[Music] which is statistical analysis um this is quite a big topic as part of the Year 11 course um I use a lot of this in year 12 um as like the basis of your year 12 statistics um so good thing to have like a good grasp on what it is um and how you can apply all these sorts of different things um so what start with I guess just looking at like what actually is probability um so your probability is just the likelihood that an event occurs um whenever you're thinking about probability you're thinking of it either as a decimal a fraction or percentage so your probability will always lie between zero and one um if you think of like that as a scale with zero being like it's impossible for something to happen so thinking about like if you've got a six-sided dice it's impossible you're going to roll a seven um and then one being like it's guaranteed certain something's going to happen so again with that dice like the probability of rolling a number between 1 and six is going to be um definite so yeah everything lies between Z and one um when you're thinking of probability you thinking of um like decimals you're not ever thinking of a whole number um so the possible events um within probability are called the same sample space so any possible outcome that you could get for a given um like decision you're making or like um Choice you're making is the sample space so again really common example you'll see is Dice so your sample space for your dice is 1 2 3 4 five and six if you've got a six-sided dice um or you know if you're flipping a coin your sample space is heads or tails those the only two options that you can get um and so yeah with this question pretty simple question but just to see that everyone's um got a bit of an understanding of what probability is I give you guys 20 seconds for those of you watching it live if you want to pop an answer to this question in the chat um and for those of you who are recording it give it a go as well and we'll just see how we go okay cool well hopefully you guys all got that um the next thing we'll look at is I guess the basis for like the mathematical applications of probability um for I guess when you're sort of having to calculate things rather than you know when you're thinking about a dice like the probability of rolling a one is one and six um and that's sort of something that use for most people you just sort of know you don't really have to calculate but for things that we're more um are looking at actually calculating we can think of the probability of an event where probability is p always always always denoted by P um of an event e um occurring is given by P of is equal to the number of favorable outcomes over the total number of possible outcomes so when we're looking at this e that will often change depending on what you're trying to find the probability of um You probably wouldn't denote it as e um so like for example looking at a dice and we're thinking okay what's the probability of rolling a one you would typically write P of one um and replace that e with a one because that's the event we're looking at um and then writing it like that just helps whoever's marking your um answers or your work really clearly see like what you're trying to find so when we're looking at the number of favorable outcomes we're thinking about what outcomes satisfy our event so with a dice um if we're thinking about probability of rolling a one there's only one favorable outcome because only one of the dice one side of the dice has one um you'd start to get a little bit more if we're thinking about like what's the probability of rolling an even number and then all of a sudden you've got three you know you've got two four and six um and then the number of possible outcomes is our sample space so what is like what how many um outcomes could happen so again um back with this dice example um you've got six possible outcomes comes so six would become your denominator and if we looking at even numbers three would be our numerator and to get the probability is one on two once we've simplified three on six down um so yeah that's just like the basic formula for looking at probability um this is also super common notation um which is really important that you understand so P of e is just the probability of an event occurring and P of e Dash is the probability that the event doesn't occur so it's every possibility but the event not occurring I mean the event actually occurring so um again we'll just look at the dice super simple example for you guys to understand but if we're looking at the probability that um we roll a one our P of e is one on six because there's um one favorable outcome of six possible outcomes and then the probability of not rolling a one is five on six because we've got five favorable outcomes because any other number We Roll other than one is going to be favorable CU we're looking at the probability of not one over six possible outcomes okay um and because we're looking at probability on a scale the probability that an event occurs and the event not occurring will always equal one right because in essence you have to have one of the other if you're rolling a dice it either rolls on one or it doesn't and if it rolls on one we're here if it doesn't we're here so um we know in total our probabilities will equal one because it's certain that we'll either get get it or we won't um when you start doing some more probability questions um and like hard probability questions I guess it becomes sometimes a lot easier to find the probability of the complimentary event so to find the probability that something doesn't occur um where this often comes up which you may have um or may not have seen these sorts of questions before they sort of come up a little bit more in your 12 but um it'll be like you know um Jack has six marbles um he's got two red and four blue find the probability that at least one Blue Marble is chosen um if he picks out like three marbles or something um and when you think of that sort of question that it becomes a lot easier to find the probability that there is not a Blue Marble chosen than there is but don't stress too much about those sorts of questions um they're the sort of thing that you get used to once you start seeing them a little bit more but yeah anyway um the next little thing that we're going to look at is um relative frequency so the relative frequency of an event is a measure of how many times you actually get the event happening if you perform the experiment repeatedly um which is different from theoretical probability so your theoretical probability is um what you think and what you can predict will occur like based on like the rules of chance right so if we're thinking about um flipping a coin it's theoretically it's 50/50 that you're going to get heads or tails okay so the theoretical probability of getting a head is 50% so this is a little bit different from relative frequency because relative frequency is looking at um how many times does it actually occur so if we flipped a coin 10 times um and we end up getting heads seven out of the 10 times our relative frequency is 7 out of 10 like yes theoretically we could get heads or tails and it's roughly even relative frequency