Time maybe we'll look at logs. Yeah is okay. Yeah, that works. Yeah but I really have questions for like pure. It's just like little questions like for the fact to theorem. Bit like when they give you, I would just give you like a super long equation with like the cubics and tell you to just like put them into cubics or will they always give you like say, Oh, x minus two is a fact of this, and then solve the rest with like will they ever give you just something, just a cubic until how you solve it? Yeah they can give you a cubic and say solve it. And then Yeah you have to trial and error one of the factors. Then you use some form of algebraic division to get the quadratic from it, and then you use your skills of quadratics to factorize the last two. Oh for the first test is just like sorry having mean the first step is just just try Yeah Yeah you use the facts theorem. So so you sub in. So you always want to try one minus one, two minus two, three minus three, four minus four. It's not one of them. You've probably done something wrong. And then once you've found one, you can divide by that factor. Oh, okay, all right, consenshould, we do an example. Yeah, Yeah, not by one, two. So do we just go with like one, two, three, four, like trying each one? Let's go with. Might just for this teacher, like we did some like long division stuff, but that's only what we have. Like they already gave us something. And then like and then he said, Oh, but but he also mentioned like the method of like just trial and error, but like my previous math teacher, like when I was still in my gseason Admati don't know why, he just told us, Oh, you can just look at it. And then you just come up. What the solution? Yeah. All, let's do that one. I just want to make sure that I'm going to be okay with afftherapy. Yes, I asked question. So we're fully factorizing that cubic and then we're told to sketch it. Okay. So as it's a cubic, there's not really a way to. Like find its roots nicely. So we need to find a factor of it first. So we use the facts that helps for this. Okay? So you plug in a number. So for example, if you plug in three, maybe Yeah if you plug in three, if f of three equals zero, then x minus three is a factor. Okay, okay. You want to try the values? I would always try -11. Well, youprobably get one and first before. So I would try one minus one, two minus two, three minus three and Carry on like this. Chances are by four minus four you found at least one factor. Oh okay Yeah I think three works three does work here see plug in through it does work okay I would if you so for this question that would Boff that top line however if it says if you get a question that says use the facts they want to show that then you want to go really really slow and ball Yeah two lots of three cubed plus three squared -18 lots of three minus nine and really really take it really really slow but there you are right that does equal zero so x minus three. Is a factor. And now we've got to divide two x cubed plus x squared -80 and x minus nine or x minus three. So you've got a couple of ways you can do this. And there's long division, there's the grid method and there's by inspection. So your teacher who's saying you sort of just look at it and you see the answer they're doing by inspection. But the other methods are fine as well. Like it doesn't really matter what you use. I would avoid algebraic the long division if you can, just because I think it's actually not that easy. I think it's easy to go wrong. Because just too many secand like it really takes off time. Yeah, it takes for really a long time. So shall I show you both methods with other methods? Would they ever ask you to like subdo a long division? Oh, okay. No, I don't think I've haven't seen a question where they asked that it's useful if you know you're going to get a remainder. Because it's good at finding the remainder, but you can also use the grid method to get the remainder. You show me the grid method again, please. Yeah, we can do that. Okay. So you might have seen. Similar thing when your times in out when you're expanding out quadratics. So all we're going to do is we're going to do so this is technically, this is the reverse grid method, because normally in grid method, you've got the bits around the edge and you fill in the bit in the middle. Now we're gonna fill in the bits in the middle. So this box here has to be the two x cubed. Okay, because the highest power box always goes there. So x times something is two x cubed. This here has to be two x squared. Happy with that then because this is a two x squared. I now know this this box down here must be minus six x squared because it has to be two x squared times the minus three. And I know overall I need plus one x squared and that be these two boxes added together. So this has to be plus seven x squared. And then this process just loops basically. So I now know I can work out this this middle one at the top, because it has to be x times. Something is seven x squared. So this must be plus seven x. And then from there I can go, okay, well, if that's plus seven x, that one at the bottom of its column must be -21x. And then again, I know I need -18x, so this has to be a plus three x. She makes this at the top. Remember that colors plus three. And then lastly, there's our