Good afternoon afternoon. I hope you guys are alright. I am well thank you. I'm going to start now. I'm not gonna wait for everyone to join if they will catch us because what I've decided with binomial distribution as well is to also do a summary of what you need to know about binomial distribution and then we go into doing some exercises so there will be a lot of explaining. I hope you also have a table, your statistical tables either from your study guide or from your prescribed book or if you have past exam paper you did bring the table, statistical tables that you're gonna use today. So we're still going to continue with study unit 5 which is part of your assignment 2 and I think you also did see the email from your lecturer. I think that he posted it on my UNISA to say the assignment 2 will only be open second or I think two weeks before the due date so you still have time to catch up on some of the work and then when the assignment 2 is open then you can start you can do your first your first submission and at least by then we would have gone through all the work that is needed for you to know and understand assignment 2 content as well. So without wasting any time so we're going to do binomial distribution by the end of the session today you should learn how some basic properties of binomial distribution you should learn how to calculate the being the variance and the standard deviation of a binomial probability distribution and also how to calculate or find the probability of a binomial distribution. So with binomial distribution because I also already told you that it's part of chapter or study unit 5 and the study unit 5 we did do the discrete probabilities and now we're going into looking at binomial distribution and next week we're going to look at Poisson you do not cover hypergeometric probability distributions and then later on as we go along we're going to do study unit 6 which includes continuous probabilities but we will get to that in a couple of weeks time. In terms of binomial distribution in terms of binomial distribution it comes from a counting process as well so you will always have a number of fixed observations I like for example when you are creating an event and you're tossing a coin if you toss it 15 times you are creating some observation or some events of those tosses and you're tossing it 15 times those 15 times are your number of trials that you are creating and each observation is categorized as whether or not the event will occur so here we're talking about a success or a failure of an event happening as well for example when you have a coin it has two sides a head or a tail when you create an event it it lands either on a head or it will land on a tail and those are the two categories that you can count and you can record on. When the probability of an event that you require or it is of your interest we call that the probability of success and it's always denoted by a pi side which is that side the pi side always denotes a probability of success and a probability of failure will be a complement of a success which will be one minus the probability of success which is the probability of failure and also what you need to also remember with binomial probability is that there will be a constant probability for the event of interest which is your probability of success for each observation for example when you toss a coin the probability that it will land on a head will be a half because there are two outcomes of tossing only one coin and if I'm tossing two coins then you do the number of trials will increase based on the number of times that you are tossing that coin so the probability of getting a tail is the same for each time you toss a coin because a coin has only one sample space and when I toss that coin that probability will always be a half because they are only two sides to that coin your observation needs to be independent so one observation should not have any effect on the other or should not have any influence on the other so they should be independent of one another and we know with a coin the number of times you toss a coin whether it lands on a head will not have any bearing on whether it will land on a tail or it will land on a head as well so the outcome of one observation does not affect the other outcome as well and also this is just for additional knowledge as well the type of sampling method that you can use to deliver an independent can be two you can do this in terms of infinite population where you do the sampling of the events in terms of making sure that you do not replace the coin after you take after you've tossed it you get a new coin and that will create an independent event in terms of how you toss the coin and then it can also come from a finite population with replacement for example using the same coin and tossing the same coin several times but making sure that the outcome that comes from there they are independent as well with properties of a binomial distribution you should be able to calculate as well the mean and the variance and the standard deviation to calculate the mean of a binomial distribution we use this formula the mean is also referred to as the expected value and it is calculated by using the number of trials times the probability of success which is n times pi and the variance is the number of trials times the probability of success times the probability of failure or one minus the probability of success and the standard deviation will be the square root of your variant it's just taking the square root of your variant how do we then do this how do we calculate the mean so if I'm given that my n for this data that I'm given follows a binomial distribution and I'm told that the n is 5 and my probability of success is 0.1 what is my expected value so my expected value will be n times pi which my pi is 0.1 my n is 5 so it's 5 times 0.1 which will give me 0.5 to calculate the standard deviation I'll take the square root of the variance which is the square root of n times pi times 1 minus pi so it will be the square root of 5 times 0.1 times 1 minus 0.1 and that will give me 0.708 and I'm just going to ask you to calculate the mean calculate the mean of this when I have my n of 5 and my pi of 0.5 what will be the mean that is your exercise calculate the expected value or the mean what I'm asking you is to calculate the expected value or the mean of this data set at the bottom it will be 5 multiplied by 0.5 which is 2.5 it will be 5 multiplied by 0.5 which is 2.5 correct calculate the standard deviation calculate the standard deviation which is the square root of n times pi times 1 by n as pi it will be the square root of sorry just made a mistake there it will be the square root of 2.5 which is the mean that we just got up there bracket 1 minus 0.5 which is 1.118 so it will