is looking at how many times you actually get it if you perform the experiment um most of the time when you guys are doing statistics questions in your exams um particularly as you are moving into your HSC year um in about a term um your theoretical probability is what you deal with a lot more um and if you are looking at relative frequency they give it to you in the question um so it's not too tricky um to sort of use your relative frequency as long as you understand that if they say in the question the relative frequency of something you understand what that means and how that's different from the theoretical probability but that's how that most commonly will come up for you guys um so then we've also got um the probability scale so like I was saying before all of our probabilities lie on a scale of 0 to one where zero is something is absolutely impossible and one is something that is absolutely certain okay so um everything is between this scale you can't have something that's more than certain and you can't have something that's less than impossible so really common thing to see when you're looking at 50/50 is heads or tails on a coin um impossible to roll a seven on a dice um if you are looking at a regular um dice with like one to six on it um something that's unlikely winning the lottery um not to say that it's impossible because it's not because you could win but it's just not that likely um considering everything that has to go in favor to make that happen um things that are then the sun will rise in the morning um like something like if I roll a dice you will get a number that pops up on one of the faces um so yeah just make sure when you're thinking about probability you are thinking of things in terms of a scale um whether that for you you find it easier to look at fractions decimals um percentages are technically on a scale because 100% lies um at one um and 0% is zero so whatever you find easiest for you to look at um to be able to compare probabilities do that you want to make everything as easy as you possibly can for yourself um but yeah you want to be able to look at the scale and think about where different things lie so you can compare them okay so the next thing we're going to look at is multistage events so the complement of event a is a dash okay so like we were saying um if you see probability with um a letter with a dash on the top all that means is the probability that the event doesn't occur Okay so we've been over that a little bit so I'm not going to um focus on that for too long um then we've got the intersection of two events A and B which is denoted as a this little thing that looks like um a rainbow and B which is the probability that both events occur um you won't see this to come in with like your basic probability questions where it's like dice coins um like picking a number sort of thing you start to see intersection when you've got two events okay so think about um if I flip a coin and roll a dice what's the probability that I get heads and I roll an even number okay so if heads of an A and rolling an even numers meant B the probability that both of those happen is equal to a and b okay so this just pretty much means and and so that's the intersection um we'll talk about this a little bit later when we look at V diagrams but you will see um commonly these notation when you are looking at vend diagrams because the probability of A and B is represented by the crossover of the two circles um on a v diagram but we'll talk about that a little bit later when we look at them um and then we've got third the union of two events A and B is denoted as a union B where either event occurs or both of the events occur okay so again if a is rolling heads and B is an even um getting an even number the probability of a union B is the probability that a occurs the probability that b occurs or the probability that A and B occur um this formula is actually in your formula sheet so don't stress too much about having to remember it but this is the formula for finding the union of two events so we've got the probability of a union B is equal to the probability of a plus the probability of B and then you do actually minus the probability of A and B Because if we don't minus it we're going to get double UPS okay because the probability of a occurring already takes into account the times where A and B both occur and then we've got the probability of B takes into account where b and a occur so we need to minus A and B so we don't end up with um any double ups and get a probability higher than we actually have um super easy way to check this if you have forgotten to minus if your number if your probability pops up and it's greater than one you've probably forgotten to subtract um this from your equation but yeah this is on your formula sheet so don't stress about remembering it but just important to know and be familiar with how to use it um and to know what each of these three terms means okay so then we've got tree diagrams so for the next little bit we'll be looking at some um visual representations of data so we've got tree diagrams are used when there are two or three choices such as a coin being tossed where the choices are either a head or a tail um you don't want to be doing um tree diagrams when you've got like six different choices because otherwise you start getting like really complicated really messy crowded diagrams um so you only do it when there are few choices available um but you can do it when there's like lots and lots and lots of steps um so like first step second step step third step fourth step so doesn't really matter how many steps there are um and I mean you could use it if there are a lot of choices it just gets a lot Messier when there are choices so it's simpler to do when there's only two or three so we've got this example here and the true diagrams already been um given to you so the example is two coins are toss what is the probability of getting at least one head so I'll give you guys all about 30 seconds or so um if you want to type your answer in the chat or if you're I'm watching the recording um just give it a go as well so I'll give you about 30 seconds to do that excuse me sorry um okay anyway um hopefully everyone's given that a go and you've popped your answers in the chat um so yeah awesome okay so then we've got conditional probability now this is probably the part of probability where people get the most confused um and where things start going a bit Haywire um when we're thinking about probability okay so we now know how to find the probability of event B occurring so how do we um find the probability that event a has occurred given that event B has already happened so essentially that's what conditional probability is it's like what's the probability of something happening given that something else has already happened and that's what this little line in the middle of a and b stands for it's the what's the probability of a given B has already happened this formula again is on your formula sheet so don't stress about um remembering it but um what it means is the probability that a has HP is happened um given B has already happened equals the probability of A and B occurring all