alright in this video we're going to review probability for the AP statistics test probability is one of those topics and a lot of kids don't feel very comfortable with there is a lot to it but for the most part I think the EP sad says does keep it fairly simple so I'm just going to try to focus on these five things and there is gonna be a separate video for the binomial model which does incorporate a lot of probability as well so number one we're going to talk about or verse and probability which definitely will come up multiple choice maybe even on a free response we're gonna look a two way table probability which is actually the easiest type of probability most kids can solve that very easily with just some common sense then we're going to take a brief look at conditional probability with tree diagrams then we're going to talk about multiplication of independent events and then finally probability with normal model so here we go start right into it so first let's make sure you understand what probability is probability is the long-run relative frequency of an event it's how often an event occurs in the long run after many many many trials remember this is what the law of large numbers says the true probability of any event will only reveal itself after a large number of trials for example we've everybody knows that the chance of flipping a coin in getting a head is 50% but what some people don't understand is what that 50% means this means that in the long run after many many tosses would we expect 50% to be heads and the short run anything is possible if you toss the coin 10 times you certainly might not get 50/50 right you might not get five tails and five heads I mean that certainly is the most likely outcome but that doesn't mean it will happen because only ten tosses you can get six heads seven heads one head two heads three heads I mean who knows so this goes back to the idea that bigger samples very less so if I were to toss a coin 1,000 times only then after many many tosses when I definitely expect to see something closer to 5050 now am I still going to see 500 tails and 500 heads probably not but it's much more likely in a larger sample than it is in a sample of homeland so the true probability of an event is how often that events will reveal itself in the long run after many many many trials technically many many many means infinite because only then will you start to see that true 50% number pop up alright let's talk about two events a and B now the formula I have up here in green is the addition rule this is probably one of the most important rules for you to remember and it is on the AP SATs for machine when it comes to probability the probability of A or B is equal to the probability of a plus the probability of B minus the probability of a and B now let's explain what we mean here the probability of A or B means that if event a happens we're happy or if event B happens we're happy but what a lot of kids must understand is that the probability of A or B also includes the probability of a and B so the idea of a or B that actually means a or B or even both and that's what the formula tells us so a lot of kids get confused by that because they see at the end of the form that you actually subtract away the probability of a and B well the reason why you're subtracting is in a way is so that that doesn't get counted twice so the best way to explain this was with a Venn diagram and I hope a teacher at some point showed you a Venn diagram if I have two events on the left a on the right B and in the middle is the overlap that's the a and B that's where both events are happening at the same time so if I simply try to find a or B I would take the whole a circle plus the whole B circle but if I stop there the problem is that overlap got counted twice it got counted as part of the a circle and it got counters for the B circle so the reason why we have to subtract away a and B is to avoid it from being counted twice so we're still including a and B in the or formula but we just don't want it to count twice because when you reference the probability of a some of that could be only a not B and some of that could be a in me same thing can be said for the probability of B some of that could be B only not a but some of it could all we be aired a so again now you can actually see that with the Venn diagram here the overlap is they a and B and if you just add up a and B you're gonna double count that which is why you need to subtract it away all right so this brings up a good opportunity for me to mention mutually exclusive mutually exclusive means that a and B cannot happen at the same time now if that's true that would look like this there's a there's be no overlap they cannot happen at the same time that means that the probability of a and B is a zero so if you're mutually exclusive and only if your meets exclusive is the probability of A or B equal to the probability of a plus the probability of B you don't have to worry about the overlap because there is none so I guess you would just be subtracting a zero which means nothing now the next topic is a and B are independent events if a and B are independent events this means that the outcome of a does not impact the outcome of B now if that is true and only if that is true that means that the probability of a and B is simply a times B but this is a tricky formula because a lot of kids blindly use this you can only use that formula if you know for a fact that a and B are independent of each other and that's a little bit tricky to understand so my recommendation is all we multiply a times B if you're thinking about and only do so if you are directly told in the problem that the two events are independent of each other otherwise you have to think conditionally which we're gonna get to in a little bit now one other important statement to make I've actually seen this pop up on the AP stats test is if two events are mutually exclusive they cannot be independent this is a one-way sentence it doesn't work any other way around it only works if it's worded exactly like this if two events are mutually exclusive they cannot be independent and that makes sense to us because if you have