A Level Maths Mastery A* Probability Distributions Statistics - Edexcel Maths How To Answer

so one of my students actually sent in this question a little while ago and this is one of those AAR students that I was talking about and this is about probability distributions which is technically a year 12 topic but it does come up quite a bit in year 13 as in in the actual final exam and a lot of students have trouble with them and I don't blame them in the textbook they give you really really easy examples then you get exam questions like this which are extremely difficult so if we're aiming for those a stars and again this is part of the a Lev mats Mastery series then we definitely need to be able to do questions like this hopefully I'll be able to break this down and show you where you can at least gain most of the marks if not all of them but yeah we'll jump straight into it but before we get started once again if you've ever found any of these videos useful do consider subscribing just to help me out and of course if you need tuition or anything like that there's my website and my email in the description but let's get started with this now first of all it wouldn't be an A- level maths question or at least an applied question if there wasn't an entire paragraph that you have to read but here's the rundown this person is playing a game using a ball cup and a spinner and the idea is is that when she throws the ball if she gets it into the cup she also spins the spinner in order to get her score it's a very weird game I have no idea how it works but whatever so the variable X is the spinner okay so I would probably in the exam label this as here's my spinner stuff okay ABC and D probabilities that we don't know so you can probably guess what the question is going to be but to play the game you spin the spinner to get a value of x so either 20 50 80 and 100 a very weird spinner and that gives you a probability you then stand that distance from the cup and throw the ball into the cup fantastic right and the event s represents the event that tism successfully throws the ball into the cup so again the idea of winning here is after you've spun the spinner you stand that distance away from the cup then you have to throw the ball into the cup and we've got the probability that she actually succeeds right the probability that she gets the ball into the cup given that she scored whatever on the spinner like she spun whatever number that is so for example X could be 20 50 80 or 100 now this probability does make sense if you look at it K is a constant and it divides by the distance she stands away from the cup as you can imagine the further away you stand from the cup the lower your probability which is exactly what this is saying here the probability that she gets the ball into the cup is equal to some constant divided by the distance so the further away she stands the lower the probability so she's 20 away that would be K divid by 20 but if she's 100 away that's K divided 100 it's a smaller probability that makes complete sense to me and again going through this when you're reading a question you're not just reading it passively you are trying to make sense of the situation so what I just did there is exactly what I do in the exam try and understand what this calculation what this equation is telling me then it says the probability of s and this happening should be the same whatever value of x is applied from the spinner is sorry is obtained from the spinner so the probability that she gets the ball into the cup and she gets whatever probability or whatever distance from the Cy stands should be the same again this is a model so don't have to worry too much about it but using the model show that Cal 8 5B now I've often said to a lot of my students that I have no idea how to how to finish questions and this is true because maths is not about knowing exactly what to do for every step it's to take it step by step so what would I do in this case well given that it's probability there's a few things that I can write down the first thing is that the all of these probabilities added up should give me one so a plus b plus C plus d = 1 now it's telling me to show that c is equal to 8 FS of B I can't really use this equation for that because there's a there's B there's C and there's D so that isn't useful yet but I'm willing to bet money that we'll use it at some point so this was still a useful step to do so what else have they told me well they've told me that this should be the same whatever value of x is obtained from the spinner so given that we want B and we want C because we want an equation with b and c in it what if I work out the probability of getting B and of getting C and also getting it into the cup so hopefully that kind of makes a little bit of sense as to to what I'm saying now the probability of s and getting that value should be the same whatever value of x is obtained from the spinner okay so in other words if I start with b the probability or getting s and X equaling 50 why 50 because again we looking for B at the beginning right that should equal B the probability of actually getting 50 on the spinner times K overx because that's the probability that she actually gets the ball into the cup after she's um gotten that probability after she's gotten that value of x so by K over 50 so hopefully that logically makes sense right don't forget when you see and in probability it means multiply so we've got the probability of X equaling 50 which is B and S so we have to times it by the probability of getting s the probability of getting the ball in given that the distance is whatever is K over 50 now we're going to repeat the process for C so s and X equaling 80 this time and we get C * K over 80 now what does the rest of that sentence say it says it should be the same in other words these two probabilities should be the same so let's write that out we have B * K 50 = c * K 80 and you might be able to see where we're going to go next I can cancel K from both sides and then I'm just going to times both sides by 80 because that gives me B * 8/ 50 = c again we can cancel that down so in other words C is 8 fths of B how fun right the next thing it says is find the probability distribution of X so in other words we need to find what a b c and d r now this is going to seem like a very long process but once you get one value the rest of the values come very trivially but if I take this and I substitute it into this original equation we have because remember we wanted an equ we now want an equation with a b c and d in it if I take this expression well actually it's an equation but whatever or an identity sub it into here what do I get well I'm going to write this in red for Part