A Level Maths Made Easy - Binomial Distribution Statistics

welcome back to another one of the as statistics videos and this time we're going on to something that is both very exciting very new and also something you can do entirely on your calculator which makes it pretty fun because it's quite easy now we're going to talk about the binomial distribution now there are other distributions that you may come across such as the pon distribution and the normal distribution all of those can be done on calculators too so the second I show you how to do this your life's going to become a lot easier which is why I thoroughly recommended getting this calculator here the fx991 ex because it does I'm not saying other calculators don't do it it's just that's the one I'm going to be using and that's the one I'm very very familiar with however if your calculator is slightly different you can probably figure it out so this comes right before we get on to probably the hardest part of a statistics which is hypothesis testing because in hypothesis testing use the binomial distribution to figure stuff out so what is the binomial distribution well probabilities in real life follow a distribution so you might have seen a normal distribution before it's a bell-shaped curve so for example the amount of people that are a certain height follows a bell-shaped curve we Peak at an average height so something like 5'9 in the UK or something like that and then towards the tail ends the number of people drop off exponentially which makes sense right there aren't many 7t tall people walking around like there's not many 4T tall people walking around so this is just another form it's a a binomial distribution now you might know what by means it means two because you can only use a binomial distribution when you have two options okay so to use a binomial distribution here's what you need first of all you need to be able to simplify your scenario into two outcomes which are usually something like success and fail so heads and tails cool but then you might think okay rolling a dice you could never use the binomial distrib ution but you can if you have for example probability of getting a six and not getting a six that's two outcomes probability of winning a lottery and not winning the lottery probability of being a certain height and not being a certain height all of these can be the binomial distribution but remember they don't they give you a success fail yes no the other thing you need to know is there has to be a fixed number of Trials so that might sound a bit weird but for example if I'm trying to work out the probability of getting six heads when I throw a coin I can't just say what's the probity of getting six heads because it will be one if I have an infinite number of throws or 100 throws or a th000 throws so instead what you have to do is have a fixed number of throws so I could say what is the probability of me getting fewer than six heads when I throw a coin 50 times that's valid because I have a fixed number of Trials likewise if I have um let's say I'm putting pizzas into an oven and I want to know the probability that out of 50 pizzas I burn 10 or more of them that's binomial as well those kinds of things the lastly you this probably is quite straightforward but you need to actually know the probability of success so you need to know the probability um that we call it the probability of success but that sounds a bit weird because it sounds like a good thing so for example the Proby burning a pizza burning pizza is never a good thing however that could be the success essentially the way I think of it is the probability that the event you're considering happens so the probability of burning a pizza is4 that's a probability of success I know it doesn't sound like success but that's what it is so now we can talk about this kind of notation here so the discrete random variable x tilder b what does that mean well the way you read this is the discrete random variable X can be modeled as or is approximated by that's what that tilder means a binomial distribution now this binomial distribution looks like this where n is the number of Trials so how many times are you doing this thing and P again is the probability of success right so just to read that again the discrete random variable X is modeled as a binomial distribution B if it's normal distribution it's n if it's a uniform distribution it's U if it's uh Plus on it's p Etc then number of Trials and probability of again success so now I'm going to show you how to actually work out these probabilities how exciting so for part A it says can you find the probability that X is exactly equal to 5 yes I can now if you ever see the equal sign you're going to be using the p d function I'm going to show you what that means right now because PD stands for probability distribution okay so you go to shift menu sorry you go to Just menu down to the right twice and you come across distribution don't click statistics although that might make sense distribution because we're looking at distributions press equals and you have all of these options go down to binomial so we have binomial CD and binomial PD CD is cumulative distri distribution and you use that whenever you have an inequality so these both mean cumulative distribution because remember it adds up okay now we want PD because it's in equals we want a precise value so four Now list and variable if they give you a number choose variable so what's X well X is the value that we're looking for which is five n is the number of Trials 15 p is the probability of success I'm done that can be a two mark question in an exam done I've done nothing I've done no intelligence I've done no knowledge just how to use on a calculator now you might think well can I still get marks or do I have to do some kind of calculation well without this calculator or without a calculat that can do this you don't need to do a calculation instead what they do is they give you a fat booklet which contains pages and pages of different values so it's a massive table and what you have to do is find for the binomial distribution n is 15 probability of 35 and then find x = 5 it is so pointless to do that it's complete waste of time but the good thing is is they tend to give you two or three marks for this because of how long it takes to look through the table so if you use a calculator you save 3es straight away you're saving time really really quick and efficient so in actuality you won't lose any marks because there's no calculation if you look at any Mark scheme there's no calculation you do it's just you either look it up in the table or you do it on a