GCSE Maths Tuition Lesson 8 Probability Part 1

foreign welcome back to another GCSE math lesson I believe this is the eighth one today we're going to be covering the first part of probability so there are two parts of this the second part is going to be the extremely difficult kind of exam questions that you can come up with uh but this first part is going to be mainly about conditional probability so things like and it's not just going to be the uh kind of probability tree diagrams so make sure you stick around for that for the first one it says there are only red marbles and green marbles in a bag six red and four green mason takes at random a model from the bag he doesn't put it back into the bag and he takes the second one complete the tree diagram so what I would like uh again in with probability kind of questions it can help to draw a diagram so we have one two three four five six and we'll put little R's in here so six red and then we have one two three four uh green ones right so the way these three diagrams work is this is the when he put his hand in to the bag and pulls out a marble right there's a chance that it's red and there's a chance there's going to be green okay and then if it's red then the next time he puts his hand in you have those two possibilities of it being again red and green likewise if it's green then these two so basically it shows you all of the different outcomes from pulling out in this case two marbles from a bag so good evening Evie how are you so essentially all we need to do here is work out before he's pulled out the first marble what is the what are the odds of him getting a red versus a green well in total when we work up probability we look at the number of ones that we want divided by the total number of outcomes so that's a very fancy way of saying okay so there are six red marbles so I know the top of my fraction is going to be six now how many marbles do I have in total well there are six red and four green therefore in total there are ten okay likewise for green well there are four greens and until there are ten now one thing to note about probability tree diagrams is a good way to check if you've done it correctly is that across the branches it should add up to one because all probabilities add up to one so six over ten plus four over ten gives me ten out of ten which is one so we're correct there and here's where it gets a bit tricky okay let's look at the first possibility where when he pulls out the first marble it's a red marble and he doesn't put it back so if we wanted to draw that out we could just put a line through here and say well he's gotten rid of one red marble so now we need to point out the second one what's the probability of him getting red and a green well how many red marbles do we have left well there's one fewer so there's five now so one two three four five so the top is now going to be five now how many there in total well we've taken one marble out right from this first section here so there's only nine left okay and again if we can't stop the marbles here we have one two three four five six seven eight nine so the mistake that I tend to see uh students make is that they don't take away one from the top and the bottom right so if a red Marble's taken then the number of Reds decreases by one and the total number also decreased by one okay now likewise with the green well there are still four greens left because we didn't take a green out I have a total of nine and again if you look across the branches here five at four is nine so we have nine out of nine which is one now what happens if you took a green marble well the number of Reds doesn't change so this is still a six but the total number has decreased to nine and likewise with the greens now it becomes three over nine another question I get asked is if we have to simplify the fractions so you don't have to with probability tree diagrams because it gets a bit tricky to realize like they need to ideally be the same so you don't need to simplify them okay and it in some cases helps to not simplify them hopefully that will make sense again I don't think probability tree diagrams are too tricky there's just a few things you have to remember um and again the questions are going to get more tricky as we go along perfect so Part B says work out the probability that Mason takes two marbles of the same color and again this is another place where students tend to make mistakes okay the first thing you need to do is write out the different ways that he could take two marbles of the same color so in this case he can either take out a red followed by Red or a green followed by a green now it might seem fairly obvious in this question but in some questions it can get a bit tricky make sure you eliminate every possible way of fulfilling this condition okay so red and a red what we're going to do is read along this probability tree so you fix a red then another red now when we go across the tree like along it here we times the probabilities so in this case we'll have 6 over 10 multiplying by 5 over 9. another way to look at it is we want the probability of getting a red and another red now whenever I say the word and in probability it means times okay so the other way of writing this would be probability of one red and so I put a times a probability of getting a second red okay 6 times 5 gives me 30. 