B we get a + b + 8 over 5 b plus d = 1 can you see the hint they've given you in the question now part A is a hint for Part B because what we've done is we've effectively removed one of the variables so instead of having a b c and d we only have a b and d so what if we repeated the process for A and D to get everything in terms of B and if everything's in terms of B when we know what B is we can sub it in to find everything else because we know that c is 8 fths of B and we'll know that a is something to do with b and d is something to do with b as well so uh just in preparation I'm going to open up my calculator because I don't want to work out fractions so we're going to repeat the exact same process well first of I'm actually going to simplify this so B plus a over 5B if you use a calculator should give you 13 over 5B plus d = 1 and I'm going to do the same thing I did here but I'm going to work it out for a and for D so let's start with a first the probability of s and X equaling 20 is equal to a * k/ 20 and again that should be equal to B so let's work that out we have K over 20 a equal K over 50 B once again we can cancel the K's and we're going to times both sides by 20 so a is going to be 20 over 50 B in other words it's 2 fths of B we can sub this back into here I'm going to do that later once I've actually worked out d as well just to do everything in one step so I don't repeat myself do the exact same thing for D probability of s and X equaling 100 is equal to D * K over 100 and once again that should be equal to B * k 50 so K Over 100d equals K over 50b so again we can cancel the K's I'm going to times both sides by 100 and hopefully you can see that D is just equal to 2B we're going to put that back into here as well so honestly spotting to do that is the hard bit but now it's just linear algebra this is an equation that year sevens and eights can solve it's not too bad but yeah anyways uh we sub that in we have a is equal to 2 fths of B so we're going to have 2 fths of B plus 13 fths of b + 2 b = 1 so 2 fths + 3 um 13 over 5 that's 15 over 5 so it's 3B plus 2 B is 5 b = 1 and all of that to get B is equal to 0.2 now we know that we can work out D we can work out a and we can work out C wonderful um yeah fantastic right so let's do it so we've worked out what B is that means D is going to equal twice of B so it's equal to 0.4 so those two are done a is equal to 2 fths of B which would be I'm pretty sure I know but I 0.08 right 0.08 and finally C I don't really actually no I do have another color that's 8 fths of B so let's do 8 over 5 multip by 0.2 0.32 and that's your probability distribution and if you notice these four numbers add up to one because it's a probability distribution and that's it for Part B that is uh quite intense I have to admit but hopefully you can see where you meant to go with that again the actual number work the actual algebra the math is not too bad but it's knowing what to do uh part C is another question which is not easy whatsoever so nav tries a large number of times to throw the ball into the cup from a distance of 100 cm he successfully gets the ball in the cup 30% of the time State giving a reason why tsm's model of this game is not suitable to describe uh now playing for all values of X so 100 cm by the way is just this distance here so it's D so we could work out the probability of getting D which is essentially what we have to do by the way and yeah so we'll start with that so they've said that the probability that nav gets the ball in given it's 100 is equal to 0.3 if we use this definition which we have back say back in the day back here there it should be D * K 100 so we have D * K over 100 equal 0.3 now I know that D is equal to 0.4 so let's do that we have 0.4 * K over 100 = 0.3 times both sides by 100 we get 0.4k = 30 K is equal to 30 0.4 which not sure actually let's have a look I'm aware that I can do the maths that gives us 75 now that in and of itself isn't illegal it's not illegal at all but if k equals 75 bearing in mind they've told me it's it's not suitable so there's something wrong with this answer but I have to figure out what it is so how do I do that well if I assume K is 75 and I use this equation here we have KX so what I'm going to do is I'm going to check for all of these if k x works well K over 100 gives me 0.75 that's a legitimate probability um 80 over 75 50 over 75 sorry 75 over 80 also gives me a number less than one so that's legit but have a look what happens when X is smaller than K well then my probability is bigger than one which can't work so if this is true if then X cannot be 20 or 50 because otherwise we're going to get a probability bigger than one so when x = 50 then our probability becomes 75 k/ x which is 50 which is bigger than 1 which is not possible and this is also true when X is equal to 20 which is also bigger than one which again is not possible in fact the only values of x that you can have is X has to be bigger than or technically equal to 75 for the model to work because again the probab is bigger than one now the most common thing I I get asked when I go through this is but that's only one Mark now here's the thing in mark schemes they do just give you the answer they just say oh well because the probability is unrealistic in fact the mark scheme for this says K is 75 therefore not valid for X is 50 and X is 20 that's a one Mark answer the reason why I've gone through it in this exact way is to show you what they mean and this is the type of thing you need to test if it's a probability distribution what things can go wrong well if you get a negative probability it's wrong if you get a number a probability bigger than one it's wrong if the numbers or the probabilties add up to a number bigger than one it's wrong these are things you need to think about is what would make this break it will break if a probabil is bigger than one or less than zero in this case we got ones that are bigger than one so sorry I had to sneeze but hopefully that makes a bit more sense when you come across questions like this the number one skill you need to have for probability distributions is being able to interpret what this means so what I did at the beginning of this video when I explained what this means that's what you need to get good at otherwise the probility questions are going to be extremely hard for you but anyways if you do want to see more of these kind of aiming for AAR or a questions I do have like about 400 billion of them so I'm more than happy to do more and more for all sorts of topics but let me know what topics you want in the comments and also if you even want more AOA Star level content without further Ado I'll let you get on with your revision have a great day