Learn Precalculus and College Algebra in this course for beginners. Over these next few Precalculus videos you will be 100% prepared to learn Calculus. I have found that most people think Calculus is difficult because they don't know everything required to easily learn it. This tutorial series will solve that.
In this video I'll teach how to :
► Solve Linear Equations
► Find the Slope of a Line
► Graph Complex Equations
► Convert to Standard Form
► Find Least Common Multiple
► Create Equations from a Slope & Point
► Find Distance between Points
► Find Greatest Common Factor
► Solve Inequalities
► Graph Inequalities
► And more
#precalculus #collegealgebra #calculus
In this Precalculus Course's video, we will learn the Addition rule for probability. Venn diagrams and the addition rule for probability
In this Precalculus Course's video, we will learn the Probability with playing cards and Venn diagrams. Probability of compound events. The Addition Rule. Common Core Standard 457 S-CP.7
In this Precalculus Course's video, we will learn the Statistical significance of experiment
In this Precalculus Course's video, we will learn the Using a list of random number to calculate an experimental probability.
In this Precalculus Course's video, we will learn how to Using a list of random numbers to simulate multiple trials of an experiment.
In this Precalculus Course's video, we will learn the Experimental versus theoretical probability simulation.
What we're going to do in this video is explore how experimental probability should get closer and closer to theoretical probability as we conduct more and more experiments or as we conduct more and more trials. This is often referred to as The Law of Large Numbers. If we only have a few experiments, it's very possible that our experimental probability could be different than our theoretical probability or even very different. But as we have many many more experiments, thousands, millions, billions of experiments, the probability that the experimental and the theoretical probabilities are very different, goes down dramatically. But let's get an intuitive sense for it. This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not. Right now, it says that we have a 50% probability of getting heads. We can make it unfair by changing this but I'll stick with the 50% probability. If we wanna show that on this graph here, we can plot it. And what this says is at a time, how many tosses do we wanna take. So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads and if you wanted to, say what's your experimental probability after that one flip, you'd say well, with only one experiment, I got one heads so it looks like 100% were heads. But in the second flip, it looks like it was a tails. Because now the proportion that was heads after two flips was 50%. But then the third flip, it looks like it was tails again because now only one out of three or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now the fifth flip, it looks like we got another heads and so now have three out of five or 60% being heads. And so, the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability diverges from the real probability. And this even continues all the way until we get to our ninth or tenth tosses. But what happens if we do way more tosses. So now I'm gonna do another, well let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here and you can see, wow look at this, there is a big run getting a lot of heads right over here, and then it looks like there's actually a run of getting a bunch of tails right over here, then a little run of heads, tails, and then another run of heads and notice, even after 215 tosses, our experimental probability has still, is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time. And what we're seeing in real-time here should be The Law of Large Numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going, we're now at almost 800 tosses. And you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%, it's getting close now, we're at 50.6%. And I could just keep tossing, this is 1100, we're gonna approach 1200 or 1300 flips right over here. But as you can see, as we get many many many more flips, it was actually valuable to see even after 200 flips, that there was a difference in the proportion between what we got from the experiment, and what you would theoretically expect. But as we get to many many more flips, now we're at 1,210, we're getting pretty close to 50% of them turning out heads. But we could keep tossing it more and more and more and what we'll see is, as we get larger and larger and larger, it is likely that we're gonna get closer and closer and closer to 50%. It's not to say that it's impossible that we diverge again, but the likelihood of diverging gets lower and lower and lower the more tosses, the more experiments you make.
Compare expected probabilities to what really happens when we run experiments.