10 times 9 gives me 90. now I'm going to have to add probabilities together so I'm not going to simplify it just yet what about green and the green well we start off at this probability here it's 4 out of 10. and multiply it by this probability which is for the second green three out of 9. now here's where some students get a bit confused they think that this is the answer but it's not okay so in order for me to get two marbles of the same color I can get a red and a red or a green and a green now whenever I say the word or I need to add okay so probability of the same color is the probability of getting two Reds 30 over 90. or so add the probability of getting two greens 12 over 90. and again the reason for that is because sorry I've got a bit of a cold so it might sound a bit uh under the weather where was I okay so because we're saying that there's two different ways of it happening we need to add those probabilities together okay so in this case it's 42 over 90 you can simplify if you want but you don't have to okay so you can say it's 21 over 45 leading divided by three as well right there's m1421 that becomes seven and that becomes 15 and that's as low as you can get either way you don't have to simplify with probability usually unless they specify and I don't think that was too tricky but again the questions will gradually very slowly get harder and harder okay so even here this is slightly more different more different more difficult this is something called conditional probability okay and I'll explain why in a second so each day Paul either wears a red or a black tie to work on Monday the probability wears a black tie 0.6 okay so I'm going to put that in right now probability on a Monday is 0.6 okay now if it says here he either wears a black or a red tie those are the two options so if the probability of in buying a black tie 0.6 the informative him wearing a red tie is 1 minus 0.6 and the reason for that is all probabilities have to add up to one okay so if Paul wears a black tie on Monday the probability that he will wear black tie on Tuesday 0.35 so black time Monday so black time Tuesday is 0.35 and again we need to make sure it adds up to one so that means the red tire is 0.65 . so the reason why it's called conditional probability is this here is a condition if he wears a black tar Monday that's a condition then the probability changes okay so the probability depends on what happens on Monday so it's a condition okay but it's really not that complicated it doesn't really matter if he does not wear a black tie on Monday the probability that he will wear black tan Tuesday 0.68 so again we can fill that in and again it needs to add up to one so all we need to do is do one minus 0.68 which is 0.32 those are our two marks are there any questions on that at all is all about really reading quite carefully oh good perfect so again they're gradually getting a bit more tricky but not too bad so far I hope and again work out the probability pool wears different colors Ties on Monday and Tuesday so again this is where students can get a bit worked up in terms of the probabilities so he has to wear a different color tie so one red one black so we can either wear the black tie on Monday followed by the red tie on Tuesday or you can wear the red tie on Monday in the black tan Tuesday now there's no other combinations right because either he wears two black ties or two red ties which isn't a different color tie okay so again we can work out these probabilities now Black Tie on Monday 0.6 so let's put 0.6 here and then because we're moving across sorry along the um tree we times that by the red tie on Tuesday 0.65 just so you're aware as well is the first question we did can turn up on non-calculator or calculator because it's fractions but with decimals it's only going to be um calculator paper okay unless they're really easy decimals but these ones aren't particularly easy I'll work those out both together in a minute red then black we have 0.4 multiplied by black which is 0.68 so again if you notice I'm hesitating when I'm saying it because again it needs to be red then black some students multiply the wrong probabilities they go red and red or something all black and black for example so now we can pull up our whoops not that Katana put up our calculator perfect so 0.6 multiplied by 0.65 gives me 0.39 and then we also have 0.4 so hey which is 0.272 but again the probability of wearing a different color is both of those things added together okay so answer plus 0.29 gives me 0.662 and that's our final answer and again that's three marks you might notice it doesn't really take three marks worth of time but again it's all about effort probabilities tends to be one of the trickier topics because people can lose marks Without Really noticing okay so this is these kinds of questions are the most common for students to kind of lose a Mark or two because they don't include other probabilities or something else okay so it's very easy to make a mistake everything okay with that one mm-hmm I think that's all okay can it gradually gets more difficult so finding this easy that's good perfect thank you Evie so again this one so you may have noticed in the previous questions it kind of holds your hand a little bit right um it takes you through it but this one is just a format question straight away you have five red three blue counters two cancers taken at random work out the probability that one of each color is taken two ways to do this okay the first way is to do what was given in the previous question and draw a probability tree the other way is I like to like draw a little diagram five red counters there are three blues so I can do the probabilities in my head fairly easily but that's because of practice but again you can draw a probability tree now if we want to one of each color either a red is taken then a blue or a blue is taken then a red right now let's have a think about the probability so I'm going to do this without doing a probability tree because we've already because I want to show you a new way of doing it because in the previous questions we've already done probability trees Okay so probably still taking a red right is going to be the five Reds we have so five over the total number of counters one two three four five six seven eight and then again don't forget we need to multiply because it's a red and a blue now if we've already taken a red we can put a line through red now what's left over well we have our three blues over seven okay so again you don't need a probability tree but again this is just a different way of doing it if you want to use a probability tree that's a valid method