minus nine. Okay. Then if if we wanted a remainder, let's say we were doing this cubic divided by x one us three, we wanted to know the remainder. And let's imagine that was minus eight. We've got minus nine. We've got a remainder of minus one. So by on as well. So that's Yeah reverse grid method, which then gives us x minus three, two x squared, plus seven, x minus three and then no plus three, sorry, not minus three. My inspection is basically doing that in your head. So by inspection you can get okay with the first the quadratic term has to be two x squared for the same reason because I know it's a factor straight away. I know that the constant on the end must be plus three because I need to get to minus nine. And then it's just try to see by inspection, essentially. So yougo, okay. Well, minus three times two x squared is minus six x squared. I need plus x squared. When I need plus seven x squared, I need another seven x squared. X times seven x goes athat. And you can check it works for the middle term as well. It's essentially grid method, but in your head. Oh okay, strange because us I've never seen that method before. I think I should to meet before when we doing adma but that's a really good method. I think we've used it mainly for expananning brackets. It can be really useful for factorizing as well that Yeah Yeah. And then for this question, we go back to the actual question that that's it for the factor theorem and the algebra division. It's then just knowledge of quadratics. So you then have to factorize the turret square plus seven x plus three. I'm not going to let you do that. I'll tell you what that is. So you're just using new facts of quadratics. So we've got x points three still. And apparently it is two x plus one and x plus three. Okay, so we only need the factor theorem defined as one factor and then we could do the rest manually. We could have found the x plus three, you know we could have plugged in minus three and see how that worked, but we didn't need to. Okay? As soon as you find one, you do your division. So that's the Batrice and then you're happy with sketching that. Yeah, no way. Yeah, that makes sense. Okay, thank you. Stuff. Trick identity, it's just like one. You know how like a graph is like that. And then. And then in which like essay section A, B, C and d and sometimes sign is like let's say splus ten, sometimes positive, sometimes it's negative. Yeah, I just like I don't really understand like this idea. Oh, this is a cast diagram, all right. Yeah, so this is when you found one value and you want to find your other values. Maybe all right, let's let's let's go for site 30. So what is sign 30? Right? No. No, it goes the other way. Sorry, it goes the other way, I'm sure. Let's go science, the minus one over half. Maybe. No 90s. Sorry. Sorry. Is it is it 30? Is it 30? Well, I tested my trade identities. Do I know exact trade values? I think it's 30, but that's what I was aiming for. But I'm down myself now. It is thirno. It is this. I'm sitting there trying to work it out using the triangle. It is thirsty. Okay. So when you're given this and you said. So you might get a question that says equals X X is between zero and 360. Okay. So we can use our calculator as we've just done or whatever to find the first 130. We then if you want to find another value, there's three methods. First one is I think what you'll talk about here with this grid, I don't like how they've labeled the quadrant. So is this how you teach as labell at abcd? Well know it probably isn't. But I was just like saying it. Okay, so we call it a cast diagram. Because of the labeling. Okay, so a stands for all. S stands for sine, t stands for tan and c stands for cos. So these are our three functions plus all. Okay. And then we start from the positive x axis and we measure it in an anclockwise direction. Okay, so the first 90 degrees, so what this is saying that the word in each quadrant tells you which trig function is positive for angles in that quadrant. So sine costant tan of any number between zero and 90 is always positive. Okay? They're all positive in the first quadrant between 90 and 180 degrees, only sine gives you a positive answer. Okay? So sine 170 is positive cos al 170 is negative, ten 170 is negative. Why is that? But I didn't really get the angles bit. So the first bit is zero to 90. So so this is zero to 90 that there would be 90 degrees if you continue around. You then get 180, you then get 270. You get 360 if you want. You could actually do as many spins as you want. Yes. So if you get your calculator and you type it in sign of any number between 90 and 180, you get a positive value. Okay, here's a cause. Of any number between 90 and 180, it's negative. The ton of any number between 90 and 180 is negative in this third quadrant. So any value between any angle between 180 and 270, ten will be positive and the other two will be negative. And then the last quadrant, same is for cause. Okay. So in that quadrant, the cost will be positive and the other two will be negative. Exactly that. Yeah. So if you do cos of I no 300, that's a positive value, but sine of 310 of 300 is negative. Okay. And then let me how this works. But in all, they're all positive. In all, they're all positive. Yeah. Okay. So what work what happens is you'll have done some