A Level Maths Made Easy - Probability Statistics

So today we're looking at probability however as you can imagine in statistics there's a bunch of different styles of probability that we're going to come on to this is specifically going to talk about generic probability and vend diagrams now both of those you should have covered in gcsc maths however of course there are a few extra pieces the generic probability is going to be virtually the same as GCS maths but the vend diagrams have a couple of extra pieces added in so we will go through them one by one nice and slowly but just so you're aware this isn't the end or B or for probability we have a bunch of distributions that we have to do as well now let's get started so we have the vend diagram below shows three events a b and c and it says write down two of the events that are mutually exclusive so first of all we should probably uh discuss what mutually exclusive means so mutually exclusive simply means that they both cannot happen at the same time so just put a box around here just so you have the definition cannot happen at the same time at the same time okay and that's really really important so what would that look like on a ven diagram well what it means is if I just kind of describe what a ven diagram shows you first we have all of our events drawn as circles outside of the circles but inside ins the rectangle is for example in this case it's the letter R is the probability that none of these events happen okay so here is the probability that no events happen okay if there is an overlap it means that both of those events happen at the same time so the number here 0.25 is the probability that just a happens it's not the probability that a happens it's the probability that that only a happens because there is a slight difference this number in the middle however is an overlap between A and B that is the probability that A and B both happen so for example it could be something like flipping a coin and rolling a dice a could be the probability when you roll a dice you get a six and B could be the probability that when you a coin you get a heads so the problem are getting heads and a six in this case is 0.15 if there is no overlap so for example the circle a and the Circle C do not overlap at all in this particular V diagram then they can't happen together A and C are mutually exclusive so an example of that is again let's go back to flipping a coin a could be the probability you get heads and B and C could be a probability of getting Tails now of course you can't get a heads and a taals at the same time they are mutually exclusive so that is an example of a mutually exclusive event so the answer for part A write down two of the events that are mutually exclusive you just look for the two circles that do not overlap which in this case are A and C so hopefully that makes sense just as a recap in the overlaps is the probability that both of these events happen so in this case 0.15 is a probability that A and B happen all of the numbers inside a will add up to the probability that a happens so the probability of a happening is 0.25 + 0.15 which is of course 04 okay so probability of a happening is 0.4 in this particular case but that 0.25 is the probability that only a nothing else happens for the next part it says events A and B are independent of each other and the probability of getting C which means in this case p+ Q is equal to 0.3 so we have to find out the values of p q and of course r as well so what's the easiest thing to start with it's definitely the uh part P okay and what we're going to do is we're going to use this thing here events A and B are independent of one another so what does it mean when two events are independent well in words it simply means that if two events are independent of each other it means that one event occurring does not change the probability of the other event happening so let's think of an example let's think of an example of two events that are independent again let's go back to our normal kind of situation let's say I'm flipping a coin and I'm looking at the probability of getting heads and I'm rolling a dice and it's a probability of getting a six now no matter what the outcome of flipping the coin it will not change the probability of rolling a six right they're independent they don't affect one another what's an example of two depend dependent of variables well imagine this situation what is the probability of me taking an umbrella to work well that depends on the probability that it's going to rain that day there you go those two events are are linked the higher the probability that it's going to rain the higher the probability that I'm going to take an umbrella to work those two events are dependent not independent whereas flipping a coin and rolling a dice those two things are completely independent because the probabilities do not affect each other one event happening does not change the outcome of the other event so what does that mean in terms of actual useful maths now well if two events are independent and this is a test that you can do then that means the probability of a multiplied by the probability of B will equal the probability of A and B remember this thing here just means and so this is actually a test for if two events are independent or not if you multiply their probabilities it should give you the probability of the