okay so we've done that possibility and again I'll work it out in a second so probably getting a blue at the beginning is the three blues we have over the total number of um counters which is eight multiplied by the five Reds we have over the seven that'll be left over if I were to cross out blue all okay with that again just stop me if you uh need any help with that 215. then we can that you might notice it's actually the exact same probability but once again one of each color car so one of each well either one of those methods works right so we're going to add we're going to take the red blue probability and add it to the blue red because it's either or it doesn't matter and that is a four mark question done in three lines so you can use a probability tree that's valid we can do it this way and is this method okay is there any questions about that at all good we're going through these pretty good yeah perfect thank you Evie perfect so Perfect all good okay now here it gets a bit more tricky right because we have three colors okay so there's a few more combinations but it's the same maths okay so again let's draw a diagram out so we have one two three four five these are Reds we have one two three blues and we have one two great again if you want we can do a probability tree but it's going to be a pretty big tree but Billy takes two counters at random from the bag work out the probability that both of the counters are from the same color good so either it's two Reds or it's two blues or it's two greens fairly standard I think I think that's okay for Two Reds well how many Reds do we have at first well we have five and how many uh counters we have five six seven eight nine ten and again because it's and we need to multiply and we get rid of one of the Reds we have four Reds left over and there's nine counters left over again just feel free to stop me if any of this doesn't make sense two Blues well we have three blues at the start out of ten then one of them goes away so it becomes two over nine so again you can see that you can start doing this a bit faster and faster and faster okay but only once you're comfortable with the greens we start with two over ten and then one over nine so again uh feel free to digest that and stop me I'm just going to work out these fractions okay now once again we're looking for the probability that both counters are the same color not that they're red red blue blue or green green right so we need to add all of these together so 20 over 90 . so 20 add 6 is 26 plus 2 is 28 90. and again you don't have to simplify if they don't specify okay so I think that's okay so far hopefully again that was a format question done pretty quickly I think okay okay so in this one they actually give us the diagram okay so I'm just going through a bunch of different style of questions and they should be in general getting harder as we go along so Helen takes a card at random she does not replace it Helen then takes another card at random so so far all of our questions have been this they don't replace okay now if they don't say that they replace it then assume that they don't replace it okay if they replace it something different happens so you need to be very careful that it says this or that it doesn't say they replace them okay there's a question coming up I'm pretty sure there has one where they replace it the probabilities change slightly like think about what would happen okay then takes another card at random both cards have the same number well either it's going to be a one one or a two two or a three three okay what about probably getting a 1-1 well at first they make two out of seven right so there's two ones out of seven cards then one of the seven goes away so we have one one left over one one out of six total cards tutu once again it's two over seven and it'd be one over six again because there's only two cards cross one out look at that three threes we have three over seven at the beginning multiplied by two over six it's gonna make sure you can see when I'm getting all these fractions from uh just while I work this out so we have two over forty two we have 2 over 42 again then we have six over forty two and once again it's asking for probability both cards have the same number so two plus two plus six would be 8 over 42. so again we're doing these questions pretty quickly now the second part this is the style of question that can get a bit more interesting okay definitely a higher paper question definitely towards the end to calculate the probability that the number in the second card Helen takes is greater than the number on the first card she takes there's a uh is that the answer to Part B or oh well spotted TV two plus two plus six is ten not eight I'm sorry I'm dosed up on a lot of medicine at the moment yeah that is completely correct that is a 10 well spotted cool uh this isn't an actual exam paper Lila so I've actually gathered a bunch of questions from different exam papers and also from math Genie okay so this isn't actually a specific paper um in terms of what paper can it come up in so it can come up in paper one two or three but mainly It's Paper one or two okay if it has decimals it's only paper two because it's calculator if it's fractions it can come up in paper one or two okay so calculate property okay see it we have a bunch of different combinations okay so either if she picks a one first then the second card must be greater than one so it can either be a two or it can be a three okay if you picks a two then the second number has to be a three and she can't have picked a three right because you've picked a three there's no number bigger than three is that okay for the three combinations we have are there any questions about that foreign card yeah perfect sweet a lot of I think I know you mean there's no question so good so probably to getting a one and then a two uh so obviously getting a one was two over seven if we remember rightly ah perfect thank you Lila and then getting a two afterwards well if we get