- Let's say that you've got a bag, and in that bag you put a bunch of marbles. So, let's say you put 50 of these magenta marbles. So one, two, three, four, five, six, seven, I'm not gonna draw all of them but you get the general idea. There are going to be 50 magenta marbles, and there's also going to be 50 blue marbles. And what you do is, you have these 100 marbles in there, half of them magenta, half of them blue. And before picking a marble out, and you're gonna be blindfolded when you pick a marble out, you shake the bag really good to, so you think, mix them up a little bit. And so, if you were to say theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta? I feel the need to write magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically there's 100 equally likely possibilities, there's 100 marbles in the bag. And 50 of them involve picking a magenta. So, 50 out of 100, when this is the same thing as a 1/2 probability. So you could say, well, "Theoretically, "there is a 1/2 probability, I just did the math." If you say these are 100 equally likely possibilities, 50 of them are picking magenta. Now let's say you actually start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again. And so, let's say that after every time you put your hand in the bag and you take something out of the bag, and you observe what it is, we're gonna call that an experiment. So, after 10 experiments, let's say that you you have picked out seven magenta and three blue. So, is this strange that out of the first 10 experiments, you haven't picked out exactly half of them being magenta, you've picked out seven magenta, and then the other three were blue. Well no, this is definitely a reasonable thing. If the true probability of picking out a magenta is 1/2, it's definitely possible that you could still pick out seven magenta, that just happened to be what your fingers touched. And this isn't a lot of experiments, it's completely reasonable that out of 10, yeah, you could have, later on in statistics we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1/2. Instead of having five magenta, it's completely reasonable to have seven magenta. So, this really wouldn't cause me a lot of pause. I still wouldn't question what I did here when I calculated this theoretical probability. But let's say you have a lot of time on your hands. And let's say after 10,000 trials here, after 10,000 experiments, and remember the experiment; you're sticking your hand in the bag without looking, your fingers kind of feeling around, picks out a marble, and you observe the marble and you record what you found. And so, let's say after 10,000 experiments, you get 7,000 magenta. Actually I'm gonna do slightly different numbers, so let me make it even more extreme. Let's say you get 8,000 magenta and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different. And now you have a large number of trials right over here, not just 10. 10 is completely reasonable that, hey you know, I got seven magenta and three blue instead of five and five, but now you've done 10,000. You would've expected if this was the true probability, you would've expected that half of these would've been magenta, only 5,000 magenta and 5,000 blue, but you got 8,000 magenta. Now, this is within the realm of possibility if the true probability of picking a magenta is 1/2, but it's very unlikely that you would've gotten this result with this many experiments, this many trials if the true probability was 1/2. Here your experimental probability is showing, look, out of 10,000 trials, experimental probability here is you had 10,000 trials, or 10,000 experiments I guess you could say. And and in 8,000 of them, you got a magenta marble. And so, this is going to be 80%, or 8/10. So, there seems to be a difference here. The reason why I would take this more seriously is that you had a lot of trials here, you did this 10,000 times. If the true probability was one half, it's very low likelihood that you would've gotten this many magenta. So, when you think about it you're like, "What's going on here, what are "possible explanations for this?" This, I wouldn't have fretted about, after 10 experiments, not a big deal. But after 10,000, this would have caused me pause. Well, why would this happen, I mixed up the bag every time. And there're some different possibilities; maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settled to the bottom, and you're more likely to pick a magenta marble. Maybe the blue marbles have a slightly different texture to them, in which case, maybe they slip out of your hands, or they're less likely to be gripped on, and so you're more likely to pick a magenta. So, I don't know the explanation, I don't know what's going on in that bag, but if I thought theoretically that the probability should be 1/2, because half of the marbles are magenta, but I'm seeing through my experiments that 80% of what I'm picking out, especially if I did 10,000 of them, if I did this 10,000 times, well, this is going to cause me some pause. I would really start to think about whether it's truly equally likely for me to pick out a red, a magenta versus a blue. Something else must be going on.
Sal summarizes the set operations that he has discussed in the previous videos.