overlap so what does this mean well the probability of a as we discussed it's not 0.25 it's 0.25 add 0.15 so that is 0.4 and what about the probability of B well the probability of B is going to be going to be this plus this plus this remember all of the numbers inside the circle so for now I'm actually just going to leave it as probability of B and that should equal the probability of A and B which is simply just the overlap 0.15 so now the probability of B is just going to be 0.15 over 0.4 and what this is going to do is give me what all three of of these things should add up to which is really useful because then I can work out what p is so let's just whip out our calculators and do 0.15 oops divided by 0.4 which gives me 38 which is 0.375 so in other words 0.1 + 0.15 + p so once again we can use our calculators to just work out what p is so it's going to be 0.375 so 0.375 subtract 0.15 and subtract 0.1 whoops which is 1/8 which is 0.125 and that gives us what p is nice and easy there you go so we've already got some marks now what about Q well this one's actually a bit easier they tell me that the probability of C so i p+ q should equal 0.3 and again that comes from this line here the probability of C is 0.3 so p + Q is 0.3 but I know what p is is 0.125 so Q is just equal 0.3 - P which is equal to 0.3 - uh 0.125 which should give you 0.175 okay so let's write out one more time just to make it super clear again for the examiner sake or your teacher sake I would recommend just putting them in nice little boxes like so cool now what about R well R is an interesting thing because there's one other Factor at V diagrams we haven't quite mentioned yet inside this rectangle everything should add up to one why is that well this box represents the entire possibility space it is everything that can happen in this situation now if the numbers didn't add up to one then that means there is a chance that nothing that something else happens other than nothing and these events basically what that means is you haven't included all the possibilities okay so for example if I'm rolling a dice and I include the numbers one to five it obviously won't add up to one because proba getting six hasn't been included so in short every digit inside this should add up to one so let's see what it adds up to so we have the 0.25 from over here so 0.25 + 0.15 + 0.1 so I've just read these three numbers P was 0.125 and Q was 0.175 and it adds up to 0.8 so R is just going to equal 1us 0.8 which is 0.2 so like so hopefully that wasn't too new I think the two things that were new were the mutually exclusive and the independent things but other than that is a very typical but difficult GCS vend diagrams question so here we have another ven diagram because it is slightly different it does look different to the other one so it shows three events a b and c and again we have missing numbers so you can probably imagine what we're going to be doing but part eight says write down two of the events that are mutually exclusive and they're actually two correct answers if you remember mutually exclusive means that they both cannot happen at the same time and it's represented in a vend diagram as having no overlap so which two circles don't have overlap well A and C have no overlap B and C also don't have overlap so the correct answer is a and C or you can put B and C don't put both both but one of those is completely fine for Part B it says simply find the value of p q and R but if you have a look at the information they give us they say the probability of a is4 and the probability of a or b is 7 so start with that point4 so the probability of a being 0.4 means both of the numbers in this circle should add up to4 so in other words 0.2 plus p is 0.4 and you can probably already imagine what p is going to be then 0.2 nice and easy what about the other bit the probability of A or B happening is 0.7 so what does a or b look like on a v diagram well I told you in the first part that A and B is the overlap but what about A or B A or B looks like this on the ven diagram so I'm going to use a highlighter it's everything inside either one of these circles okay so everything inside that figure of eight looking thing in other words it's 0.2 + p + Q so 0.2 + p + Q equal 0.7 and we already know that P is 0.2 so that's 0.4 + Q = 0 .7 and that finally means that Q is equal to 0.3 just from subtracting 0.4 from both sides so so far pretty good right three marks and it's only been 2 minutes and 30 seconds not bad and then it says for the last part States giving a reason whether or not the events A and B are statistically independent and if you go back to part A the first part sorry independent means that if you multiply the two probabilities of the events happening it should give you the overlap so let's try that out we're going to do the probability of a multi by the probability of B and see if we get whatever p is so we're going to see if we get 0.2 and I'm going to write that over there so what's the probability of a well keep in mind it's not just 0.2 it's 0.2 and P p is 0.2 as well so in other words 0.4 so 0.4 multiplied by and what is B well B is Q + P so it's going to be 0.2 + 0.3 which is 0.5 right because both of those added together and that gives me that gives me uh that should give you 0.2 as well let me delete this 0.2 so in other words words PA a multili PB does equal A and B