calculator and personally calculator wins every single time it's so much better in terms of uh kind of how many decimal places well this is kind of a test question I was giving you so it's not a kind of an exam question although this is an exam question that could come up um it's normally going to just be give it to three significant figures is what I would suggest now of course it's not all sunshine and rainbows instead we do have a couple of other starles of question funnily enough part C is easier than Part B now we're going to I'm going to do part C first and that might be a bit weird but let me show you why so as you can tell it's an inequality so instead of binomial PD we're going to go to binomial CD okay and then variable now you in this case you may if you're very eagle-eyed you may have spotted that one has an equal sign less than equal sign this one has a less than sign just so you're aware CD only gives you less than equals to it doesn't give you greater than it doesn't give you greater than equal to it doesn't give you less than it only gives you less than equal to so if they ask you for greater than you have to manipulate it to give you less than equal to as you need to manipulate your answer if it's a less than you need to manipulate it so you get your answer okay but for this case part C it's less than equal to so now I'm just changing this to a 10 done 997 so the question is how do I manipulate less than four for me to be able to use on a calculator well less than 4 is the same as less than equals to 3 because remember this is a discrete variable right so it takes fixed numbers integers so all I have to do is change that to a three less than equal to 3 is the same as less than four when you're dealing with integers right if I ask you to write out all of the integers that are less than four and all the integers that are less than equal to three you would write the same list of numbers so equals to go back change this to a three done 0.173 for this one cuz it's three significant figures and that's it I'm not even kidding this kind of question has come up before and was worth five marks right it's not that bad but it does take practice and I would thoroughly thoroughly recommend that you get this calculator using the tables is a pain in the neck it's not difficult it's just tedious and it takes a lot of practice and the reason why I say that is because you'll know roughly where certain values are but that's not exactly a good form of knowledge right so I would thly recommend getting that calculator I'll put a link to it in this post but again bear in mind that's an Amazon link so you might be able to find a cheaper version in person or somewhere else online or buy it second hand I'm just saying that that's the calculator that you kind of want to get so you can probably imagine we're going to do for the rest of this video and if you looked at the length of this video it's a lot of practice we're going to keep doing this however of course there's going to be different inequalities that we're going to deal with now first things first if you notice they haven't actually given me the binomial distribution and they haven't even said if we should use a binomial distribution interesting because part of uh a level statistics and I will say this is more A2 when you come across normal and plus on as well is you sometimes have to decide which one to use now of course because this is a binomial distribution video you probably can tell it's going to be binomial distribution so but let's think of why so the probability of Harry being late for school is 0.1 over a term of 30 days find the probability that Harry is late one time more than four blah blah blah so why is this a good thing to use for the um binomial distribution well because there's only two outcomes he's late or he's not late you can't be kind of late right it's late not late that's it cool and we have the probability and we also have a fixed number of days so a fixed number of Trials so you can probably see by the way why the wording of the definition that I used earlier is a bit vague because number of Trials can mean a lot of things 30 days is technically a number of Trials and so is um being late for school being 0.1 being a Proby of success so how would I write this well the first thing that you write if you remember is the number of Trials so we're going to put 30 there and 0.1 as our probability of success so exactly one time what does that look like well that looks like the probability that so I should probably put the X in the tilder that X is equal to 1 so what we're going to use take a second to think about it if you guessed probability distribution PD then you are correct because it's an exact number of values it's one each exactly one time so go down to B PD or just choose that variable because we know how many we want so X going to be 1 n is 30 now and the probability of being late is 0.1 0.141 again to three significant figures because they haven't really given us any other indication that's the why I will say that when we do hypothesis testing we sometimes go to something like five significant figures but you know that that's for another day so more than four times what does that look like as an inequality well that looks like X is greater than four so we're going to use the cumulative distribution because it's an inequality but remember what I said we need to write this in terms of less than equal to so how can we do that well we want more than four right well if I look at a number line this might help me so if I just make a straightish line like this and then we put the number zero 1 2 3 I know it's not evenly spaced it's completely fine how can I make this less than equal to we want the probability that all of the numbers are greater well we want more than four but the problem is we can only go from from zero upwards so what we could do what we could do is work out all of these values here so all of those values and take it away from one so y one well if it's not less than four it has to be greater than equal to four right so uh that means we would actually have to include this value as well so if it's not less than four it has to be greater than four right if it's not less than equal to four it must be greater than four so what we're going to actually do is do 1 minus the probability that X is less than equal to 4 because that will give me the values I want now when I uploaded this video to YouTube once um not this video but another one like it there was some confusion about using this overlap so I'm going to write out in a different way we're looking for probability of X is greater than four so we're looking