two events that are mutually exclusive like this picture shows right here two events and B they have no overlap the moment that a occurs B is impossible to occur because they cannot happen at the same time which is what music exclusive means so that means that the moment a occurs it certainly affects the probability of B and E's the example I use in class is getting an A and getting a B on a test you cannot get an A and to be on a test you can only get one or the other so the moment you get an a on a test the probability of you getting a B drops to zero which means a affected B that is not independent remember independent means they do not affect each other so this is a pretty important statement to understand if two events are mutually exclusive they cannot be independent now if two events are not mutually exclusive that means they have overlap well sometimes you're independent sometimes you're not it all depends so that's why the only statement we know for a fact is that the two events are mutually exclusive they cannot be independent all right let's put that formula to work all right the probability of event a is 0.2 424 percent chance event a happens and the probability of event B happens is 31 percent what is the probability of A or B all right now we know the formula for a or B is the probability of a plus the probability B minus the probability of a and B all right well let's answer this question but I actually cannot answer the question unless I'm told something so the first question is okay let's find the answer to this if they're mutually exclusive all right if an be a mutually exclusive what does that tell me remember if a and B are mutually exclusive that means a and B is empty impossible they cannot occur at the same time so if that's true to find the probability of A or B it's actually quite easy I'm just going to take the probability of a point to four plus the probability B 0.31 and I'm just going to add them together what about the back part well there is no back part because remember the probability a and B is zero so I just add these together and I get point five five so this is a very common AP question I see this a lot they'll ask this type of scenario and they'll say what happens if you exclusive all you have to remember is that that back part a and B is zero all right what if they're independent okay well I'm trying to find the probability of A or B if they're independent well I'm still going to use the formula the probability of a is point two four plus the probability B is point three one - now if you're independent what do I know about the probability of a and B well if you're independent only then can I multiplies the probability of a and B would be point two four times point three one now point to four times point three one is point zero seven four four okay that means that the final answer for the probability of A or B is 0.24 plus point 3 1 - point O seven four four or point four seven five six so I hope that makes sense now a couple kids say well what if I don't know you're independent we're out to be honest if you don't know you're independent then you really can't answer this question you do need to know one of these two things if you're mutually exclusive the a and B is zero so all you got to do is add a and B if you're independent well then they're you know for a fact that the probability of a and B is a times B all right let's move on now let's talk about conditional probability here this is where the outcome of one event does impact the outcome of a second event now we write this with a line so right here we say what is the probability of a given the back part is always the given given that B has already occurred now this is where we are not independent because this means that B affects a so if the probability of a changes based on the fact that or that B occurs that is not independent so the formula to find conditional probabilities a really easy formula on top you find the probability of a and B and on the bottom you divide this by the probability of the condition B so it's a really easy formula also given two on the AP SATs for machine but I have to something a lot of kids say okay I know how to find the probability to a and B just multiply a times B right only if you're independent so please do not multiply the probability of a times B unless you know they're independent that's only true if you're independent so please keep that in mind otherwise you have to hopefully be told or somehow find those probabilities and we're going to explore that right now with this to a table I love seeing two-way tables on the a piece at sets and so should you because they're really easy to work with especially when it comes to probability so here's a two way table that looked at 2500 people and it asked them their level of education college grad high school but not all college graduate and then not high school graduate and then we asked where do you get your news from newspaper local TV cable TV the internet or you don't get any news all right and then we have all of our numbers here in this two-way table all right first question is what is the probability a college graduate is selected all right I want to choose a college graduate that's the only thing mentioned in this problem so how many college graduates are there well if I look at the college graduate column there are 693 college graduates divided by 2500 total people to pick from there's my answer 693 divided by 2500 and you get point two seven seven two so that's a twenty seven point seven two percent chance a college graduate gets picked awesome what is the probability that a college graduate is picked or someone who gets their news from the internet okay I see that word or so I got to use that or formula right the probability of A or B college graduate is a or getting your news from the internet is B all right so I'm gonna follow formula okay I'm gonna find the probability of college grad all right the probability a college graduate is picked easy we just got done fighting that 693 about of 2500 or I'm gonna add the probability that I get my news from the Internet okay the probability I get my news from the Internet 687 people get their