therefore they are independent a neat little trick to uh kind of solve those types of questions now this should give you warm homely feelings from GCSE maths uh assuming that you did it somewhat recently anyway uh because this is pretty much a gcsc math question but we we've learned how to use ven diagrams but now we should learn how to draw them ourselves so we have Rim asks 50 people which sport they watch they can choose from football Golf and hockey and they give you a just a boatload of information that to be honest doesn't make sense to a human being right it's very hard to kind of look at these numbers and understand exactly what's going on which is exactly why we have ven diagrams so we have to draw a vend diagram for this information and if you notice the first thing it says all three Sports and then there's also overlap between football golf golf hockey between all the sports too so what you want to do is draw three overlapping circles which I always kind of struggle with uh at least on paper but thankfully computers kind of help me out here so I want to draw draw your circles relatively large or ovals in my case so that way there is a big enough kind of consistency to it so you can actually have a big overlap so I like to draw mine pretty damn large them being the same size doesn't matter because it doesn't you know we don't really care about kind of scale in terms of vend diagrams uh this is pretty reasonable to be honest this is actually not bad one and then please don't forget to draw The Big Box around the whole thing and now what you want to do is label your circles now you don't need to write football Golf and hockey you can use letters they you know the examiners are relatively intelligent and they'll be able to know what you mean assuming you don't use some strange kind of notation for it and you might notice something in our previous Vin diagrams we had decimals right because they represented probabilities but ven diagrams can also be used to just represent whole numbers because we can work out probabilities from whole numbers and proportions so what do we do I always start from the very middle the overlap between all three and then I work my way out so for example it would be the overlap between all three then the overlap between any two and then the uh kind of just leftover ones and then finally the number on the outside remember there has to be a number on the outside even if it's zero there still has to be a number otherwise you'll lose a mark so write something it might be zero it could be any other number assuming it's positive anyway okay but you need to put a number there so luckily in this case they actually wrote them in order that we're going to be doing them in pretty useful so five watch all three Sports there's no harm no foul here just stick a five in the middle now with football and golf it says eight people so you might be tempted to write this you know the overlap between football and golf eight but this isn't quite correct because remember it's saying eight people watch football and golf but there's five people that watch football Golf and hockey that includes people that watch football and golf right so you actually have to do eight minus five there's three people that only watch football and golf so just as a reminder there is eight people in total that watch football and golf right because that includes the five people that watch all three hopefully that makes sense but there are three people that watch just football and golf and you use that same logic with golf and hockey so that is this one over here seven people watch golf and hockey well five people already accounted for so that means there's two more people people keep in mind that this number in the middle could be zero so they might say five people watch all three Sports and five people watch golf and hockey you would just write a zero there there should be no blank spaces always put the zero if you need to so I actually like to tick these off as I go along so we've done these three next we're doing football and hockey and it says nine people so there's already five from here which means there's going to be another four here 31 people watch football so what what that means is everything inside the circle which is f so all of these ones add up to 31 how many do I have already I have three 7 12 I have 12 already and there's 31 people that watch football so all we need to do is do 31 minus the ones we already have so you could do if you wanted to just do 31- 4 + 5 + 3 either way you'll get the same answer of 19 and that goes in the football Circle and let's s it off 13 people watch golf well how many do we have already we have three 8 10 13 - 10 is three so that's three people that only watch golf no other sports so let's tick that off 17 watch hockey well we have four + 5 is 9 plus 2 is 11 so that means there's six more people that just watch hockey now you might be tempted to leave it though cuz we have no other information but we have more information we have that there are 50 people that were asked that is information so what we want to do is add up all these numbers in the ven diagram and see what we get if we get 50 we'll put a zero here if we get 49 we'll put a one out here because those are people that don't watch football golf or hockey so let's add all these numbers up and see what we get 19 + 3 + 4 + 5 whoops so that's uh football