take the square root of the answer and that gives me 1 2844. Are we all happy? Are we good? Are we following? If you are still lost, you speak now. So in a nutshell what we've learned also is in order for us to calculate the probability and from now on going forward we you must remember this because even when we do uh hypothesis testing or when we do confidence interval not the confidence intervals sampling distribution or when we do normal distribution all these questions will come in handy and even including later on when we do binomial and poison distribution you have to remember this. So you have to know the mathematical symbol and what does that mean and if they were if they use the word phrasing and what does that mean? So for example uh the the equal is easy. It's exactly the less than will be less than or fewer than or below. The greater than will be greater than or more than or above. If they use words like this then you will always remember that the less than or equal to it's always going to be at most or no more than. So you need to know and remember all those terminologies. Sorry. Uh and then the greater than or equal will be at least or sorry at least or no less than. And then they sometimes can ask you to calculate the between. So when they ask you for the between, the generic form of the between is always going to be inclusive. But sometimes they will tell you that it has to be exclusive. So if they don't mention inclusive, so always use the less than or equal. So it means it must be between value A and value B. So if they say calculate the value will be between this and this you include them because it says it's inclusive. If they say x then it just purely the value lies between this and this but it does not include those values. So where it has a equal sign it means it includes those a and b. If there's no equal sign it does not include. Right. So, choose the incorrect statement and I'm going to leave you to answer this and write in the chat whether the answer is number one, number two, number three or number four. Africa check knows that around the election time the number of daily fake news post about politician follows the following discrete probability distribution. So some days there are no fake news, some fake post, fake news post. Some days there's only one, some days there are two, some days there are three, some days there are four. And they give you also their corresponding probability. And they're telling you that let X be the number of days fake news post appears because X be the number of daily fake news. And when there are four daily fake news post they didn't give it's a question mark. So you need to calculate what that is. Which of the following statement is incorrect? So they are looking for the statement that is not correct. So they say statement number one the sum of all probabilities should equals to one. So it means if I add all these probabilities should give me one. The second one the probability of x = 4 is zero. So you would have already calculated this to know whether it's zero. The probability that X is less than or equals to 4 is 1. And the probability that X is greater than 4 is 0.3. Three. I'm not sure if you are answering exercise three. Please re look at it again if it's exercise three. They are looking for the incorrect value. That's the most important part. Okay, without wasting more time. So the first thing especially in the exam or in your assignment the first thing you need to do is when you get a table like this and you recognize that this is discrete probability. We know already the properties of a discrete probability all probability should be between 0 and one. The sum of all probability should be equals to one. So it means if I add all these probabilities I should get one. What will be this probability? So you add all of them, you subtract them from one and you will get that this is 0 comma 3 0 because all of them when you add them together is 0 comma 7 0. It means when there are four fake news post they posted the probability will be 30%. So, so since we know that we are able to go and answer the whole question. So, number one says the sum of all probability should be equals to one. I've tested that because I said all of these subtract from one. So, it means if I add all of them, they will be equals to 1. That's 0.70 + 0.30 3 0 is = to 1. The probability that x = 4 is 0. That is the probability of exactly that. That should be 0 comma 3 0. So it means that is the incorrect one and you could have stopped right there if you are in the exam. But in the assignment for the sake of the assignment because it's practice time you continue the probability that X is less than four less than or equals to 4. So it means it's from here to there you add all of them. So it will be the same as the probability of the sum of all of them because you have to add all of them because it's less than or equals to four. The probability that X is greater than or equals to 4 is the same as exactly the probability of X= to 4 because there is no nothing beyond four. So it means it stop right there at four. So that will be equals to 0, 03. I know that usually they do put this none of the above statement in most questions. uh you only resolve to that if you can't get the answer but you can always signal any question that has the none of the above because most of the time the answer is there. So and that's how you're going to answer the questions