horrible trick and you'll end up at something like this where you get an answer, you get one answer for x. Okay. And you're looking. That so that's positive. So so sine of x is a positive number. Yeah, we add sine x equals third. No, didn't we had sine x equals a half? Okay, that tells us the value of x is going to be somewhere between zero and 188. Isn't that one of those first two quadrants? Okay, we can use our calacdata to get one of them. It's 30. What you do is you then draw that angle. On here. Okay. Oh, okay, I know what for some reason I felt like this graph is like the sgraph and the cost graphs and ten graphso. That's that's a different method. You can use that one as well. So what I found, in general, students do not like cast diagrams. So this is not usually the preferred method, which hurts a bit because this is the method I use as as a kid. It was the one that I was taso. It was quite quite shocking to me when I you know started to teach and all the kids were, no, I don't like that. So we we found some of the methods basically. Yeah. So once you found one value, you can then use symmetry in this cast diagram, find all your other values. Okay, if you're in a positive quadrant, you use symmetry to get to the other positive quadrant. If you're in a negative quadrant, use symmetry to get to the other negative quadrant. What does that do? So we're a sign and it's positive. So we're in one of the positive sign quadrants. We're in the orquadrant. The other positive sine quadant is the sine quadrant between 19, 180. So we use symmetry of this whole graph and we effectively get 150, 150. Yeah. So we get that there. But the angle we're actually interesting is the 150. Okay? So x, the two values for this one are 30 and 150. If we change the question, if we change the question to but now 720, we've then got those two values plus we've then got whatever that is. So one loop out of it. So thatbe 390 and then again, one loop all the way to. That there we go about that talk that bit. Not 5:10. Yeah I eventually took these two and added 360 to them. Oh, okay, okay. Because it's just it's just a full lap plus another like whatever it is to get to that one. Oh, okay, okay. So that's the cast diagram again. I quite like them because it's the same diagram every time. But in general, students aren' T Fan because it's that they're they're weird. Okay, second method Yeah is you can use trig graphs. So I think we've done a bit of this one in the ad mats. So for example, I think I'm okay with with that. Yeah loso, that's the second method. You draw the graph and you use the symmetry in the graph to find your other values. Okay? That I would say is the favored method in general. The third method is you can just memorize the rules to find new. Got new values. So for example, the rule for sine is that sine of x equals sine of 180 minus x. Cos of x equals cos R here we go 360. Minus eps and ten of x is sound of 180 plus appi think, let me, let me just check them, because I don't, I don't like this method because it is just a memory game. So the problem with this one, as I've just said, is, yes, it's a memory game. And if you forget it, you forget that rule, you'll you'll get it wrong. So I would pick either the graph or the cast diagram. Probably the graphs realistically whereit's going to come up angles in all four quadrants, page two of three. But the using the graph and the cast dager, actually you effectively use this rule. If you go back to how we got that 150, we literally did 180-30. That is literally what we did. Same with if you had the graph if you drew the graph out, youget the same thing. Wear on your nose. Is a more complicated version of it. 对对呀，我说。Okay. So the three I wrote down were. That one, that one and that one positive ones. There are then equivalent ones for negative just speaking that how much are this in the formula? But. Do we get given any of this? I don't think we do it really. We might get a little bit. I don't, I don't. I think so. I'm not sure. No, all the trigs here too trick. They've been really nice with the formula books because they really have like basically all the equations on it, especially for the or that really nice that they give or Yeah Suthat and bionial expansion. They're really, really generous. Same with the if you're doing differentiation from first principles, Yeah, I think we we did it at math actually. Yeah, I think we did it, but I don't think our class has done it yet. Okay. Yeah, they're last in the forof book as well. Yeah, so really Yeah like they give you the the the initial line, like the top the top line that youstart. But Yeah, with that, that's the hard bit to remember. And that's yes, just given, which is nice. Yeah I guess it's good because that doesn't really turn it into like a memory. Yeah I mean Yeah it just feel a bit mean when if if you've not got the greatest memory, it does feel a bit mean if you've got memorized just formulas. Yeah but what but that also means I'm just going to give you really, really, really, really hard questions. Yeah, that's that's the flip side. If it's not a member game, the questions are asked. Last year's adma test was, Oh my God, like the last few questions I've just got like no clue. It was something about something about triangles and circles together and like that's just