be the square root of 5 times 0.5 times 1 minus 0.5 which is equals to 1.118 are there any questions any query questions something that you don't understand this should be straightforward right I have a question yes do we use the calculator to calculate this yes you use the calculator the stat mode nope you use your normal calculator this is just the normal maths calculations thank you what type of a calculator do you have it's a sharp do you have a sharp yeah you use a normal calculator so you know how to use your calculator to find the square root then all that right yes to find the square root yes my question is okay looks like the pi is always given or it's changing because I noticed that there's a pi time calculator so I guess that's a question this is the symbol it's not the pi this is not a pi this is a symbol we use for the probability of success it will always they will give it to you or you will have to calculate remember if they haven't given it to you your pi will be the number satisfying that outcome divide by how many they are it's always the same because it is a probability it's a probability of success so if they didn't give you they would have given you some records or outcomes and the number of or your sample space where you can calculate that probability so if you still need to remember what we've learned when we learn the basic probabilities as well okay so that is just the symbol it's not the one on your calculator where you select the pi always remember that let's look at another example which is more of your theory than calculations because you are also expected to know some theory about statistics so you not all the questions will always be calculations so which one of the following is not a property of a binomial we just touched on some of this thing so I'm not expecting you to know everything but let's try and answer it we'll go through each statement together and see if it's true or false and then we can answer because we're looking for not the property of a binomial so each trial has each experiment has n trial is that correct correct that is correct because if I toss a coin 15 times 15 will be my number of trials that I am creating the n trials are independent of each other that would be true that is also correct because we said all outcomes or experiments needs to be independent each trial has two possible outcomes that are mutually exclusive a success and failure true that it is true because we also spoke about this that is for success the probability of calculating that will be a pi and for a failure it will be one minus a pi the probability of that and we know that they cannot both happen at the same time so therefore they will be mutually exclusive yes because a head and a tail cannot happen at the same time the probability of success remains the same for all trials not true that will not be true because for example if I have a die the probability of success for a die will be different because a die has six sides so a probability of getting a one will be one divide by six a probability of getting two will be one divide by six because there is only one side with a two and when I toss a coin the probability of getting a tail will be one over two so a one over two which is half is different to one over six which is what is one over six I should have used um one divide by six which is 0.17 so they are different for each trial so that the probability of success remain the same for all time will be incorrect because they will be different the probability of success is always a half because of the outcomes there are two and here we're talking about when we're looking at the basic probability of binomial distribution where there are only two outcomes correct that will be correct because a success and a failure it will always be one over two for those ones but for all the trials will be different so you must also pay attention because here we interchange through we have trials and then we also have outcomes because within a trial you get outcomes right okay so let's move on to the next one some of the question might look like this Africa check found that the source of fake news on facebook are mostly ghost profile suppose that 20 percent of the profiles on facebook are ghost profiles suppose further that we randomly select 20 facebook profiles and check whether or not they are ghost profiles so before I can answer the question I must just go back and reread the statement and make sense of what is happening here this is what's going to happen when you answer the questions in the exam you need to be able to recognize whether is this question a binomial is this question a basic probability is this question a normal distribution question because they are all closer to one another one thing for sure that I can make you at ease is that the questions you receive in your exam follow the same pattern as the chapter or the study unit of your material for so for example the first question will come from chapter one chapter two chapter three probably because they are mostly like linked the third and fourth question might come from the study unit three and study and there can be two questions per study unit and then the question number six and seven will come from study unit four like that like that so they will always follow the same structure of your study guide when you writing your assignment as well because you're only using two chapters to write your assignment the first few questions will be from one study unit to the next to the next so you just need to make sure that you understand the question asked in order to apply and know from which study unit what kind of formulas will you will need to apply and so on but we will get to that when we do the exam prep as well so in terms of this question let's go back and read and understand what the question is saying in relation to binomial distribution africa check found that the source of fake news on facebook are mostly ghost profiles so therefore it means somewhere i'm just gonna highlight the mostly ghost profile so somewhere ghost profile it's either a success or a failure suppose that 20 of the profiles on facebook are ghost profiles so therefore it means my ghost profile are my probability are my success so suppose that 20 of them so therefore yeah this 20 percent is my probability of success and i can say it is 0.30 that is the probability of success for ghost profile so the probability of failure in this regard will be not a ghost profile that will be a failure because if the researcher is interested in ghost profile that is the probability of interest which is your success probability uh suppose that we randomly select 20 facebook profiles so we select 20 facebook profiles so that is our end and check whether or not they are ghost profile so which one of the following statement is