in the exam or the assignment. What happens if they give you in a not in a mathematical format but in a a text format? We're not going to go through this one. You can take a screenshot of it but I've also gave you u those who are already on the WhatsApp group I've already shared the notes with you but you can take a screenshot of it and uh do it later so that we can introduce the next section. What is the most important about this is you already recognize the table is discrete probabilities. The most important thing is reading the sentences that they give you by the keywords at least. We know that at least means it's x is greater than or equals to one. Okay. And with this one it says uh the probability that the speech will consult with one child at any given time. So one child so it's exactly so. So the probability of X is equals to one right. Oh sorry I didn't mention the one child not that one there but this is the with one child. All right. So this one is looking for exactly this one says the probability that the therapist will consult no child. So therefore it means this one they looking for the probability that x is equals to zero because there won't be any child that they consult with the probability because every question starts with probability. So it makes it easier for me to put the P in front. All right. The probability that the speech therapist will consult more than five. So what is more than? So you need to go back and remember the signs. A more than it's it's greater than. Right? So the more than five child will be the probability that X is greater than five that is more than what is the expected value. So yeah they are asking you to find the expected value which is the sum of your X values times it crossbonding probability. That's what they sorry that's what they are asking you to calculate. So you can take a screenshot and then answer it on your own. Remember also the chat in the chat there is um your register and if you're going to leave the session early the I'm going to repost it again here but it's also pinned on the chat. So, if you go back to the chat pin, at the beginning of the chat, there's always the message that is pinned uh there always pinned it on the chat. So please make sure that you complete the first is the attendance register and then at the end of the session don't forget to complete the evaluation form as well. So remember we're on the evaluation form. My name is Elizabeth Buie. Okay. So now let's move on to the next part of the session which is the binomial distribution. and binomial distribution. There's um your experiments that happens in a sequential number uh where for example if you toss a coin um and you toss it 15 times or you uh you have 10 light bulbs in the warehouse that you want to go and take for testing. uh because you can count there 1 2 3 4. The same way as when you toss a coin, you can count how many times you are tossing it. You tossing it 15 times. So there are sequences that are happening and the outcome of that experiment or that event you are creating will always be two. So they that's why it's called binomial. By means two. So there will be two outcomes for possible trials that you are doing or experiments that you are doing and we always deem those two trials as a success and a failure. So, for example, on your on your coin, you will have a head or a tail. Irregardless if I've got a dye and I'm throwing a dye, and a dye will land on either side of that dice, whether it's a one or it's a two. If my success, if I claim that the success of when I roll a dice, it needs to land on a four, that's my success. So every time I roll a dieice and it lands on a four, it's a success. If it lands on any other number, it's a failure. So there are two outcomes. It's either there's a success or failure of that outcome. And the probability of a success, we always going to denote it by the mathematical format or formula or symbol called pi. And the probability of a failure will be 1 minus uh the probability of a success. So which will 1 minus pi. So if I if I have a coin a coin has a head and a take. So the probability of a success of it landing on a head will be 50%. Or 0a 5. And the probability of it landing on the opposite will be 1 - 0a 5 which will be 0a 0a 5 again. But if I've got a die, a die has six sides. The probability, sorry, today's session. Uh actually I'm not feeling that well. Uh so what I was explaining is a die has six sides. So if my probability of a success will be when it lands on one of those sites will be be 1 / 6. So my probability of success will be 0 comma 1 666 or 1 167 if I put it that way. Then the probability of failure will be 1 minus 0 comma uh 167 right that will be the probability of a failure. So um the two categories will be mutually exclus because it's a discrete probability. So one cannot belong into two groups. It has to be mut mutually exclusive and it also has to be collectively exhaustive. So all everything in the sample size must or sample unit or sample size must be exhaustive must be included in the whole calculation and all events needs to be independent. So one does not depend on the other from happening. So they need to be independent. So let's look at this. Which one of the following is not a property of a binomial? So after I've explained all of that, so let's see if we can pick up from this statement. Each experiment has an end trial. Yes, we know that they all have an end