for the probability that it's 5 6 7 a dot dot dot but the problem is I can only work out the probability from left to right right so as you can see I I want to get rid of all of these values so how can I do that well if I work out the probability of getting these values and then subtract it I'm taking it away hopefully that makes a bit more sense the probability of getting all of these values is of course one because all probabilities have to sum to one so this is where things get a bit different first of all we need the cumulative distribution and again I hope that makes a bit more sense with practice I I it will make more sense so we're going to cumulative dist distribution so that's one variable and we want four and the rest of this is the same so we have 0.825 but remember that's not our actual answer so we want to do 1 minus 0.825 now one trick by the way is this is still saved as the answer button so if I go back to calculation I can do one minus answer and get the value 0.175 which is the actual real answer and then for part C it says less than three times so that is actually a much easier one so X is less than three so that is going to be the probability that it's less than equal to two because remember it's a discrete variable think about it can he be late 1.5 times or 1.36 times or 10.24 51 times no he's either late if you count the number of lates it's always going to be an integer so once again back to the distribution and it does usually save and it's cumulative again it does usually save oh never mind it didn't this time but it does normally save the values and N was 30 and the probability of success is 0.1 so equals a couple times you get 0.411 again six mark question the thing that's difficult here is definitely this middle part here so really make sure you understand what I was doing with this kind of number line and with getting rid of these values okay really try and understand where it comes from but again with practice it will make a hell of a lot more sense and now comes on to the hardest inequalities that we'll have to do which is when they have a between sense so here we have it's less than seven less than equal to seven that bits easy to do then we also have to have it greater than three so how is this going to work well once again we can visualize this using a number line or by writing out the numbers so I'm going to do both so first of all with the number line again let's just draw a straight line so 0 1 2 3 4 five 6 7 8 okay now the values that we want are seven and greater than three so what we can do using our inequalities because again remember we have to go from left to right is we could work out less than equal to 7 which would look like this and then get less than equal to three so this is the probability that it's X is less than equal to 7 and this is the probability that it's less than equal to 3 and subtract them from each other so we could effectively do this minus and that will give us our answer perfect right now let's show you in the other way because I'm willing to bet that didn't make full sense to everyone so if we want between three and seven here's the values that we want we want four five 6 and seven now if I just go from 0 to 7 then match them up again what I want to do is effectively get rid of all of these ones so how can I do that well if I work out what all of these are then I can subtract these ones so in other words I want all of the numbers that are less than equal to 7 which is why we have this bit less than equal to 7 and I want to get rid of everything that's less than equal to three so I subtract less than equal to three hopefully one of those methods makes sense because these are really good for just gaining three marks once you understand it again this is very confusing for the first time you've never done something like this before ever in your life so it's completely understandable that you are a bit confused so once again we're going to be using the good old fashion cumulative distribution I believe I was on the cumulative distribution before yep by no CD so I'm going to put the seven in first and if I remember n is 20 and the probability is41 so we have 0.380 let's just call it for funsies and we're going to subtract change that to a three 0 013 let's call it I know that technically that's not a significant figure but what I'm going to do anyway is I'm actually going to round it to three significant so 0.38 minus answer and then I'll round this to 36 fig anyway so 0.367 which turns out to be the same anyway so it doesn't really make much of a difference does it and that's it hopefully it starting to make a bit more sense um again this is the trickiest part of this whole thing once you get this once you know what you're looking for this becomes Child's Play I could get get a 9-year-old to do this right the actual putting into a calculator right but getting to that bit is where you actually get your marks that's where you prove your knowledge and it's not the easiest thing in the world so the best advice I can give you is make it make sense if it doesn't leave a comment I will help you out maybe even we'll do a kind of like Microsoft teams quick go through of it or a Discord call something like that I will help you understand this but take time yourself to really break down what I'm doing why am I doing it and kind of understanding it there I explain it in maybe two or three different ways but if they don't make sense let me know and I can help you out one that's probably not too bad considering the last one that we just did but the discret random variable X is modeled as a binomial distribution with a number of Trials of 30 and a probability of success as 058 find the probability that X is greater than equal to 12 so once again if we just draw our little number line out and I'm going to make it a bit smaller by starting off at eight because if you notice we just care about greater than 12 so to be honest I probably should have started at 12 but you know what live and learn and we want all of these values so in other words we want the values 12 13 14 15 and 16 and this goes on forever by the way and the problem is is we can only go less than equals 2 so what we want to get rid of is these values so so dot dot dot we have 8 9 10 11 12 13 dot dot dot so we want to get rid of all these values so the best way to do that is simply put just do the probability of getting less than equal to 11 and do one minus it cool back to the cumulative distribution so number seven and then it's cumulative again normally is cumulative but just don't get caught out with it because it isn't always as we've seen today and some of the questions that