incorrect so it means we need to evaluate each statement with regards to the information given to us at the top so we're gonna do this also together right the given information the given information described a binomial experiment with possible outcome ghost profile or not ghost profile is that correct or incorrect that is correct because we have two outcomes ghost profile and not ghost profile yes number two the number of trials is 20 correct that is correct because our end is 20 we just established that the 20 trials are independent of each other correct that will be correct in terms of ghost profile and not ghost profile they cannot influence one another so they do not affect each other the probability of success or ghost profile is 20 trials incorrect correct is it correct or incorrect i had two correct correct are you saying correct yes you need you need to pay attention probability is between zero and one here it says the probability of success or ghost profile is 20 trials incorrect that should that should have already given you a hint that if this value is bigger than one therefore it's not the probability so it will be incorrect and that is something that you always have to remember when we work with probability if the question was asking you calculate the probability and you get more than one then you know that it is not correct so it needs to be one or hundred percent or less so this is the incorrect one the probability of failure or not ghost profile is 0 comma 8 correct that is correct because this is our probability of failure which is one minus zero comma two zero which will be zero comma eight zero because we know that the probability of success is zero comma two zero so this one should have said zero comma two zero not 20 trials and also yeah by mentioning trials makes it incorrect okay so it should be easy like this to answer questions relating to binomial distribution based on what we just learned calculate the mean based on this information so it's still the same as what we had so the probability of success is zero comma two zero probably and then this is our end the question here is calculate the expected mean let me write it the way they wrote it here it's the mean which is the new they're asking you to calculate the mean and they are also asking you to calculate the standard deviation which is the square root of n times the probability of success one minus the probability of success so that is what they are asking you that is your information the mean is four and the standard deviation is one point seven nine and how do we calculate the mean it is 20 times zero point two zero 20 times zero point two zero which is equals two four right yes and standard deviation square root of four two okay we just substitute the values as we see them okay it times zero point two zero times one minus zero point two zero and the answer that is four times zero point eight square root of four times zero point eight one point seven nine which is option number two happiness are we happy are we good yes happy yep great okay so those are the basic properties of a bimomial distribution if you have any questions before I move on are we going into how do we calculate their probabilities do you have any question relating to the basic basic properties of the bimomial probability distribution hi Lizzie did you also get one point seven nine because I got one point eight four four is the same double check your answer I can also double check from my side so it is two square root of 20 times point two times it's one comma seven eight eight eight eight five four three eight two which if you're rounded off to two decimal it's one comma seven nine so just double check that you substituted correctly and you multiply correctly and you took the square root correctly okay thank you all right let's look at how we calculate the probability of a bimomial distribution so we dealt with counting rules as part of the basic probabilities as well a bimomial distribution also uses some counting techniques in its formula and we remember those counting techniques like our factorials um combination permutation and multiplication rules and so on and so forth so a bimomial distribution also uses some counting techniques so suppose the event of interest is obtaining heads on a toss of a fair coin you add to toss the coin three times and if we need to find out in how many number of ways can we get these two heads if that is the question that was asked and the possible ways will be either when you toss in the three coin one time it can land on a head a head in a tail or a head a tail in a head or a tail a head in a head so those three points can have those scenarios there are three ways that you can get two heads remember there are other times that you can get different but not get it the head like you can toss all those three coins and they all land on heads therefore it means there are three heads so yeah we're checking how many times how many ways can we get two heads regardless of the what you call that order remember yeah they are not talking about other so that is why we're going to use what we call a combination because combination has no order to it so and this will be the formula that you will use normally when you answer the question in terms of the tail in terms of combination so we will use the ncr or ncs or we can use the formula that looks like n factorial divided by x factorial and minus x factorial so that we can find the number of ways the three coins will land on at least two heads i'm not going to do any exercise in terms of this because we did without with them in the basic probabilities you can go and watch how we how we calculate the counting rules i was just introducing it so in terms of a binomial distribution formula which you need to know how to calculate it but it's not necessarily that you need to use it as you calculate the probabilities but you need to know how it looks and how you calculate the binomial distribution probability distribution you need to know that because sometimes in your exam or assignment your lecturer can just give you the formula as an option and with already populated values that you need to be able to say oh this is a binomial formula and decide the correct way of substituting the values or this is the correct way of calculating a binomial probability distribution so with a binomial probability distribution we've used or it has two parts to it the first part which is the counting process and the second part which is the probability process so here we want to calculate the probability of how many number of ways you can do certain things for example i like to use the loto because loto has money so if you want you can count how many number of ways you can win loto and you can also use this to calculate the probability of you winning the