trial. They can be 20 of them or 10 of them, 15 of them or 30 of them or 100 of them. The tiles are independent of each other. Yes, we said they are independent. So, one should not influence on the other. Each trial has two possible outcome. We said it. Yes, there is a a failure and a success and they are mutually exclusive and they should be collectively exhaustive. The probability of success remain the same for all trials. No ways. coaster. If I toss a coin, it only has two heads and a t. If I toss a um if I toss a uh what do you call that? Now, if I toss a uh a oh sorry, if I roll a die, it has six sides. So the probability of success will differ with um uh with different tracks with all the trials that you run. So depending on whichever the trial you are running the pro so number four will not be the correct one. The probability of success is always a half because there are two outcomes in a in a sense we can always assume that there will be a 50%. It's a 50/50 chance because if we think about a um what do you call that? uh if you think about uh a a a 50/50 chance a success or a failure. So that's that's what that it says. So there are two outcomes out of it. It's not like there will be three outcomes or four outcomes and all that. So, it's always going to be a 50/50 chance um in a in a way. So, this is your exercise. Which one of these is incorrect based on the statement? Africa check found that the source of fake news on Facebook are mostly ghost profile. So it means our success in this is for us to identify the ghost profiles. That's our success. Suppose that 20% of the profiles on Facebook are ghost profiles. So already they are telling us the probability of success. So we have our probability of success which we said we denote it by high right. Our probability of success is ghost profiles 20% of them. Suppose that we randomly select 20 profiles. So that is our trials uh and trials 20 profiles and we check whether or not they are ghost profiles. So we check whether a success or a failure. Which one of the following statement is incorrect? You can unmute and answer anyone who wants to answer. Number one, you can say whether it's correct or incorrect. Number one, given the information described or given the information describes a uh given the information describes a binomial experiment with possible outcomes, ghost profile or not ghost profile. True or false? Incorrect or the given information describes the binomial experiment with two possible outcomes. A go ghost profile or not ghost profile. True or false or what do you call it? >> That's that's correct. Right. So that's true. This is correct. The number of trials is 20. Our n is 20. Is it true? Incorrect or correct? >> True. >> That's correct. The 20 trials are independent from each other. It should be because we're dealing with binomial distribution. They are independent from each other. No, none affect the other or influences the other. So that is correct. The probability of success or ghost profile is 20 tries. Is it correct or incorrect? >> True. >> Huh? See where it's cycled? Are they the same? 20 trials means count. Probability means either it's a percentage or a decimal. Right? So is it incorrect or correct? Don't get confused with 20 trials of Facebook profiles. So these are 20 trials and these are 20% profiles. >> Probability. So is this correct or incorrect? >> Incorrect. >> That's the incorrect one. It should have said 20%. Or 0A 20. That would have sufficed. The probability of failure or not a ghost is 0a 80 because the probability of failure is 1 minus pi. So we have 1 >> -.20 >> 1 -.20 which is 0.80 which is double. So you must pay attention to this little tricky situation here with the trials and the percentage and all that very important. So what are the properties of a binomial distribution? So similar to what we did with a discrete probability distribution with binomial also you can calculate your your mean which is your expected value. It's your number of trials times the probability of success and your uh your variance will be your number of trials times the probability of success times the probability of failure and your variance will be or your standard deviation will be the square root of your your variance. So it's easy always remember that. So how do we calculate all of them? So let's assume that we have this question says a student is taking a multiplechoice exam and there are four multiplechoice questions with each question having four choices. Now each question has four choices. So it means only one question only one answer is correct and the rest are incorrect. So it means I need to be able to calculate my probability of success which will be 1 / 4 and that will be equals to 1 / 4 is 0a 25. Come on my pen 0 comma 25. All right. So with that I know there are four multiple choice questions. So my n = 4 already that it's done. Find the mean. The mean is my number of trials times 0.25 25 which is my probability of success which is equals to 4 *25 is equals to 1. My variant is 4 * 0.25 * 1 - 0.25 which is equals to 0.75 and my variance oh sorry my standard deviation will be the square root of my standard deviation I don't have to repeat again which is the same as the square root of 0 comma 75 which is 0 comma 8 7 or I can leave it then for decimal 0a 8 6 6 0. So depending normally with probabilities in your assignment or exam they always like to leave it in four decimals. So you can