news from the internet / 2500 but now I have to subtract away that and now don't we use any fancy formula to find and just look at the table how many people are both college graduates and get their news from the internet that would be these 245 people right here now I'm gonna subtract away the and those are the people that are both college graduates and get their news from the internet now I want to emphasize this again I'm not getting rid of those people but if you think about it those 245 people were counted among the 693 college graduates but those 245 people are also counted among the 687 internet news source people well those 245 people definitely deserve to be counted but not twice that's why I'm subtracting them away so 693 plus 687 693 687 subtract away the 245 I get 1 1 3 5 out of 2,500 remember I got to come to Dominator there so I'm saying the numerators divide by 2500 I get point four five four so again all you got to do is use a little bit of common sense there and think your way through it all right next question what is the probability somebody who gets their news from the internet and is a college graduate is picked so what is the probability that somebody gets their news from the internet and their college graduate Oh where'd he talked about that that is a very specific 245 people that's it easy where do you know that it's said and don't multiply right I'm need to multiply I got this table in front of it shows me everything I need I'm looking at this table and I clearly see 245 people that get their news from the internet and our College crash that's it 245 out of 2,500 don't make it harder than it is 0.09 8 all right next question what is the probability that college a that X article type of their what is the probability that a college grad is picked given he or she gets the news alright now I see that word given that's conditional because I'm told that they give their news from the internet that's a given so I'm trying to find the probability I'm try to find the probability that they are at college graduate but I'm given that they get their news from the internet see this is using that conditional formula I know something about this person preview prior to picking them I know that they get their news from the internet now I actually don't need a fancy former for this I just need a little bit of common sense the fact that I'm given that they get their news from the internet means I'm only allowed to look at these 687 people who get their news from the internet that goes in the denominator because again if they're given that they're good from the news from the internet everybody else I need to ignore because they don't get the news from there I'm only allowed to focus on the 687 people who get their news from the internet now of those people 245 of them get there are upside 2040 five of them are college graduates so again this is using the formula on top goes both college graduate and get their news from the internet that's the turn forty five people we've already mentioned a couple of times now the denominator is just getting your news from the internet and that is the 687 people that get their news from the internet and that is conditional probability so conditional probability your denominator changes because of that condition so now q45 out of 687 missed point three five six six all right final question is being a college grad independent of getting news from the internet all right this is actually a really easy question to answer now okay the probability of a given B right that's conditional probability this means that there is an event B that is happening prior to event a now if this equals the probability of a what this tells me is that I'm independent because this tells me that B did not impact a on the right side all I care about is a and if that is equal to caring about a after B has occurred and if I could show that those two numbers are equal then what means is that the probability I'm sorry what that means is that a and B are truly independent of each other because this means that B did not impact a let me briefly go over this one more time let's think about this real quick if you're independent if you're independent what do we know about a and B we know that the probability of a time's the probability of B is a and B only if you're independent well if that's true in this formula the probabilities of B are going to cancel and we get the probability of a given B equals the probability of a thus verifying that only if you're independent is the probability of a equal to the probability of a given B so if I want to show that being a college graduate is independent getting news from the internet I need to show that the probability of being a college graduate given I get my news from the internet is exactly the same as being the process the probability of getting a college graduate see what I want to show you is that getting your news from the internet doesn't affect being a college graduate well let's find out all right what's the probability of your college graduate that was actually the very first question we answered it was all the way back up here being a college graduate 693 college graduates out of 2,500 point two seven seven two so I'm just gonna fill that in right there point two seven seven two that means there's a twenty seven percent chance I get somebody who's a college graduate now what's the probability I get a college graduate given you get your news to the Internet well I already found that as well that was 0.35 six six again the denominator was getting your news from the internet six hundred eighty seven people get the news the internet the numerator is both news from the internet and college grad 245 now these two numbers are not equal which means we are not independent and this actually makes sense there's a 27% chance that somebody is a college graduate however if you sure news from the internet the chance of you being a college graduate goes up to about 36% this means that getting your news from the internet does impact or there is an association with you being a college graduate you are more likely to be a college graduate if you get your news from the internet so this shows