loto by using this formula so this part the first part will tell you how many number of ways you can win the loto combining the number of ways you can win the loto and the probability of the binomial probability formula you can calculate the probability the chance of you winning the loto and that is this probability function okay so we will use this to calculate the probability of an outcome whether is it a an outcome of one or an outcome of two but those outcome needs to belong to a a try and how many number of trials will that there be and also you will be given the probability of success in order for you to be able to answer the question so in the question you will be told how many number of x of outcomes you will have to calculate for but they will give you in the statement the number of trials and the probability of success if they didn't give you the probability of success probably they gave you your outcomes that satisfy the probability of success and the sample space and you can calculate that probability of success let's look at the examples what is the probability of one success in five observations if the probability of an event of an event of interest is 0.1 what is the probability of one success in five observations if the probability of interest is 0.1 so our one is our x we want to know the probability of x in five observations which are the number of trials that we have which is our n and the probability of success is 0.1 using our formula we just substitute the value our end is five five factorial divide by your x factorial which is one your n minus x factorial five minus one times the probability of success which is 0.1 to the power of x x is one one minus the probability of success of 0.1 to the power of n minus x which is five minus one and solving this equation then we get five multiply because if you solve the combination function you will get a five 0.1 to the power of one is 0.1 one minus 0.1 is 0.9 five minus one is four and solving this you will get the probability of 0.3 to 805 and that is how you will calculate the probabilities of a binomial distribution now the challenge with this formula is that if you need to calculate the probability of greater than or the probability of less than it then becomes difficult because if yet they would have said find the probability of less than one therefore it means to find the probability of x less than one we know that that is the probability of x is equals to one plus the probability of x is equals to oh let's start with zero zero and x is equals to one oh but then I said less right let's say less than or equal let's put less than or equal sorry about the confusion so if I need to find the probability of x less than or equal then it means I need to calculate the probability of x is equals to zero and the probability of x is equals to one so I already calculated the probability of x is equals to one so it means I need to calculate the probability of x is equals to zero so that for I can use a shortcut and the shortcut will be my n c r pi x 1 minus pi n minus x and my n is 5 my r is 0 my pi is 0.1 my x is 0 1 minus 1 minus 0.1 and 5 minus 0 and that will be what is my n c r 5 in terms of the calculator r 5 second function n c r 0 0 and that is equals to 1 so that will be 1 times 1 times 0.9 because 0.1 to the power of 0 is 1 right any number to the power of 0 will be equals to 1 and this will be to the power of 4 and the answer will be 0.9 to the power of 4 which is equals to 0.6 5 isn't it to the power of 5 because it's 5 minus 0 oh yes it's to the power of 5 yes you're right oh gosh let's bring it on a Sunday that's to the power of 5 let's fix that to the power of 5 which is 0 comma 5 9 0 4 9 and because I'm doing plus so I'm just going to add both of this answer so that I can find my probability of x less than 1 and that will be 0.5 9 0 5 I'm just 1 bit off to it is it at 5 decimal I'm gonna leave it at 5 decimal 0 4 9 plus 0.3 2 8 0 5 which is equals to 0.91854 as you can see if I had it's greater than it's greater than 1 therefore it means it is 2345 so I need to calculate this five times in order to answer the question which is time consuming and that is the reason why when you deal with binomial probabilities you need to rely on the tables and that's what we're going to do just now so this is another example I'm not going to go through this example with you Liz sorry before you continue yep so when you were calculating the ncr right then it was 5 which function is it when you're doing the 5 cr on the calculator it is called ncr you must look for it do you using a sharp right it's on I'm using a cascio using a cascio but the one above the division sign those with the cascio is the one above the division sign yes so you first put the 5 and then you press the shift and then press 0 if for example on this one on number one we needed to do as five shift and then you will press the ncr and then you will press this side is what is this one you had one right one and then you press equal you will see that it will give you five for this site it will be 5 shift and cr and you press 0 and you press equal and the answer should be one thank you all right so I'm not going to do this exercise which is almost exactly the same as the one that we did the probability of success here is 0.02 um our x is 2 and our n is 10 and you just calculate the probability I want to go to the table now with your binomial distribution there are two tables don't worry if you can't see the tables I just want to explain the table on this slide and then we'll go to the table tables so on your tables the your tables that you have in front of you probably they have two sides so there is the bottom and there is the the the top the bottom four is on one page and then the sorry the top is on one page the bottom is also on another page but if you combine both of those two files and pages and make one be next to the other underneath the other one and match them you will see that they will look almost like one table which runs from um your n values of two up until your n values of 10 on one table there are also other values like n values of 11 12 dating and then they are 20 and and so on there are also probabilities at the top of your table as you can see there so how you read the table the probabilities at the top of your table corresponds with the n and the x on your right so the top probabilities corresponds with the n and the x on your left so top with left the bottom probabilities if you look at the bottom of this table when you look at your tables you will have the bottom so because your table your first table which i'm referring to as the top table has also probabilities at the bottom so this page combines um so the bottom of the page there are probabilities as well if you look there so at the top here it's zero comma zero