plus three 2 and six for the other plus 3 + 2 + 6 gives me 42 I want 50 which means there are eight people who decided they didn't watch any of those Sports hopefully that makes some sense for Part B it says that two people are selected at random find the probability that they both watch football so what you want to do is work out how many people watch football in total so that is going to be 19 + 3 + 5 + 4 31 people watch football in total so I'm going to write that number 31 down and how many people in total are there there are 50 people in total so there's a 31 in 50 chance of picking one person who picks foot who likes to watch football but we want to pick two people in a row so you might be tempted to write the exact same thing again but this isn't quite right because you can't pick the same person twice right should make sense let's say if I had a group of five people to think of maybe members of your family if I'm picking two of them I can't pick the same person twice right because those automatically get removed so we actually subtract one from the total number of people so it's now 49 and one from the number of people who like to watch football that's your final answer the reason why we times is because we're picking one person who likes football and another person who likes football and when you say and in maths it means times so let's work out what that's going to be again we can use a calculator no harm easy peasy lemon squeezy over 49 93 over 245 which is indeed the cor correct correct answer perfect so five marks in total not bad now the way it works for drawing vend diagrams in terms of marking is a bit interesting you actually start off with three marks that sounds pretty good right however you lose a mark for every number that is either missing or incorrect so you start at three and if you miss out two numbers let's say you miss out the number on the outside that's two marks now and let's say you get another one wrong that's one mark So effectively you get three mistakes okay so just keep that mind when you're doing ven diagram questions so we're going to go through some kind of Rapid Fire questions on drawing basically ven diagrams and how these different things will look based on V diam so a bunch of different scenarios so pretty quick questions but it's a nice change of pace from The Fairly long ones we've been doing up till now so for part A again two events are independent prob if a is at point4 and B is.3 find the probability of A and B well if they're independent then the probability of a times the probability of B will equal the probability of A and B so simply put all we need to do is multiply them so 0.4 * 0.3 which you can use a calculator but it is just 0.12 and then it says just draw a vent diagram to show this information so once again you're going to have two events so it only have two circles this time makes it slightly easier for people who are incapable of drawing like me good good Lord that's disgusting and put a box around it of course and label it A and B so don't forget the labels don't forget the box of course and then the probability of a is 0.4 if you remember the probability of both was 0.12 so I kind of made that a bit small 0.12 0.4 minus 0.12 gives me 0.28 because remember we still have to have numbers inside the circles themselves and 0.3 - 0.12 would be 0.18 so now we have those three numbers and are we done no we need to have a number outside the bracket remember so what do we do well they need to add up to one in this case because it's probability so we're going to do one minus then all of these numbers 0.28 - 0.12 - 0.18 and that gives us 0.42 it should go outside side of the both of the circles so please please please remember to label the circles and also to have a number outside both circles but inside the rectangle don't draw outside the rectangle either so another rapid fire question so two events A and B are mutually exclusive probab of A is4 and B is 0.3 so write down the probab of A and B well that is going to be zero why because they're mutually exclusive that literally means that they can't both happen so the probity of A and B happening is zero okay perfect so again it could be something like flipping a coin and getting heads and tails that's not possible so drawing a v diagram for this information so what is that going to look like well it's going to look slightly different it's going to look like B and C do here there should be no overlap so we're going to draw two circles and we're going to draw them separate from one another A and B we're going to draw a big old rectangle around them perfect and what was the probility of a it was 0.4 and B was 0.3 and once again we need a number outside of the both of the circle so 1- 0.4 - 0.3 gives me 0.3 that's the number that's outside okay nice and easy nice and quick I I hope one more rapid fire question and then we're moving on to two of the hardest exam questions I found keep in mind and this is something I haven't mentioned in the previous videos which I probably should have um just for those of you wondering because I did get a few questions are they all of the past paper questions these are not all the past paper questions because and for a very good reason is at the end of the course at least the the as stuff I'm going to do all of the as past papers and at the end of the A2 course I'm going to go through all of the