rid of a one we still have two twos left out of six that would be two over six likewise if we have a seven we have uh sorry a one and three so that'd be two over seven times and we actually have three threes left so that'd be three over six I may not have left myself from most amount of space but yeah then we pick a two first so that's two out of the total seven multiplied by will we get rid of one of the twos and it has to be a three so three out of six so that would become four over forty two I'll try and make sure I can add basic numbers now so 6 over 42. and this will always be six over two and then probability um I'm not going to write the whole thing out it's going to be 4 over 42 plus 6 over 42 that's ten so be 16 over 42. this time I didn't make a mistake should be it it's getting slightly trickier in terms of what they can ask you this can be anything okay so you have to be um if there's one thing to take away from the lesson make sure you understand my logic because this kind of question won't turn up but they'll change what the condition is and unfortunately there's basically an infinite number of conditions they can give you mm-hmm okay uh this one's interesting so 50 people asked if they like tea coffee and hot chocolate and everyone I liked at least one of the drinks 17 likes or three blah blah this type of question right when they give you information like this okay it's always so if I if I draw blocks here you have to draw a then diagram you have to this information is way too complicated to try and piece together yourself okay hopefully you can see why Nutri exclusive okay so the word mute sorry the term mutually exclusive actually I don't know why I'm writing mutually exclusive just means that both events cannot take place at the same time okay so the word exclusive means that it's kind of apart from everything right like an exclusive Club means that certain people can't come in mutually means that it can't happen for both of them okay so mutually exclusive means that both events cannot take place at the same time so for example if the probability it's raining tomorrow is 0.4 probably if it's sunny tomorrow 0.6 those two events are mutually exclusive because they can't happen at the same time for example hopefully that clears up your question so we have to draw a Venn diagram uh is everyone okay and this is a five month question is everyone okay with what a Venn diagram is I hope so okay okay so what you need first is we need a circle for each of these three things and they need to overlap so this is where things get a bit messy so we say that's for t then coffee has to overlap and then hot chocolate needs to overlap as well page so T is for T C is for coffee and H is for hot chocolate in fact let's do HC for hot chocolate maybe that'll make it a bit more meaningful cool now we have to read this uh one by one so with Venn diagrams you always start from the middle and work out Venn diagrams tend to be something quite easy for students but they always make mistakes in the exam so again those of you that are tuning in I'm willing to bet you're pretty clever right and you can understand maths with a bit of help okay it's fine the problem is is human beings tend to get a bit arrogant right and when we find something easy we rush it and we make mistakes when you come to sit your exam or your mocks or whatever you'll find that you'll answer every question but won't get a hundred percent because you've made mistakes okay so with this be it's so easy to make a mistake with this it's unbelievably easy okay really really easy so you start from the middle of the Venn diagram and work outwards okay so you've worked from the three-way intersection then the two-way ones then then the outer ones okay now the good thing is is that this information is written in order that you should add it okay actually no it's not never mind uh you'd have to work out your way out from from this stuff okay well I'm glad that I've managed to give it to you if you work from the outwards in you won't get it correctly so for example 31 people like hot chocolate right that's what it tells me so all of the numbers in the hot chocolate Circle should add up to 31. so we're going to have a number here here here and here all four of those numbers should add up to 31. so starting from the out is basically impossible right I can't write 31 here because that's not true the total is 31. yeah good so you need to look at the the unions the interceptions okay so just be very aware of that try and work your way from the innermost to the outermost so the innermost is all three people sorry one person likes all three drinks 17 people like all three drinks there we go now let's try and work out how many people like tea and coffee so okay actually coffee and hot chocolate but not tea so they like coffee and hot chocolate so that would be this intersection over here is two okay then seven people like tea and hot hot chocolate but not coffee so they like tea and Hot and Hot Chocolate so that'd be seven over here so if I was like okay so this question is a bit tricky because they haven't actually given us some of the um information so sometimes they're quite nice and they'd give us tea and coffee as well but they haven't been very nice okay so we don't know this number and we can't work it out just yet right actually we can tea and coffee can work it out okay so again sometimes they give you directly what tea and coffee is uh like together but we can actually work it out pretty uh easily so third coffee so it says here so I also like to tick off the information as I use it right so I've used this I've used this I've used this so now all I'm looking at is these three pieces of information so again I like to start from the end and work my way out so let's look for the one that says and which is this piece of information here 21 people like tea and coffee right so what some students do is write 21 here it's not correct okay for tea and coffee we're looking at the intersection between these two