one at the bottom here is zero comma zero zero comma nine nine because they are complements of one another but the ones at the bottom corresponds to the n and x at the bottom and now you need to also pay attention when you read this table at the bottom the n starts from 10 and it goes up as well but also the x value starts from the bottom and goes up they don't start at the top you can see they write they stop at the bottom and they go up every n value has the x observation that corresponds with the last x observation will the last x outcome will be the same as your number of your trials so you can see that x is up to two there therefore it means your n is also equals to two three equals n is three four n is four and also here it would have been nine and n is nine because I cut off the values uh I cut off some five six seven eight from my table so that it can feed on here and ten corresponds with the x value of ten so you need to be able to read this table correctly right always remember the top values which are your smaller so the top values are the small probabilities of success so the small probability of success are at the top they correspond with the left hand side n value and x value so let's say let's say we are on n is equals to 10 on the table when you scroll to the next table you will notice that there won't be any probabilities at the top but always remember in your mind that here at the top it is zero comma zero one and here at the bottom of this table it is zero comma nine nine it doesn't matter regardless of where you are on the table the top has the bottom they correspond the sum of the top should be this equals to one uh the sum of the top and the bottom probabilities those probability of success the top and the bottom should give you one okay so how do we read this remember our exercise that we used uh but we used n of five we're gonna go to the table just now just want to go to the actual table and answer the same question that we have remember this is the table that I'm referring to and it's also always going to be called table E6 that's the table remember our question that we were answering our example n was equals to five x was equals to one and the probability of success was equals to 0.1 right yes if I remember correctly and we know that we were looking for the probability of x is equals to one so because my probability of success is just zero comma one I go to the top of the table look for zero comma one and then I go to the side of the table and I look for n of five and my x is one where they meet that should be the probability that I'm looking for and we find the answer is zero comma three two eight zero so let's go back to that question which was this one zero comma three two eight one see is the same so you can use the table or you can use the formula to calculate but the table saves you time than the than the formula so let's look at the one that we didn't do which is this one it said the probability of success is zero comma two n is ten and x is two so we go probability of success is zero point zero two our n we need to look for n this side n is ten scroll up until we get to ten and this is our ten and we know that we are in this column column number two right that's where we are at column number two n is ten and what is our x x is equals to two so we go to x of two and that is zero comma one five three so what did we get when we were calculating zero comma one five three one and that's how you will use the table now let's do more examples before we do more examples I also want you to remember that in the questions they might not give you the symbols they might give you wet phrases you need to know how to interpret the wet phrases into a symbol in order for you to be able to answer the question correctly if they ask you sorry what is the probability that fewer people are between the ages of 30 and 40 you need to be able to say fewer means less than if they say at least you need to know that they mean greater than or equal things like that so they still are applicable whether they say it's between or inclusive and exclusive all those things we still need to remember them as we do activities related to binomial distribution even next week when we do poison this is very important as well so let's look at another example that we can do together if x follows a binomial distribution with n of six and the pi of zero comma two what is the probability that x is less than three those who wants to use the formula you can use the formula those who wants to use the table you can go ahead and use the table we know that with the formula we need to go and find regardless of whether it's the formula or the table we need to go find the probability that x is zero plus the probability that x is one plus the probability that x is two because yeah it says it's less than three so it has to become a two and anything below two below three and we know that the probability of a binomial is given by n dr of pi to the power of x one minus pi n minus x and that is the formula you can use if you want to go and calculate this manually you can use this formula so you will calculate for x of zero x of one and x of two separately and add them together when you are done so when we use the table you need to go to the probability of success of 0.2 which will be at the top of the table n of six and find the probability where x is zero so let's go there we're looking for let's delete all this we're looking for 0.2 we're looking for n of six and we're looking for any value less than three so it means I'm going to draw a line here so that I don't miss a thing and I know that I'm in this color okay so you need to add all these values I will give you time to write them those who haven't who don't have a table in front of you I'm giving you a chance to write the values down which is 0.2621 plus 0.3932 plus 0.2458 are you done yes okay give me those values because I need to write them here I don't have them what is x of zero 0 comma 2621 2621 for x 0 comma 3932 3932 and 0 comma 0.2458 yes which is equals to 0.9011 option number two that's how easy it will be if you use the formula but sometimes remember like I said it might be a little bit tricky so maybe this number maybe this is not the answer that is on there maybe this one also is not there but here they gave you a formula that looks like this and say the answer is oh but then it will not work because here we have less than or equal but probably they will have something like this I'm just going to take a try and say 3 ncr or ncr 3 c 3 that will be 6 3 and they have here 0.2 I'm going to leave the option here s current is like that but I'm just going to add another option which looks like this even put one minus let's make it interesting and say this is 0.8 and they have here five six minus three as well so something