pretty. Yeah, that was horrible. Yeah, a little bit I think, but I think the is also like a bit lower than you. So I found out Yeah, it was just a hard paper there. I think so I'm not sure what they're trying to do because last year I feel like they changed a lot of like the staff questions compared to like previous years of say, especially for my at excel, like it was just really not expected. I hope they judo that again this year. Now they're a bit more rigid in the Oh, that that's good. Yeah, the boundaries have been they've been climbing recently, but the style of questions are all pretty similar. But that's fine. You you go over a year to wait for that anyway. Sheonly have, I suppose you might do mocks this here. Yeah but mox is like the key part, right? Like you get the grades and then you apply for unyes. Yeah your moare really, really important because that that's what your teachers will use this year or predicted grades if you look at it doing University. Yeah have you thought about that? You thought about what you want to do after after finishing sixfour? I was gonna I want to apply for law and like all the other subjects I do are like essay subjects it's just maths as though new like science you bit so I kind of find math like hard and everything else because it's so different Yeah everything was I study so I do history that's completely just like your and politics Yeah same as well and then you can't look a little bit different like gegraphy Yeah like like a little bit little bit of graphs maybe Yeah there's not a massive amount a couple of formulas and then Yeah issuplanned demand graphs and all that fun stuff. We looked at sorry, I said it's quite a mixture of subjects. Oh, okay. Oh Yeah, quite actually quite a lot of people do that in my school because a lot of them want to do ppe, okay? If they want to be, they kind have to do pc on and that's Yeah similar subject choices. That's right. Brilright what we doing next? Oh, sorry. Should we look at the statbit again? I forgot hypothesis testing. Yeah, Yeah. So hypothesis testing is the last chapter of stats in year one, chapter seven, right? I should already have that downloaded. Let me upload that. Okay, thank you. But annoyingly I don't I don't like the way the textbook teaches it. I think they teach in a weird order. So we're gonna to bounce around a bit. Yeah, okay, but I'm Stathis textbook because ultimately they've got some really good examples. I would just order that in a different order. All it's just just loading got slow. You've got one of the graphical calculators on here. Yeah I'm not sure how to use it. That's what I could show you for this one. I think in fact, now you won't have to do this, but because you had done it before, you've done binomial distribution. Yeah, I think Yeah. Have you done that chapter? Otherwise, we did wait. Let me just check binomial distribution that that sounds like pure. So binomial expansion is pure. Oh, sorry. One of the distribution is the stats part of that. No for apply we've only done like vectors so or similar motion have to annyour first Oh sorry, I'll have to do the binomial first then Oh okay Yeah sure because underwhile people all the physics part first maybe okay, I mean the order doesn't really matter Yeah like the stats and mechanics are completely separate. So you can actually just you know what? We'll do this one. Here one sts on chapter six that was going to take just to initiis it, initialize what it says. Converting once wisdom, just converting it. There we go down. That one, we like that. Oh, Yeah. Okay. So we've got we've got a choice now because if he's not doing this chapter, we can look at all of this chapter or we can look at just the bit we need to do hypothesis testing. We first look at the bit that we need for hysis hypothesis testing, please, and then maybe then do that. Thank you. Okay. So familiar though. I think we did in ma. We might have Yeah, I can't. I'm trying to remember if we did the probability. So I know we definitely did the expansion stuff. Yeah, there's a good look at it now. Yeah, we might look at it and go, Yeah, remember this? In which case we can move on straight away. Okay, so this is chapter six of the year one applied book. The first part we're gonna to skip because that is just using simple discrete probability distributions, doing discrete uniform distribution. We don't care about that. We care about binomial als. So that's the next three bits. Okay, because hypothesis test sts will use mainly this last one actually the hypothesis test will use okay. But to do the last bit of this, we need to have we need to have looked at those first two as well. So we'll skip 6.1 and then on six quite two. So this should be quite quick because you'll have seen some of this before. So binomdistribution looks at a model that does this essentially. So we've seen with binomial expansion that you'll have had, what did we do, distribution. I don't know, I can't remember annoying that I've taught it to other students, but I don't know if I've Tait, were you I it we'll pretend we haven't and if it comes back to you, we'll skip it. Isn't that isn't that just like. I mean not no money would just look good as example and maybe I can't remember. Okay, so you'll have seen you'll have seen stuff like x plus five to power eight. You'll