that the two events are not independent if they would have been equal it would show that they are independent I have seen questions like this based on a two-way table come up quite frequently on the AP stats exam so please be ready for them all right now let's talk about conditional probability again but this time through a Chi diagram this is another very common ap stats type question studies showed that two percent of dogs get heartworm a new test is used to help diagnose dogs with heartworm if the dog has heartworm it will give a positive test result ninety-five percent of time see that if there is given right that's that's a big conditional thing on the condition that a dog has heartworm will they get a positive test result ninety-five percent of the time if a dog does not have heartworm again conditional if the dog does not have heartworm it will give a negative result eighty-five percent of time so what's best is to create a little tree diagram here so let's have a little dog here right little dog yeah and there's this tail okay now there is a two percent chance that this dog has heartworm I'm sorry that's unfortunate ninety-eight percent chance they don't I just use a little bit of common sense there hope you're okay with that now the first statement said if you have heartworm so I'm gonna follow the heartworm branch if you have heartworm there is a 95 percent chance of a positive test result which means there's a five percent chance of a negative test result now that would obviously be bad a negative test result would tell you that your dog does not have heartworm when they really do now the other statement was if a dog does not have heartworm he will give a negative result 85 percent of the time so if your dog does not have heartworm so I'm following the knot heartworm there is an 85% chance of a negative test result which means there's a 15% chance of a positive test result now that would also be bad because that means you tell that you're saying your dog has heartworm but they really don't so what we see here is we see what's called a false positive it's a positive that's wrong and we see a false negative that's a negative that's wrong all right now what could I ask you now that I have this beautiful tree diagram drawn up well one thing I could ask you say what's the probability a dog has a positive test result well there's two ways you can be have a positive test result heartworm and positive know heartworm and positive okay all I got to do is multiply heartworm and positive test result or no heartworm and positive test result those are the two branches that lead to a positive test result and again I'm allowed to multiply because it's conditional this 95% was on the condition that you had heartworm 2% chance this 15% was on the condition that she did not have heartworm ninety-eight percent chance so all I got to do is multiply that point oh two times point nine five plus twenty nine eight times point one five and I get point one six six so that is the probability that a dog has a positive test result all right another question that's very common is this given a positive result what is the probability the dog has heartworm now this is an important question because notice you could get a positive result and not have heartworm or you could get a positive result and have hard work so we're saying aren't Givens this is conditional given that you have a positive result what's the probability that you have heart work so I'm trying to find the probability that a dog has heartworm given a positive test result all right well let's use the formula for conditional probability on top goes both heartworm and positive result heartworm point oh two and a positive result based on having that heartworm point oh two times point nine five that's the top the bottom goes only the condition a positive test result what we already calculated that a positive test result is 0.166 now remember that was two things because there's two ways you can get a positive test result heartworm and positive no heartworm and positive so those are the two things that would go on the bottom but we already calculated that so now I got to do is a little bit of math here 0.02 times 29 five divided by the point 166 and I get point one one four five and if you want to confirm on your calculator you can so that means that if I get a positive test result for my dog there's only about eleven percent chance that my dog actually has heartworm now a lot of you say well wait a minute it's a positive test result doesn't that mean they have heartworm yes but remember very few dogs I'm sorry I'm sorry most dogs don't have heart work which is why that fake positive is probably what's happening here now obviously they're still 11 mr. Chancellor dog really has heartworm but the point is since heartworm is a pretty rare thing most dogs don't have it anyway so if you get that positive result don't panic your dog might not have heartworm so there's eleven point four five percent chance that you actually have heartworm if you get that positive test result but this is a classic tree diagram problem where you see these if statements so please make sure you understand that that leads to a tree diagram make sure you not a draw one and use one to help you out all right next up here I want to talk about multiplying for multiple events now the point I want to make here is that when you're talking about multiple things happening okay now I'm not talking about both I'm not talking about both here I don't mean that you are both a boy and you have blonde hair okay I'm talking about doing something and then something else and then something else and then something else so I'm doing things multiple times right multiple events you just multiply but all you have to do is make sure you think about conditional probabilities as long as you think about it there's no special formula here other than to multiply so anytime you want to find probability of one thing happening after another just multiply as as you consider if the outcome of the first event might have changed the outcome the second event and we're going to see that right here all right so a room has four boys and three girls so seven