like this they might have an answer like that looks like this so if you need to be able to know that this is correct if only the question was just p of x is equals to 3 then that would be correct in that way so you just need to make sure that you know how to use the formula not only to use the table alone so right and that is something that you need to get used to when you are practicing try to do the answers using a table and also some of them using the formula as well just for practice papers let's say another example before we go in to doing more exercises if x follows a binomial distribution with n of six and the probability of success is 0.75 what is the probability that x is less than 3 which is the same thing so we are told x is less than 3 which then it means we're still going to have to find the probability of x is 0 plus the probability that x is 1 plus the probability that x is equals to 2 all right so it means coming back yeah somebody's uh this new is that okay so we need to go find the probability of success now is 0.75 right and our n is six so we go to the table remember at the bottom of the table bottom of the table which is this one we are able to identify where 0.75 is at but if we come to this uh side of the table there are no values here at the bottom you can write them because if we know that the sum of the top part and the bottom part are equals to 1 so therefore it means under 0.25 there is 0.75 right this are our probability of success and here is 0.80 and 0.85 0.90 and on and on and on and on and on and on so those i know that right so i'm looking for n of 6 we're looking for x of 0 x of 1 x of 2 that's what we have right and our probability of success is 0.75 so we need to go to 0.75 which is at the bottom then it means we need to use the right hand side now the right hand side not the left hand side so we need to come this side and go and find our n of 6 and we're looking for 0 1 and 2 so those will be the values and the way i'm going to stop it'll be at that so those will be the two values that we are looking for the three values so you need to take those three values i'm gonna give you time to write them and then you can give them to me are we done yes okay so for 0 that is 0 comma 002 plus 0.0044 plus 0 comma 0.330 which is equal to 0.0376 which is option number five and that's how you use the table are we happy are we good or are we great good and that concludes what we supposed to learn about binomial distribution so we're gonna take the next 30 minutes to just go through some activities but just to recap on what you have learned you've learned the properties of binomial distribution you've learned how to calculate the mean the variance and the standard deviation and now lastly you've learned how to calculate or find the probability of a binomial distribution either by using the formula or by using the table now let's go into doing some more exercises your first exercise which one or which of the following statement is incorrect so it means we're going to evaluate each and every statement we can do this together for now let's do that i will for the ones where it requires you to do some calculations i can give you some time to look at that so let's go with option number one says if the value of a variable depends on the outcome of a experiment the variable is called a random variable this is also related to what we have covered last time i'm not sure but wait wait wait wait wait um i'm gonna read all the statement just to give you time to to think through all of them and then we'll come back in and answer the question number two says a discrete random variable takes on a any numerical value with a within a certain interval number three says the number of outcomes obtained when two dice are rolled is equals to 36 number four says the mean of a random variable that follows a binomial distribution with the parameter n and pi is equals to n times pi number five the probability of an event is always on the range of zero and one that is the probability of an event lies between zero and one we are looking for that one incorrect statement okay we're going to do this together but i'm going to direct you in terms of each statement which one we're going to answer first so we can start from the bottom and go up so let's start at the bottom because i think at the bottom it's easier the probability of an event is always a range between zero and one is that correct or incorrect correct that is true because we know that the probability lies between zero and one number four the mean of a random variable that follows a binomial distribution with the parameter n and pi is equals to n times pi is that correct correct that would be correct because we know that the expected value or the mean is equals to n times pi that is what the question is asking you number three the number of outcomes obtained when two dice are rolled is equals to 26 how many outcomes when we have two dice how many outcomes will they be i'm not sure if i'm right but i think it's true because six times six i'm just guessing okay your guess is is correct in the way that you are looking at it but also you can also say um when you're looking at the outcomes anyway you will eventually have to use the multiplication rule as well because if i have a die first die on the on at the top having all these sides three four five six right and on this side i can write the outcome of this second die three four five six and i can count the outcomes way because i've got two dice right i can count how many outcomes will the first die land on a on a one and the second die also land on a one so that will be one day and the first die lands on a two and the second die lands on a one that will be a point there if i add all of them all these outcomes because it talks about outcomes if i will add all these outcomes there will be 36 of them because there are six out of and there will be six on this so six times six will give you 36 if you want to use multiplication otherwise like we did with the head tail head you can also do the same to count how many there are but it will take you forever because there are 36 outcomes that will happen here so you will count one two three four five six one two three four five six one two three and there are six out of six rows and six columns as well so there will be 36 rows and columns that you're going to count as well so that is correct there will be 36 outcomes number one says if the value of a variable depends on an outcome of an experiment the variable is called a random variable is that correct or incorrect if you create an experiment an outcome that will come from that experiment will be a random will be at random right it will not be something that you just picked it will be a random and therefore it means the variables or the values that will be