have seen this you know where you go. Oh, okay, well I've got x the power eight plus and would have gone like eight choose seven. No no, 827728 got some more those days. Can we doing this? So this is Bonan expansion. Sorry, isn't it? Did I have it right the first time? No, but like isn't it eight through seven? Yeah, sorry, I'm wait. Yeah, should be eight. I'm not sure. Wait. Yeah. Do you get eight? Which one is it? Hseven, I think I think Yeah. Hseven, I was what, the first? Another one myself. And Yeah, so this is but of an expansion. Okay. Luckily, we don't really have to do much of that nonsense for this because that calculate is going to do a lot of the nasty MaaS for us. So there are there are some links between them in that in binoman expansion, you've got a binary also. There's 22 parts to it. There's the x and the five in that example will just I'll put it back. Yeah, let's let's look at bononial distribution. So a bononial distribution is a distribution that satisfies these four things. So you can model it and the notation, we use this b bracket np and you can only use one omdistribution if the thing that you're modeling has a fixed number of trials. So that's the first thing. So n is the number of trials and it can't vary. So for example, you might go, I'm gonna to flip a coin ten times and see what happens. So ten is your n. If it's fixed, it's not gonna to change, okay? It's binomial als. So it has to be two different outcomes, either a PaaS or a failure because this does not mean it has to be like a coin where it's just heads and tails. We could do a dice roll, however, wehave to define success and failure ourselves. So for example, we might be rolling a dice looking at six es, in which case weconsider six a success and all other for all of the other numbers as a failure, okay? So it doesn't have to just have literally two outcomes, but you have to be able to group the outcomes into success and failure. Okay? Again, we then need a fixed probability of success. So if we're on the dice, it's going to be the same chance of getting a six on every roll. It won't won't change. So it's not like you know those questions where you picking sweets out of the bag and the sweets stay out the bag. It's not like that. The probability stays the same throughout. And then the trials are independent. So this just means they don't affect each other. So viro 16 are not more likely to get another six next. Okay. So other than the name, to be honest, there's not that much that's similar to binomial expansion anymore because you can use your calculator to do it for you. Okay. So in all of a level maths, you're going to see a few distributions. You'll see some of those more simple ones that we've skipped over. There's the binomial distribution and then there's the normal distribution. So binomia will be the one that you get the most familiar with because you'll use this the most. And then this is where it starts to look like the expansion. However, as I said, we don't actually need to write all of this out because our calculate all of this math for us. Okay, so if we got runvarial, x has this binomial distribution with n trials and p is our probability of success, but its probability MaaS function is given by this. This looks really nasty. This is basically saying the chance, the probability that x equals R is that. So this is our n choose R P to the power R. So the probability success to the power of however many U we want, and then one minus p, the power of n minus R. So back in, I got rid of it again, put split back. So instead of x and five. These become. A and one minus p. Okay, you see that? Yeah. And then if I want I'd know that say we want what's the probability that x equals seven would just want. That once would go p power seven y minus p power one. Can you see how what is it going? See how that lines up with that? Okay, so so is there is as I say, I have to keep saying there is a relationship between distribution and expansion. However, you don't need to do this anymore. The back genuinely back when I was a kid, we did have to do this, but you don't have to know you calculate or do all of this automatically. Okay? Sometimes n is called your index, p is called your parameter. But in general, its number of trials, probability, success. So let's see if I can ever Hato this on your calculator. So get your calculator in front here. Done this. That is that is the permutations and combinations. Yeah, we said it comes up in mail. Oh, okay, Yeah. So it's not it's not quite as in depth as the ad math stuff was. So you never asked if I've got three blue balls and eight White balls and this like there's never how many different combinations either to that, but it will come up with in binomial. Okay. See, as I said, you can use your calculator to work these out. So you need to go into if you've got like a like a stats mode. I can just, I can just click my H and do probability and just all comes up on your Press probability. Let me full screen here. Let's set up. How do I full screen here? I've definitely done this before. So we want. I don't want any of them. They're just like short functions instead of mats mode. Is there lack of stats mode or a probability function mode? Are there you settings? How do I do it on beefy graphical? What's the model of calcula? Is