students in a room four boys three girls what is the probability that four kids are picked and they're all boys so all I got to do is walk through this all right I need four kids to be picked what's the probability that the first one is a boy well there's four boys out of seven kids to pick from but I now need another boy see I'm gonna multiply because I need another boy well one boys already been picked so there's only three boys left but one kid has already been picked so there's only six kids left all I did was use my brain to think conditionally because the outcome of this first boy affected the outcome of the second boy now we need a third boy well I've already picked two so there's only two left and there's only five kids left you get the idea and then now I need a fourth boy well there's only one boy left out of four kids remaining in the room and now I'm done all I got to do is multiply that out now you can multiply that out on your own it's not that hard to do on a calculator to get your final answer all right I'm going to move on the next question here the teacher wants to select a girl build a probability model for how many kids have to be picked to get a girl all right so I want to build a probability model for how many kids can be picked to get a girl so the moment I get a girl I'm done all right so X is gonna be my random variable for how many kids need to be picked all right let's kind of just walk through this a probability model has the probability of each outcome beneath it all right so could I get a girl with only picking one student absolutely what has to happen to get a girl with one student well I need to pick a girl there's three girls to pick from out of seven boom I'm done could it take two kids to get a girl well for it to take two kids to get a girl that means that first one how to be a boy so that'd be four over seven to get that boy and then I have three girls to pick from out of six kids remaining to get that girl on the second kit so again to have the second you'd be a girl that means the first one was a boy and then the second one was a girl could it take three kids absolutely that would be boy boy and then a girl so four out of seven for that first boy three out of six for that second boy and then finally on that third kid I get the girl three girls out of five kids remaining hopefully you're getting the point here could it take four kids to get a boy hops are to get a girl sorry four kids yeah that would be boy boy boy and then I get that girl all right so once again four out of seven for that first boy three out of six for that second boy two out of five for that third boy and then finally from the fourth kid I get the girl and there's three out of four kids that are girls left in the room all right could it go to a fifth kid absolutely that would be boy boy boy boy which at that point I've exhausted all of my boys because I only had four of them and then I must get that girl so that's four out of seven for the first boy three out of six for the second boy two out of five for the third boy one out of four for the last boy and then there's no there's no boys left which means I got a three out of three chance which is a hundred percent chance to get a girl now that's it I will never have to go to six kids because that's impossible right why would I need to go to six kids there's only four boys so if I pick all four boys I'm guaranteed 100% chance three out of three to get a girl in that fifth boy so you guys can multiply those out to actually get the probabilities but there are the different outcomes for I need to get a girl I can get a girl in the very first kid second kid third kid fourth kid or I might have to wait all the way to that fifth kid and again that's how you build a probability model and all I had to do is think conditionally I didn't any special formulas I just had to use my brain now there are some events that are independent I mean that they don't affect each other for example if you are spinning a I'm sorry if you're gonna throw a right the probability of any outcome of a die is 1/6 well the outcome for one die has no impact on another die so if I said what's the probability you get four fives in a row so four fives in a row okay well I got 1/6 chance to get a five I got a 1/6 chance to get another five I got a 1/6 chance they get another five and I got a 1/6 chance to get another five so there's my four fives in a row and I didn't have to change the probability because they're independent that die doesn't lose a five after it rolls on it's not like a classroom where you're picking kids so those are independent so here's my five here's another five here's another five here's another five there's my four straight fives 1/6 chance for each multiplied together you got your answer so the point is when you're doing multiple things always multiply just be careful to think conditionally if need be all right the last thing I want to leave you with here is that many questions that ask for a probability directly relate to the normal model recall the problem must say the normal model or that the normal distribution is being used and then you will need a mean and a standard deviation then all you got to do is use normalcdf on your calculator never be afraid of a problem relating to the normal model that asks for probability because normal CDF can be your best friend all you got to know is to use these scores now if the problem deals with a sample size of n don't forget that the standard deviation for a sample mean or a sample proportion will change based on the appropriate formula police check out the videos for normal bottles and the video for sampling distributions for more examples of these types of problems but the point I'm trying to make is that the probability can be used in these settings as well but those are pretty easy and I've actually already covered them in those other videos so check them out please also be aware that there is binomial probability right there's gonna be a separate video for that as well so if you're interested in binomial probability which is really great and very common ap questions as well please check out that video for that alright that's it guys