circles which is both of these numbers here so 17 plus the number in this bit add up to 21. so in this case it's going to be 4. that's probably the first thing I've said that might be pretty tricky any questions about that so do you mind me you're okay perfect Lila and Evie are you good with that it's a bit tricky so tea and coffee is this intersection here that adds up to 21 okay perfect wonderful so now we've done all of the inner bits we have to now work out the outer bits uh there is one thing we can say every person likes at least one drink which means the number outside the circles is zero because everyone has one everyone's in at least one Circle good I'm glad that that was okay so far the next thing we realize is let's look at the number of people that like tea so 34 in total right so everyone in this circle here should add up to 31. okay so the number in here is going to be 31 minus 4 minus 17 minus 7. okay so 7 17 that's uh 24 28 so it's going to be 3. hang on if I 7 and 17 is 24. 28 yeah also it's 34 people my bad my bad so add-on three that should be a six my math was right I just chose the wrong number so the number in there is going to be equal to six showing you the um the way I'm working I was helping but again stop me the second you don't understand so we've used that piece of information now one more piece of information 31 people liked hot chocolate let's do the same thing 31 minus seven minus 17 minus two so 7 and 17 is 24 26 so that should be equal to 5. and there'll be an X in the middle of 17. uh in the middle of 17. as in the middle of this one or when I was calculating it so as in in this middle 17 here should there be an X or when I was calculating it just now hmm okay so while you tap that I'm just going to continue so in the last bit we need to work out how many people are in here ah good question Ed yeah so this counts as working out okay so if you look this is a five mark question if you manage to do the probability by hand as in writing out whatever that is you'd get four marks but this is all method marks so you get three marks for this Venn diagram a mark 4 of the probability calculation and another Mark for the right answer okay good question really good question you get marks for the Venn diagram okay because this does take a long time you get up to three marks for it so with the coffee so we use that information where Everyone likes one drink so if we do 50 take away all of the rest of these numbers so for we will get our final answer you can still get method marks so for example if I've read this as like 40 instead of 50. there's something called error carried forward something called ECF which you'll see in your mark schemes where if you make a mistake they can reward method marks for the following parts okay so you don't just lose all the marks if I read this as 40 though they wouldn't give any marks they might give some error carried forward marks but you'd lose a lot so um yeah so you know what I'm going to be lazy to do this on a calculator uh so 50 minus 4 minus 17 minus 7 minus 2 minus 6 minus 5. 9 perfect that makes my life easier cool so we've gained most of the marks already so we need to do now is work out the probability that they both like coffee if 50 people are chosen at random so the people number of people that like coffee are all of these people in the diagram here so we have nine plus four plus Seventeen plus two thirteen Thirty thirty two people probably said they both like coffee it's going to be 32 over 50. now they won't choose the same person again so we subtract one from the number of people that like coffee and one from the total number of people okay and once again should be a calculated paper question but if it's not it doesn't matter 992 you can do this by hand I know but I don't think you want to watch me just do like you know column method multiplication so 2450. and that is five marks long question not gonna lie but it is five marks so it's expected you can do this Venn diagram a lot faster than what I did bear in mind I spoke through everything I did so I'm willing to bet you guys can eventually write this pretty quickly any questions at all about that that's okay good yeah uh another question which is pretty pretty annoying it's another Venn diagram question but you know it's good to kind of practice apples bananas and oranges so let's just real quick blast this one at least would start a bit faster than this um sorry for stumbling over my words like I said I'm a bit sick so apples it's a nice I'm trying to make sure that my overlap is big enough that you guys can actually read what's happening this box okay so once again we start from the middle so it's roughly an order of when we're going to use it 11 people like all three so that makes our life slightly easier and again take off the piece of information 30 people like apples again I like to work my way out so I'm gonna skip that six people like apples and bananas but not oranges so apples and bananas is here not oranges so that should be six there right because again we're not including the people that like oranges so they shouldn't overlap Okay so 15 people like bananas and oranges so again that's in total so this should be four because again it doesn't specify that they don't like apples five people don't like any fruit so those people are gonna go straight outside the Venn diagram and all 25 people who like oranges like at least one other fruit so that means all three numbers here should add up to 25. 