I've taken you might notice it's only part A because Parts B through D sometimes e are hypothesis testing which we obviously haven't gone through now the reason why I split it up like this is simply put it's hella hard otherwise okay and what I mean by that is it's too much to know at one time so sometimes schools will literally go through all of it at once the whole thing hypothesis testing are you going to learn hypothesis testing the binomial distribution you're going to learn all of it that's too much in my opinion it's better to get a good grasp of the binomial distribution and then do the hypothesis testing remember we do have time we have quite a bit of time so 0.985 before our exams take it slow it's better to gain some if you rush things you'll gain a couple of marks everywhere if you take time you'll gain full marks in most places which is better okay you want that efficiency you want that quality so you have three more questions and therefore three more examples I would recommend now going through the worksheet yourself doing the last three and then coming back and checking your answers if you haven't been doing that already remember this is active learning I need you to be doing these questions as well or at minimum doing them again tomorrow for yourself seeing how much you retain but try and do one now at least so this one is probably a good one to do okay but for those of you that have already done it let's say company produces pens the probability that it's defective is 08 sample of 15 is taken find the probility that two or more are defective so again we're going to do this as a binomial distribution what is the sample size it's 15 what is the number of Trials it's 15 and the policy of success is 0.08 0.08 chance that a pen is defective and we're looking for defective pens and they're asking me to find the probability that X is greater than or equal to two two or more means greater than equal to now you might be able to do this in your head now it's going to be X is less than equal to 1 cuz think about it greater than equal to 2 means 2 3 4 5 6 7 blah blah blah so we want to get rid of zero and one so that's exactly what we're going to do getting rid of 0 and 1 means getting rid of everything that's less than equal to one so seven and then go down one variable uh we have one o careful and 0.08 so that's 0 all right maybe I'll go to four decimal places for this one 0. 6597 actually no I don't need to do that one sec uh six five dot do dot then need to do one minus it anyway so then I can round that perfect one minus answer [Music] 0.340 nice and easy technically you don't need the zero but if you're giving it to three SigFig then you do need to include the zero Andy plays tennis the probability that Andy will get one of his serves in the court is 60% again that means you're just going to be writing it 0.6 however ever you can actually put 60% in your calculator you can even put fractions if you wanted to any of these will work so you don't even need to do anything okay you can just use the normal convention remember to get a percentage on your calculator it's shift and the answer button that does indeed work and he serves 20 times so we have again a binomial distribution absolutely shocking in a binomial distribution video 20 times 60% again again I'm just going to write it 0.6 and then for part I it says exactly 15 serves so have a think what are we going to do we're going to do the probability distribution so menu 7 binomial PD because we have an exact value it says exactly 15 equals 20 equals 0.6 equals and that gives me 0.075 now if you're wondering why this seems kind of low keep in mind this isn't cumulative so even though there's a good chance he'll get one of his serves in it's exactly 15 serves which is a very specific number it doesn't say he gets less than or equal to 15 serves or more than 15 serves however the next question does say more than 15 serves so we want the probability that X is greater than 50 so once again using your number line hopefully you can see that that'll be 1 minus the probability of getting less than equal to 15 okay so we want the probability of getting more than 15 serves so we take 15 and below and get rid of it which is exactly what that says now of course we need to change to the cumulative distribution so seven down 1 variable 15 again so the rest this is normal so that's fine 0.9 49 but again we're going to be using our calculator so we don't need to be too precise so back to one one minus ANS 0.051 let's call it so there's about a 5% chance that more than 15 would be in the core and this makes sense CU he misses 40% of the time so hopefully this is becoming a bit easier for you and I thought i'' be kind end with hopefully a question that is quite easy for you right now so again a dice can use the binomial distribution even though there's more than six outcomes because if you look at the wording it's landing on six and that's it so it's either it lands on six or it doesn't so again we can absolutely use a binomial distribution so it says here the number of times they roll it is 20 and the probability of getting a six is4 so this question is literally just ask asking me to find the probility that X is equal to 5 which again I decided to give you a little bit of a soft ball just a little bit to finish up we're looking for PD because we want exact values variable so we're going five and it's roll 20 times and it's 0.4 0.075 let's call it there you have it binomial distribution now if you hated this this then I'm sorry because the next video is going to heavily focus on the Boma distribution and in fact a lot of these kind of part A's we're going to be redoing them in the next video as well because I'm going to include the whole thing um just for practice purposes so even if you don't decide to redo the worksheets which I recommend we're going to redo them anyway okay you don't have a choice in that but once again I would recommend that tomorrow within 24 hours ideally within 48 at worst redo these worksheets have a look through them and try and do them all yourself remember maths is active it's like drawing I can tell you how to draw but it's not going to make you better at drawing until you actually start practicing drawing math is actually the same way okay you need to practice it yourself you need to get used to it yourself I know it's a lot it's very painful but you have all the answers you have the mark schem you have these videos you have basically all the support you can ever ask for again leave comments if you get stuck and I will respond to you but other than that I hope you have a wonderful day and I'll see you in the next one