also leave it look at the answers in your on your answer sheet how many decimals they want to they want you to leave your answers. But if you need to round it off, you need to know how to round off. Any number bigger than or equals to five to the left of that number you rounding up to or the position of where you rounding up to, you add one. If it's less than five, you do nothing. You need to know those rules. Here's your exercise. Africa check found sources of fake news on Facebook and uh and they found that most of them are ghost profile. Suppose that 20% of the profiles on Facebook are ghost profile. Suppose that further than that 20% of Facebook profile and check whether or not they are ghost profile. So we know that our probability of success and we know our n they're asking you what is the mean and the standard deviation of s. So the mean which is your expected value is your n * your pi and your standard deviation is the square root of your n * i * 1 - i And this will be 20 * 0.20. And you get your answer. And this one will be the square t of your 20 * 0 2 0 * 1 - 0 2 Which option? Option one, two, three, and four. Are you okay? 20 * 0.20 2 0 it's four and this will be 08 *20 * 20 gives me 1 7 8885 1.7 888 8 five but I'm I can see here it's either one or two decimal so I'm going to round it off to two decimal so the number the side is bigger than five so it will be 7 9 if it was three 8 three I would leave it at zero uh 7.8 eight but because 8 is bigger than five so I'm just going to add one to eight then it's 7 9 so it means it's option two need more information to determine both values mm we have enough information um I think uh because we're not left with so many minutes Now we only have 15 minutes. I think we will only cover only the binomial. We will see if the next session we can start with the poison because we only have 15 minutes and I don't want to rush. But let's see uh how we do with the with the binomial questions because it need more time as well. So always remember this is very important. I'm just flagging it. I'm not going to go into detail. We will be using them to calculate the probabilities. So now with the binomial distribution uh even though we're going to be using the table easy to use a table than go and calculate the probabilities because then it's going to take you forever uh to calculate the probability of greater than and less than or equal or the between right uh if we're going to use the formula but we need to also to know the formula. So the formula to calculate the binomial distribution is we use the counting times the probabilities. So this is your combination which tells you how many number of ways you can arrange things times the probability. Right? So our n factorial means uh n if n is 20 it means is 20 * 19 * 18 * 17 * 16 times until you get to 2 * 1 that's that n to the exclamation. Uh the same with x exclamation if x was 3 it will be 3 * 2 * 1 uh and this one also when you subtract your sample from your x uh sorry your x from your sample uh exclamation so that will give you that or you can use the combination uh on your calculator your NCR on your calculator uh it looks like this NCR because then you have your n and you have your x, right? So if n was 20 and x was was three. So you just use that formula instead of populating it there. The probability of success to the power of x time the probability of failure to the power n minus x. So we've and that will give you the probability distribution. So how do we then if we want we can then use the table because in the table all those probabilities are calculated with every trial value corresponding with its own x value and the probability of success are written here at the top. So those are your probability of success at the bottom. It also have your probability of success uh at the bottom. So you've got your right, your left, and you've got the right. So how does this table work? Easy. The values at the top works with the left. So those two work together. The values at the bottom works with the values on the right. So if I'm reading the table from the values at the bottom, therefore it means I must use the n and the x from the right. So your table has two sides and both sides they combine. So depending on how you view your table, they both sides should combine uh at the middle. So it has your left hand side you will see also in the bottom side it has the number of trials uh also from there. So everything on the left will read with everything from the top. So if you think about it the top value probability at the top there where it says probability of success at the top it at the bottom of it it will be your probability of failure of that probability of success but at the bottom is also a probability of success for the bottom and the right and the probability of failure for the top part. So, I don't know if I'm confusing you or you getting it. What I'm trying to get to is if I need to know the probability of 0, 09, because my n is 10, my n will appear on the next table somewhere at the bottom, right? So I will know when I'm going to that table because then the it's cut off here at the top. You don't see those values. They won't appear here these values. So I will know that 01 corresponds with 0a 99. So this if I'm reading from this side I should be reading from 0 comma 01 based on that because this will be the opposite of that right because the values here are missing and I'm looking