11 plus 4 is 15 10 uh so where does the four come from Lila okay so it tells me here 15 people like bananas and oranges okay now all the people that like bananas and oranges are in this intersection here those two parts so those two numbers should add up to 15. does that make sense Lila okay perfect uh does the 10 make sense I did hurry through that explanation just to get to Lila's question so cool sorry for that noise by the way I've muted it okay now we can start working our way out right so 33 people like apples is where that six is not quite it would be a and then the N symbol B and not C because you're saying it's or Nazi right so it'd be something similar to this so foreign come from okay so if you look here Jemima it says that 15 people like bananas and oranges so if we look at where bananas and oranges we're looking for this the overlap between bananas and oranges which can you see as this overlap here so the bit right in the middle and the bit over here those two numbers should add up to 15. so I knew that 11 plus something had to add to 15 that something had to be 4. does that make sense so I hope that makes sense okay so 33 people like apples so everything in this circle here should add up to 33. currently I have 10 plus 11 which is 21 plus 6 which is at 27. so to get to 33 I'll need another six I hope that makes sense okay now if you won you can actually write a zero in where this part is because again everyone that liked oranges liked other fruit as well okay so we there's zero in just the oranges so now everyone else needs to be in just the banana column column uh Circle so we're doing 50 minus 5 minus six minus six minus 11 minus ten and minus four and that will give us everyone that's left over which is in the banana um Circle so 50 minus five minus six minus six minus 11 minus 10 minus again you can do this in your head I'm being a bit lazy and that's our Venn diagram we did it a lot quicker but we can do it even faster this is here to the 50 people are chosen at random work at the probability they both like bananas so how many people in total like bananas well we have eight plus six plus eleven plus four okay so six plus four 10 21 29 people so you have 29 out of the total 50 multiplied by we subtract one because they're not going to pick the same person twice over 49. and let's just use a calculator again you can do this by hand and if you want me to show you how to do it I'm more than happy to um but I I think everyone here is okay with column multiplication 58 over 175. I think that's too bad any questions about that at all I think that's okay okay all right perfect so we only have 10 minutes left and I actually split probability in two okay probability can be one of those unseen super hard questions the ones where they have like algebra in it and ratios and all this other stuff we're going to do that next week there's no point of me doing it now because we only have 10 minutes and it gets a bit more intense but the stuff we've mainly covered today is probability trees and the Venn diagram kind of questions so those should hopefully be okay but next week is the hardest probability questions you'll find okay so I'm not looking forward to it but again if you get good at it it's going to be um yeah it is exciting but probability is really easy to lose marks on because again you can miss out on one combination and you get the wrong answer you'll get method marks don't worry about that but it is where pretty much all students will lose at least one mark okay so if you get like 98 in your test you'll know why it's probably a probability question but again only 10 minutes I don't I don't think it's fair for me to just shove this in now you guys have worked really hard so we can finish slightly early but next week will definitely go through the harder probability [Music] um thank you Lila I really appreciate it um honestly at the moment it's just a bit of a it just hurts to talk so everything else is pretty much fine um oh it's worth studying yeah but honestly so this is going to sound a bit cringy but probability is more of a mindset okay it's less equations and more thinking logically about what they're asking you to do what are the combinations what are the situations you're in Okay so yeah it's a bit more a bit more of a mindset than it is uh calculations okay yeah it's one of the tricky ones after probability I want to ask you guys what would you like to do next so vectors is probably the hard actually factors is one of the tricky ones as well so we can do vectors for sure I know it's in all three papers Evie and it's definitely likely to come up in at least two so that like the length of a line um we can do that we can do that yeah it's I think they just call it coordinate geometry that's just what they call it but that has like everything in it just everything like mid points um equations of lines perpendicular bisectors all that also includes like lengths of lines as well yeah do ratios yeah so ratios length of line yeah cool so I'll write those down but we will cover everything but some topics are pretty some people just don't care about them so no one really Tunes in so I don't bother doing them just yet I will say that soon I'm going to be doing topic summaries of the whole there are seven topics in GCSE maths I'm gonna do big topic summaries of all of them so you have like algebra numeracy ratios Etc yeah okay again I think vectors is pretty tricky but we can save that for a bit later this covers the next three lessons now so it's fine okay if that's everything I hope you guys enjoy your evening you're probably on half term at least I am as well uh so enjoy your half turn take a bit of a bit of a break okay don't work too hard and yeah very welcome but yeah definitely don't work too hard burnout is very real okay you can drive yourself a bit crazy I appreciate that thank you both for your kind words and thank you for tuning in as well it's always good to see you guys yeah then definitely take a break okay definitely if you get burned out you're going to do really badly in the exams even if you know what to do some time off okay I know you've probably got to set a ton of homework but yeah take a bit of a relaxing a relaxing break although the weather's terrible very welcome yeah perfect I will see you guys yeah so I'll see you guys next week um again enjoy your half time you've definitely earned it and yeah we'll cover the harder parts of probability and then we've probably gone to length of lines or ratios and things like that yeah see you guys later