that the is that the is that one ti 84 plus cnine one? Distribution. Have a look. Okay, Press second, then vase. So this activates the distribution menu above the vast key. Yeah, okay, brilliant. And then scroll down using the arrows. And we want binnupdf. By pdf Yeah Binon pdf. We got Yeah a pdf for parrot. That's the one. Yeah. Okay. So what this is. So binome stands for binomial, and then pdf stands for probability distribution function, and the cdf stands for a cumulative distribution function. So pdf will tell you the exact probability of getting that value. So click on that one because that's what we're doing right now. And then how do I unfull screen you? Yeah, there you go. And now we input what we have. So if we're looking at example four a, so our number of trials, so n is twelve, our probability of success is a sixth and the value x that we're interested in is two. So you plug them in, you should be able to Press equals, and I'm hoping you'll get 0.296. Wait one sec. I think I did something wrong. Twelve P1 over six x 52 was paste. What is paste? Paste would you mean paste? I didn't see a paste. I have no idea what it's paste. Does that like enter? No, because like if I Press enter and then know, just give me that, let me full screen you again. Now I Press enter again. Yeah, there we go. There we go. Yeah. The last time it gave me zero. Let's see. What did you do in the last one? I know when I entered the zero last time I know, apparently I clicked zero before. Oh, this is zero. Yeah, this a zero before. Yeah, okay. Okay, yes. So but that function essentially does all of this matfor you you don't have to worry with the choose function, youdon't have to worry with bactorials. You have to do any of that nonsense you calculate, or does it all for you. Okay? So well, back when I was at school, you had to do the long way. I had to do this. So I would have written all of that when I was at school. And then I tuped into my calculator to get 0.296. You don't have to do that yet to Press what was it? Second? Second and vase Yeah second and then vase and then bonmeal pdf so you can use this to find anything. So b isn't any harder. You just change you two to a nine apparently you get that nasty number. Okay, they thing can be a bit mean and they can do this. The way I would have done it is you work out the probability of zero, we work out the probability of one and you add them together. I'm not going to make you do that because I'm going to show you, cdf in a minute, a question. Sorry, just a really good question. Like how is this different to like the permutations and combinations? So it's not that it's different, it's that the maths that you yourself have to do is different. So the combinations bit is this bit in the yellow. So it's that that tells you how many different combinations there are. So for example, we've got twelve trials. It's a one sixth. So that says wrong with dice. We've got there six sided dice. And I want to know what's the chance that I get two falls? Let's say the chance of success is the pro go to get a four. I'll just picks a number randomly. So if I if I want two out the twelve to be a four, I don't care which two are four, it could be the first two. It could be the first and the fifth one. It could be the eighth and the twelfth one. There are this many combinations. Okay. So we no longer care about what that number is. Because he calculates the defor you as a twelve choose two is a amount of different combinations. Back at ad maths, the question will be worded differently. So the question might say, back in ad maths, if you roll a dice twelve times, how many different combinations are that of getting exactly two fours and you get, okay, well, I could get a four, the first one and the second one for the first one and the third one, first one and the fourth one, first and the fifth one and there's twelve, choose two. The shortcut I'm telling you now, it's just twelve, choose two. However, I still don't care about that actual probability anymore because my calculator does that for me. Yeah. So you don't you don't have to worry about any of this. So that that one sixth squared times five, six, part ten, that is the chance of going four, four, not four, not four, not four, not four, not four, not four, not ten times. Okay, the twelve choose two, that is the amount of different combinations that that goes well. Okay, well, I could have the two fourscfirst they up the second position and the fifth position. It is combinations. But you don't have to work them out yourself anymore. If that makes sense. Okay. So so so the bit in the yellow, it is that that is the similar soft stitutento ad maths. But now you calculate or deal with all that fun stuff for you. Okay. So the underlying maths is the same, but the way that you use it will be different now. Okay so that essentially you calculated as it all for you if you can remember to go second verse binome me pdf put in your n you p and your x you good. All of these questions are like they then start throwing wordier ones like you so example five it's just a wordier version but it's can you pull out that information so you have to go okay there's my probability I've got n is 20 and then part B I want exactly seven. So x equals seven. Okay, if we look at the the actual solution, you see that's