divided by the probability of B okay and that's how you find um the probability of a given B has happened um so like I was saying before this A and B is represented by our intersection we've got the probability of B and then the probability of a as well on either side of our Circle okay so that's all you really need to know about for conditional probability is how to use this um this formula actually comes in as well when your trying um the question might ask you to find the probability of event b um and in the question they'll give you um probability of a given B and the probability of um A and B and then ask you to find the probability of b um and a lot of people get sort of stuck on that question because they don't really know where to go from um when they've only given you like these two pieces of data um but as long as you can remember this formula and remember how to use it you can use those things and then just like manipulate this expression move things around to find the prob ability of b as well okay so just some more terminology um about statistics so you've got two sort of events um that commonly come up in questions so we'll talk independent events first so this blue bit um your independent events are events that occur without being affected by anything else um if you think about like flipping a head on a coin um the probability that you flip ahead is the same whether or not you're only flipping a coin whether you're flipping a coin and rolling a dice whether um you're flipping a coin and picking a number the probability of flipping that coin doesn't change if you're doing something else with it so um the way that you can work that out if something's independent is if the probability of a given B is equal to the probability of a so essentially what this is saying is that even though B has already happened that makes no difference to whether a or not is going to happen or not okay and then we've got mutually exclusive events so these are events where the occurrence of one event will exclude the occurrence of the other okay so um where P of A and B is equal to zero so like an example for this is if the probability of rolling a one on a dice is a and the probability of rolling a b um a two on a dice is B and you're only rolling the dice once you you can't get a and b right so you're not going to get one and two when you R the dice if you're only rolling it once so essentially um with these two terms it's in the name independent events means the events are independent of one another that that don't affect each other and then again in the name mutually exclusive they're exclusive if one happens the other cannot um so if you ever get stuck thinking about what's an independent event and what's a mutually exclusive event think about it in terms of their names okay so just got a little bit more on statistics before we move on to functions um alend diagrams super common um representation of data super common thing to come up in terms of questions um or even super common things that sometimes your statistics questions will ask you to draw um so super important to be familiar with them um so our vend diagrams allow us to see the relationship or overlap between two or more events okay now something that a lot of people um miss or forget to do is when you're drawing a vent diagram in an exam or even you know like when you're practicing for exams and stuff you have to include a box around it for the probability that neither event occurs Okay so we've got a dash which we know is the probability that a doesn't occur and B Dash which is going to be the probability that b doesn't occur and we've got a and b so this is just the probability that neither event a nor event B is going to happen Okay so even if it's zero on the outside even if it's guaranteed that either a or or B or A and B will occur you do need to include this box on the outside and write zero um somewhere in that box okay same thing if it's you've got more than two events again this is the probability that a does not occur and B does not occur and C does not occur okay just want to make it super clear that you understand that nothing will happen outside of this range or that there are things that are going to happen that aren't a b or c okay so with your vend diagrams most of you would have seen ven diagrams um before um so we'll just quickly look over this um so this um probability that a occurs probability that b occurs A and B where the two circles intersect um pretty simple same thing if you've got more than two circles um well there's no crossover it's just a just b or just C between the crossover of two circles you've got the probability of two events occurring and where all three circles overlap you've got the probability that all three events occur um you could theoretically have vend diagrams that have you know five six seven circles that'd be super hard to draw and you guys will never have to draw them um You probably wouldn't get um a vend diagram with more than three circles so don't stress but just be super familiar with what each of the intersections between the circles mean okay so just a little bit more in the v diagrams um Union of a or b is all of it because we know that Union Means A or B or both um we've already talked about a um and B is just that part in the Middle where the two circles intersect um the probability of not a so the complement of a or a dash depending on um how it's given to you in the question is everything but the a circle um the probability of a is just a and not B so you don't want to think about um that part in the middle and then if we've got mutually exclusive events it's neither you're going to have nothing shaded okay so just give you like a little bit more time just to read through that if you need it okay and then the last little bit of Statistics that we're going to look at today is what these two terms mean and if you can take away anything from statistics today take away this bit here because for most people this is probably probably where you will go wrong um when you're doing your statistics questions particularly in year 11 when they're not too tricky just yet um so in the question if it says and it means multiply if it says or you want to add your probabilities okay so if you want to have the probability of getting two heads over two toses so think of that as heads and heads you want to multiply if you want to get a head or a tail on the first toss you want to add them okay so if you're running notes or anything if you're going to take anything away please take this away it'll make your life so much easier when you start to do um more probability questions um and it'll be super good for you guys in your 12 if you can already get a grasp on what this means so and is multiply or mean add okay so multiply your probabilities or add your probabilities um this rule obviously is for um like multi-stage events so you'd only do this if like you've got two events occurring um because otherwise if you only have single stage event you only have one probability so you can't multiply or add anything but yeah please just take this away and you'll be so set um for statistics okay so that is statistics all done um for today's lecture if you guys