for the probability of success or pi is 01 and n is uh 10 and x is 5. Let's assume that which means it will be somewhere here but anyway let me use two n is 2 because then this is the value that I'm looking for in that sense right so that will be the probability of x is equals to 2 for where the probability of success is 01 and my n is 10 so I go there n is 10 was this side n and X they won't appear there because they will only appear at the bottom with its corresponding probability for the bottom part. But I know that this row where at the bottom is 099 is the same as my left values correspond to 01. So that probability will be 0 comma. So you need to look at that table or the two tables from your from your textbook or from past exam papers or from your study guide or from the prescribed book that you have. It's called table of binomial probabilities and we're going to be using that. Okay. If we're going to calculate the probability of a binomial distribution using the formula. So what is the probability of one success in five observation if the probability of an event of interest is 0a 5. So we know that one in one success. So it means x is equals to 1 our n. So x n and the probability of success I can already classify them. So I can identify them from the question itself. So one success in five observation and if the probability of an event of interest which is the probability of success is 0a 1. What is that probability? So I need to calculate the probability of x = 1 uh where the prob n is 5 and the probability of success is 0a 1. So I'm where I see n I'm going to put five. Where I see x I'm going to put one. Where I see pi I'm going to put 05. And then I substitute into the formula and calculate. And remember on your calculator you do have an n exclamation or x exclamation function. So if for example I'm calculating 5 is the same as 5 * * 4 * 3 * 2 * 1 or you can say 5 exclamation and press your function which is okay my calculator uh changed from being a scientific calculator to being a it's no longer a a scientific calculator for me. change. I don't know when did it change. Now I can't use it the way I used to use it. It's back. We found it. So, uh, in on most cases, the x exclamation is on the x to the power of a negative number button on top. It's written in orange. So, if I press that, you need to press shift. uh you will press five and then you press shift and then you will press the button that looks like this on top of it. It will have the exclamation mark. If you using a casual it will be somewhere I don't know you must just look for it. Either it will look like this or it will look like this on your calculator. So if it's written in orange you press shift first and then you press that and that will give you 120. So that means this is equals to 120 and this will be 1 5 - 1 which will be four exclamation will be the same is 4 shift that will give you 24. So this will be 24. So it will be 120 / 1 * 24 * 0a 1 to the power. So I'm just showing you how to if you're using the formula how to calculate this. Otherwise this whole thing you can also write it as n c r ^ x 1 - ^ 1 - not 1 - n - x and instead of doing the whole formula there you just say 5 C 1. So if I look at this whole thing, the answer is 5 because it's 120 / 1 * 24. Let's see what do we get. 120 / by 24 is equals to 5. And if I use the NCR function, so you press five and you look for NCR. It will be on the divide and it's also written in orange. So you're going to press five and then you press shift and you press the divide function and you press one and the answer will be equals to five. So you can either do it that way. So you will still get 5 * 0 1 to the^ of 5 will be to the^ of 1 will be um will be 0a 1 and 1 - 0 1 to the^ of 4 will be 0a 9 ^ 4 and when you calculate it will give you 0a 3 280. So let's use the table. So we calculating it manually. How do we get the same answer on the table? So uh if I go oh do I have it there? I have it there. Let me see if I have on this table the actual table. N is five. I don't have um I don't have five. I only have four. Okay. So because my table cuts off, uh I was going to use this if my table was not cut off there. So let's use this one that we have here. So since my table cut off, I'm using the n is equals to 10. Uh and I think we chose the probability of success to be 0a 35. Um 0a 35. And wait, let me see if I have a past exam paper somewhere that I can open. Nope, I don't have a statistics paper that I can open. Uh, just give me a second because I want to demo the table itself. Oh, I do have. Let me see. Just give me a second. Oh, no. No. This is not a statistics paper. Uh just give me a second because I want to demo the the paper that if I can get the exact how is it possible that I don't even have One past in paper is T. Let me see if I can open this. Yes, I do have a an example of papers. Okay, just hold on. I'll be just there with you just now. Shame. share my entire screen with you and the slide. Okay, so we were here because I want to demo this same that we can get the same number or the same answer uh as that. Okay, so I must go and look for I'll share this table on WhatsApp with you guys as well. Uh you'll see that they've got different tables. uh you will see them the same way as in your uh uh textbook you will have them. So this one now I do have n of five. So let me write the values. So what did we have? Uh we have x = 1, n = 5 and = 0a 