what they have done on 20.15x equals seven. Okay. So that that in itself is a skill. Can you put that info out it out of a piece of text in general? You can just skim to the numbers in general. But Yeah, that that's what six besix bejust loads of this. It's just typing stuff into your calculator. Then it looks at cumulative probabilities. So in the formula book, tha load of tables like this at the back and back when I was a student, you used a table because I could only work out individual probabilities. So if I wanted, let's say the question was this one here, probability x is less than or equal to seven. So less than or equal to seven means it could be seven, could be six, could be five, it could be four, three, two, one or zero. So the only method I've got pre calculator is work out the probability of it being zero, add it to the probability of it being one, and add it to the probability of being two, and do that all the way up to seven. And been there for ages. So what they did is they created these tables and youessentially youlook up seven in the table, and it will tiyou the cumulative probability up to that point. You can still do that. There are tables in the formula book still, but we don't need to do that. We're gonna to use the cumulative distribution function on the calculator. Okay, so all you're doing is you go in second and vase and then you go and bind on cdf and this time you'll put so n is still 20, p is 0.4 they don't change. And then this time you're going to put in x equals seven. In fact, ours, yours might go a bit more advanced than this. 0.4158. Can I see the screen? Thank you. Okay. Yeah. Okay. Yours is insane. Prepthat's one. Brilliant. So if you can get the question to be in the form of x is less than or equal to a number, you just type this into your calculation and as you've just shown, you get the number. The difficulty arises now in that the question won't always be less than or equal to something. Say for example, b, it's x is less than six. So if you put in six, you calculate it will include the chance of getting six successes as well. We don't want that. We want we need the value of less than or equal to because it's discrete data and it can only take integer values. You like, you know, we've looked at it, roon the dice or flipping a coin or whatever, doesn't really matter what it is. But I can't get a decimal amount. I can't get half ahead. I can't get 0.6 of a four. So they have to be integer values. So the probability of been less than six is the same as being less than equal to five. So youthen use your calcullator around five. So x would be five value. Okay, so this bit you still go to do, and calculators don't quite go that far just yet. Imagine eventually if they will yoube able to type in literally that, but at the moment they don't. Okay, then the last one, c is what it gets, really fun. Because this is a greater than okay, it has to be a less than or equal to. So again, we're just going to manipulate the maths so that we get it with a less than or equal to in it. So if we want greater than or equal to 15, we want 15, 16, 17, 18, 19, 20. Okay, so we've done where did the one come from? Like the whole just like the whole thing itself. Yeah all the probability out to one. Yeah all all of the that all the like numbers of different numbers of success from zero up to 20 out to one. So we do one minus all the ones we don't want. So we don't want 14, 13, twelve, eleven, ten and so on and then that bit we can type into our calculator. Yeah okay. They didn't look at so you could have also had if I make a new question, up we go. D probability x is greater than eight. You then just have to manipulate it again. So that is okay. So that's I want 9:10, eleven and so on. So what would that change to? I'll change 21 minus P X equals pick up or equal to seven. Wait, wait, sorry, sorry, no bigger or equal to wait, sorry huh wait, wait, wait six we want a last than or equal to eventually. Sorry, Yeah lesser or equal. Is not. Is it not eight? Yeah, is eight here. Yeah, brilliant. Okay. You might find it helps to go, okay. Well these values are and literally right are nine, ten, eleven and so on. And then you can go, okay, so if I'm doing a one minus, I need it to be not those numbers and less than a over to eight is eight, seven, six, five, four, three, two, one, zero. Okay. And then but that that is it if you can. Here we go. Actually, there's a table showing them up. If you can go from the phrase to an inequality to one that has a less than are equal to, and the calculator does the rest for you. Okay and then there's another example now it's a wordy one and then Yeah the wordy so the phrases Oh don't want to do that. The phrases are quite mean. You've got to understand what that bit there means what that bit there means and so on. Okay but then that then is essentially it that chapter and then we can use that for hypothesis testing. I've lost where we got to where early we were at that bit. Okay, okay, what next lesson can we go through? We go through the theory of this again, like the original method, like that, the cativmethod. And then can we just Carry on with hypothesis? Toaplease ever do that? Lovely. That is Friday, pretty down for. Yeah, okay, lovely. Brilliant. Well done as always. Brilliant. And I'll see you on Friday. Bye.