from those variables will also be a random variable will come from a random variable right and that is correct and that is what chapter six or study unit six is all about it's about using random variable that comes from an experiment number three number two sorry number two it says a discrete random variable takes on any numerical value within a certain interval what is discrete variables they are their whole numbers they are whole numbers so therefore it cannot be from an interval because they come from an accounting accounting process not a measuring process as well so that will be your incorrect statement exercise two autism South Africa has found that 50% of the people with autism spectrum disorder struggles with social interaction assume we randomly select six people living with ASD what is the probability that only three people live with ASD what are we given in this we are given the pi the x and the n as well we are given the pi which is 0 comma five five and our n all right six and in the question they are asking us to find the probability that x is equals to three because it says only three people so it's exactly three three yes so you can go ahead and go and calculate those who prefer to use manual calculation you can use the formula n factorial x factorial n minus x factorial pi x 1 minus pi n minus x or you can use n c r pi to the power of x um so n minus pi to the power n minus x or you can use the table table is six it's up to you what is the correct answer those who are experienced on the table it's zero comma three one two five okay so we go to zero comma five now when we look at the table we do have two zero comma fives all right you will have a zero comma five at the bottom and have zero comma five at the top you can choose whichever one you want to use um i always prefer to use your n of zero comma five at the top instead of the one at the bottom for easy of calculations so what else was there probability is zero comma zero five any six x is three zero comma zero five six any three and the answer let me delete this one first any six x is three and we just scroll to the end of the table that is the answer even if we had to used n of that pi is zero comma five any six we just go to six x is three we just come it will give us the same answer so it's up to you whether you want to use the top especially for zero point five the top or the bottom you can choose whichever one you want to use and the answer is zero comma one two five zero comma three one two five which is option three western cape education department is interested in addressing shortage of teachers in rural area or schools previous study have suggested that one in every four rural schools have a shortage of teachers suppose 10 rural schools are selected independent of each other to check whether or not each school has a shortage of teachers which one of the following statement is incorrect before you answer the question you need to go back to your statement because they are having given you a lot but they have given you as much so we don't have we need to find what is the probability of success we need to identify what is our n and we can then the rest of the questions will come from our statement so what is our probability of success in terms of what's from this statement one in every four because that is the previous study they have given us our x and our n we can calculate x divide by n would give us one over four which is equals to zero comma two five which is our probability of success our n it tells they told us that they selected 10 schools so that was right yes which one of the following statement is incorrect so we know that our probability of success is shortage right shortage of teachers is our probability of success because it's what the researcher is interested in number one the probability that only three out of ten has a shortage meaning find the probability that x is equals to three where n is ten and our pi is zero comma zero two five so you will go to the table this identify your zero point two five n is ten n is ten is right at the bottom so because at the top of the table i am missing some values i'm going to go to the bottom and look for zero point two five cross point two seventy five right so that will be the column that we are using this one so this is the column that we're going to be using our n is ten x is three is three it's zero comma two five zero three zero comma two five zero three let's see zero comma two five zero three this one says zero comma two three three six which is not incorrect the probability of a school having a shortage of teachers is zero comma two five that is correct that is correct because that is what we just calculated there and while that of those not having a school shortage is zero point seven five the two outcome possible for each trial are a school having a shortage of teachers and a school not having a shortage of teachers all right that is correct because that's what they told us they as well in the statement we checking whether or not the school has a shortage of teachers so there will be two outcomes the expected value of the number of school with the shortage of teachers is two point five so they are asking you to calculate the mean of n times pi our n is ten our i is zero point two five five so ten times zero point two five is two comma five which means this is correct the probability that only two out of ten have a shortage so here we're looking for the probability that x is equals to two so going back to our question x is equals to two is just the value above the one that we just found which is two zero comma two eight one six which is correct and that's how you will answer the multiple test questions right by just evaluating all of them but we know that number one was the incorrect one i'm not going to do this one for you you can answer this we have eight minutes suppose that 10 of the butterflies have damaged wings if a random sample of five butterflies is selected what is the probability that none of the butterflies have their wings damaged what is none it's a zero it's a zero so what is the probability that x is equals to zero where our pi is zero comma one comma one and n and it's five it's five i'm just gonna go there and leave it for you to find the answer give me this it's zero comma five nine zero five we know that our what is our pi pi is zero comma one zero right and our n is five and one x is zero okay happiness happiness which is option number one right i forgot the number we got zero comma it's zero five nine zero five zero five you will also pay attention when you're using different books you will have different different values so this answer was you oh uses five decimals probably because they calculated manually or they used another table that had five decimals so you just need to pay attention