A2 well the overall a level past papers so don't want to end up doing all of those questions before we actually get on to doing the past papers so I take a selection of some of the really hard ones and then I'm going to go through the whole paper slightly later on and that way we actually have some practice uh exam papers to do as opposed to just having the exam questions already done for us so let's uh go through this so it gives me probative a b and the probative draw a v diagram so all we need is we need two overlapping circles this time I'm going to make the overlap a lot bigger call that a call that b start from the middle like we did in the previous questions so 0.4 probability of a is 0.6 so 0.6 minus 0.4 gives me 0.2 and 0.5 - 0.4 gives me 0.1 don't forget the big old rectangle because that means we'll have a number on the outside of both circles and how do we find it out we add these up 0.2 plus 0 .4 is 0.6 + 0.1 is 0.7 so 1 - 0.7 gives me 0.3 and that's it for the rapid fire questions let's get on to some pretty dang hard past paper questions this was actually a really tricky question for a lot of students so if we look here it says two bags A and B each contain balls which are either red yellow or green a contains four red three yellow and N green and B contains five red three yellow and one green bow is selected at random from bag a placed into B and then a a ball is selected from bag B and placed into Bag a and it says the probability that bag a now contains an equal number of red yellow and green balls as P given that p is greater than zero so essentially given that this is actually possible find the possible values of n and p so let's go through it first things first it says the possible values of n and p now we can't really work out P without knowing what n is so let's write down all of the cases for n and let's have a think so essentially we're going to take one bag from a and add one of any color to a okay how can we make them all the same well what we could do because the number of Reds is bigger than the number of yellows we could take one red and then replace it with something now we don't want to replace it with a yellow because then we'd have four yellows and three Reds so we could add one to the green that will only work if n is equal to two so that's one possible case right because if n equals 2 we take a red that makes it three Reds and we add a green which will make them all the same color okay so in other words we take a red from an A so red from a and green from B that is one possible case let's work out the probability of that happening what's the probability of taking a red from a if we have two green balls well how many red balls are there there are four and how many are there in total so 4 + 3 is 7 plus 2 is nine cool now if we take this red ball and add it to bag B we now have six red balls right so how many green balls do we have we have one and how many balls do we have in total we have 6 + 3 is 9 + 1 is 10 so that gives me 4 over 90 which is the same as 2 over 45 so that's one value of P so what's another value well let's think of another use case what we could do is make yellow into red and what I mean by that is make all of the values equal to four so if you've noticed what I'm doing is I'm looking at the Val the numbers that I actually have and I'm making them the same as each other so in the first case I made the four into three by taking one away from the red in this case I'm going to make the three into four by adding one to yellow and the only way we can do that is by taking one away from Green and we want the number of green balls to end up as four as well to make all three numbers the same which which means green has to start off as five cuz 5 minus 1 would give me the four that I need so we have what a proba so we're taking one green from Red a green from a and we're going to take a yellow from B so it's prob of taking a green from a well we have five green balls in this case so there going to be five out of how many well it's going to be 5 + 3 is 8 + 4 is 12 so 5 over 12 multipli by so again we've added one to this green ball here so we have two now we're taking a yellow so let's do three over 5 + 3 is 8 + 2 is 10 so 5 * 3 is 15 over 120 we could divide them both by three we could also divide them both by 15 actually which gives me 1 eight and that is my second set of values that is five marks which I think is very generous however of course as you can see why this was a tricky question it was a tricky question because you have to imagine these different scenarios and the best kind of advice I can give you is a practice as many questions as possible so become familiar with it but B understand what probability is okay it says here that they're an equal number of red yellow and green B we need to make these numbers same but how can we do that well we're lucky because the green balls we can change so realistically this question is actually asking us how can I make the red and yellow balls the same because like I said the green balls we change so I can write that the green balls can be any number of balls so I can ignore the green balls at the beginning I want to make the number of red and yellows the same how can I do that well I can take one away from the red or I can add one to the yellow and then I can work out what the number of green balls is so hopefully that makes a bit more sense