1 so that we can use the same the same number and I can just write the answer that we got uh we had 32805. So we know that the probability we looking for of x = 2 is 0 comma 30 did I do 32 805 32 805 So that is what we aiming to to get. So how do we get that? Let me make this bigger. Okay. So, we know that we're looking for uh the probability of success is 0a 1. So, I must go to the table and look for 0a 1. You can see that it has all the probability not 01 0a 1 0. So 0a 1 and where n is five. That's what we said. Our n is five. So I'm going to go to where n is five. So because I'm using the top table, that's why I'm using the left. And my x is one. So x on the end of five is one there. So I'm going to say where they both meet and in that column and you can see there is the answer 0 3 2805 or actually they left it two decimals in four decimals and there it is. Right. So that's how you use the probability table. So just to also uh demo what I just spoke about. I said also if you look at this you can see the table ends here. There are no values here at the bottom but there are values here on the left. So I know that here it corresponds to at the bottom will be that this one will be 0 comma 55 and and so on and so on because this side will be 0 comma 99. If I have to use this site where n is 6 and the value of uh probability of success is 0a 99 I need to be able to know that that will be the value there or if I go there this will be 0a 92 because I know that it's eight there so it will correspond to this because the sum of those two values should be equals to one. So then I can use the left and the right and go up that that way. Okay. So let's demo the table. Also the second part of the table you can see that there are no values here at the top. The probabilities are missing here and you still needs to use those probabilities with these values there. And it only has the probabilities from the bottom going up. And this table also continues cuz you can see there is the next other table. Right? So going back I said we we are over time but I just wanted to show you uh how the table works. So uh I'm just going to conclude right here because I'm can give you this exercise to do and say go and do this exercise on your own. So you can take a screenshot and um and do this exercise by yourself because also for example if I can show you one little bit of a trick with this whole thing is because it says here the probability of success is 20%. All right. So we're going to go and look for 0a 2 0 which is at the top. So it means we're going to use the left hand side the same. So if I remove the ink, let's remove the ink. So we're going to use the 0 comma 2 and all the other values that are here. So the other thing it said uh then n is 20, right? So you see there n is 20. So it means on your table you're going to go You're going to look for n is 20. So which will be the next one which is this. And I will need to turn rotate. Uh how do I rotate? Okay. Need to rotate it. Okay. So n is 20 and pi is 0a 2 0. And I should be able to answer any question. And then the first question they asked was the probability that all ghost profiles are equals to zero. So it means the probability that x is uh x is 20 or ghost profile is equals to 220. So you go to where x= to 20 and see if it's zero. Uh where no profile the probability that x is zero. uh the probability that only 13 probability that X is 13 and the probability that only 12 probability that X 12 profiles are good goes profile I'm just going to answer one let's take the 13 one uh X is 20 and 13 just take that and the probability is 0 0 0. So let's see what did it say the probability that 13 and they when they are looking for the correct one and I just gave you the correct answer or they're looking for the incorrect answer. So that one I've already proved it that it's zero. So you can prove the others as well. And that's how you use the tables to answer the questions. Okay. Uh on that note uh we are 9 minutes past 11 which is we are over time. Please don't forget if you have any question, any query, you are still stuck, you need more assistance, you can email me at boymunisa.ac.za. Z A uh to have that one hour one consultation. They are all free. So it doesn't matter how many times you ask for assistance if I've got availability. And nowadays I've got availability every day between um 5 and 8. You can book a session with me. Uh please do not forget to complete the register as well as the feedback form. And on that note, we are done. Have a lovely even weekend and see you next weekend. Same time. We'll start firstly with the uh actually I do have a slide that shows you all of that because we didn't Oh, discard. We I was supposed to show you this slide. Uh so next week we were going to start with normal probabilities. We the for 15 minutes and then we'll do the normal distribution for the rest of the session or we might take 30 minutes with poison and then we do the normal distribution. And then the 22nd of August we'll do the sampling distribution. The 29th of August we'll do the confidence interval. The 5th of September we'll do the hypothesis testing and then the 12th of September we'll oh this was supposed to be exam prep. We'll bring couple of exam papers and go through some of the difficult questions that you're still struggling with. It's not confidence intervals will be exam preparations there. On that note, toodles, thank you for coming.