to the tables that you are using as well using the binomial distribution if n is five the probability that x is one is equals to zero point one three two three the probability of success is are you gonna panic because yeah they're asking you to find pi no we're not gonna panic because we are told this really comes from the table right so we can go to the table where n is five x is three and we're going to find what the pi is n is five x is three let's go there and we're going to use our n n is five x is three and we're looking for zero comma what are we looking for zero comma one three two one three two three we're looking for zero comma one three two three we're going to stop when we get this value from that row zero comma three three two three and there we found the value that we we were given and all what we need to do is just go out to the top to go find our answer is zero comma three zero that is what we were looking for zero comma three easy right so it means when you are in your assignment you will not panic a lot when it comes to this type of questions suppose that an admission test for a certain university is designed so that the probability of passing is 45 percent find the probability that among five candidates who take the test more than three will pass so here is the thing because it says more than three what does it mean x of greater than three right so we know what our n is n is five our probability of success is zero comma four five so if n is if n is five and this says greater than three so therefore it means we need to find the probability that x is four plus the probability that x is five so with binomial distribution it's easy to to use the table because we know that our n and the last value corresponds to one another so we cannot go on and on and on and on so you just go to the table where n is five pi is zero comma four five and find the answer so let's go there n is five pi is zero comma four five and we know that we're looking for the last two digits on that column so those will be the two values zero comma zero one eight five so that is zero comma one one two eight one one two eight and zero comma plus zero comma one zero sorry zero one eight five one eight one eight five yes zero eight five and the answer is zero comma one three one three which is option three you we left with one minute there are more questions that I have on the notes that you can also go through on your own exercise seven also asks you the first option asks you to calculate or actually let's start with the statement the statement says suppose five the students were selected independently check whether or not each student has a shortage of weight previous studies suggest that one out of every four so you are given x and n you can calculate the probability of success there which one of the following is incorrect you need to find the variance of a binomial we need to calculate the probability of at most you need to calculate the probability of at most nine at most one at most nine you need to calculate the probability of at least and at least so those were the question asked exercise eight given your autism again 50 percent they're asking you calculate the expected the variance and the standard deviation and then answer the the question are they all only a correct or b correct or only c correct or a and b are the only ones that are correct or is it a b and c which one is the correct one an exercise nine looks also at africa check facebook ghost you just need to calculate the probabilities yes well probabilities that at all which is exactly exactly exactly exactly and none of that you must calculate the probabilities and answer the question and that will be the last question are the probability that at least 11 profiles are ghost profiles so you just need to calculate the probability that x is greater than or equals to 11 because there are 20 profiles so it will be 11 12 18 14 15 16 17 oh until 20 so the table has there is n is equals to 20 you should be able to find that from your textbooks or from your study guide and can use that to answer the question with that it concludes today's session thank you for coming are there any questions any comments any queries anything you want to know just one thing i noticed they often gave us a percentage will they ever give us say for example like half of like or a third uh no they will give it to you as a percentage or if they don't give it to you as a percentage like we did with the previous one way is that one like with this one where you are given your x and n and you can calculate the probability the same as what i've given you here is another question there is your x and n you can just use that to calculate your probability uh so they will either give it to you in this format or they will give you the actual probability or they will give you as a decimal like 0.2 or 0.02 or but they won't say a third or fourth or no non-confusing like that okay okay Elizabeth um i don't know uh just um and uh what if this i would like to ask this if it's possible maybe to upload the notes um before the end of the week if it's possible on your side just so that by sunday i am ready i for one uh it's not possible and i don't want to lie and i don't even want to promise that i do my notes on day morning when i'm free um okay that is why i only i only give you the notes it yeah on the day of the session because that's when i i get free time to do this in the morning okay all right now that's i will try maybe yeah i will i will see okay thank you that will be really appreciated act um that doesn't stop you from looking at because most of this are from the summary notes that are posted on this yeah on on on on my unisa um i just i just go through some of the questions and revive some of them or add some of the slides to those summary notes but most of the most of the things that we go through are already uploaded on my unisa it's only just some the questions that we use in the activities like now that i only try and find some during yeah in the morning when i'm doing my preparation for the session okay no that's fine no i noticed that there was a difference between the notes because of downloaded everything all the notes and the sessions that we go through so i just noticed that okay fine there were more on this one compared to the other ones yeah so yeah so also remember that the notes that are summary notes that are on there that are preloaded correspond with the videos that i give i give them to you a week before the session so that you are able to follow the video and the notes because they correspond all right um it's only for the session for now that the notes will be different to the rest of the other notes as well yeah okay thank you no blocks um are there any questions otherwise uh we can enjoy our sunday and i'm gonna stop there