Let's now use our understanding of some of the operations on sets to get some blood flowing to our brains. So I've defined some sets here. And just to make things interesting, I haven't only put numbers in these sets. I've even put some colors and some little yellow stars here. And what I want you to figure out is what would this set be, this crazy thing that involves relative complements, intersections, unions, absolute complements. So I encourage you to pause it and try to figure out what this set would be. Well, let's give it a shot. And the key here is to really break it down, work on the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. And then it should hopefully make a little bit of sense. So a good place to start might be to try to figure out what is the relative complement of C in B. Or another way of thinking about it is what is B minus C? What is B if you take out all the stuff with C in it? So let me write this down. The relative complement of C and B or you could call this B minus C. This is all the stuff in B with all the stuff in C taken out of it. So let's think about what this would be. B has a 0. Does C have a 0? No, so we don't have to take out the 0. B has a 17. Does C have a 17? Yes, it does. So we take out the 17. B has a 3, but c has a 3. So we take that out. B has a Blue. C does not have a blue. So we leave the Blue in. So let me write down-- we leave the blue in. And then B has a gold star. C also has a gold star. So we take the gold star out. So the relative complement of C in B is just the set of 0 and this Blue written in blue. So let me write this down. Let me write that down. Now, it gets interesting. We're going to take the absolute complement, the absolute complement of that. So let me write this down. So B-- the absolute complement of this business is going to be-- let me write it this-- the set of all things in our universe that are neither a 0 or a-- and I'll write it in blue-- or a Blue. That's the only way I could describe it right now. I haven't really defined the universe well. We already see that our universe definitely contains some integers, it contains colors, it contains some stars. So this is all I can really say. This is the set of all things in the universe that are neither a 0 or a Blue. So fair enough. So we so far we figured out all of this stuff. Let me box this off. So that is that right over there. And now we want to find the intersection. We need to find the intersection of A and this business. Let me write that down. So it's going to be A intersected with the relative complement of C and B and the absolute complement of that. So this is going to be the intersection of the set A and the set of all things in the universe that are neither a 0 or a Blue. So it's essentially the things that satisfy both of these that has to be in set A and it has to be in the set of all things in the universe that are neither a 0 or a blue. So let's think about what this is. So the number 3 is in set A and it's in the set of all things in the universe that are neither a 0 or Blue. So let's throw a 3 in there. The number 7, it's in A and it's in the set of all things in the universe that are neither a 0 or a Blue. So let's put a 7 there. Negative 5 also meets that constraint. A 0 does not meet that constraint. A 0 is in A but it's not in the set of all things in the universe that are neither a 0 or blue because it is a 0. So we're not going to throw 0 in there. And then a 13 is in A and it's in the set of all things in the universe that are neither a 0 or a Blue. So we could throw a 13 in there. So we've simplified things a good bit. This whole crazy business, all of this crazy business, has simplified to this set right over here. Now we want to find the relative complement of this business in A. So let me pick another color here. So we want to find the relative complement of this business in A. And I'll just write out the set-- 3, 7, negative 5, 13. Actually, let me write out both of them just so that we can really visualize them both right over here. So A is this. It is 3, 7, negative 5, 0, and 13. And I could write the relative complement sign. Or actually, let me just write m-- well let me write relative complement. I was going to write minus. And so in all of this business, we already figured out, is a 3, a 7, a negative 5, and a 13. So it's essentially, start with this set and take out all the stuff that are in this set. So this is going to be equal to-- so you see we're going to have to take out a 3 out of this set. We're going to take out a 7. We're going to get a negative 5. And we're going to take out a 13. So we're just left with the set that contains a 0. So all of this business right over here has simplified to a set that only contains a 0. Now let's think about what B intersect C is. These are all the things that are in both B and C. So this is going to be B intersect C. Let's see, 0 is not in both of them. 17 is in both of them. So we'll throw 17 in there. The number 3 is in both of them, the number 3 is in both of them. Blue is not in both of them. The star is in both of them. So I'll put the little gold star right over there. And so that's B intersect C. And so we're essentially going to take the union of this crazy thing-- which ended up just being a set with a 0 in it-- we're taking the union of that and B intersect C. We deserve a drum roll now. This is all going to be equal to-- we're just going to combine these two sets. It's going to be the set with a 0, a 17, a three, and our gold star. And we are-- I should make the brackets in a different color-- and we are done.
Sal explains the difference between a subset, a strict subset, and a superset. He also explains the notation behind these ideas.
Let's define ourselves some sets. So let's say the set A is composed of the numbers 1. 3. 5, 7, and 18. Let's say that the set B-- let me do this in a different color-- let's say that the set B is composed of 1, 7, and 18. And let's say that the set C is composed of 18, 7, 1, and 19. Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset-- and this is a notation right over here, this is a subset-- B is a subset of A. B is a subset. So let me write that down. B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further and we could say that B is a strict or sometimes said a proper subset of A. And the way you do that is, you could almost imagine that this is kind of a less than or equal sign, and then you kind of cross out this equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member A, but everything that's in A is not a member of B. So let me write this. This is B. B is a strict or proper subset. So, for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see. C contains a 1, it contains a 7, it contains an 18. So every member of B is indeed a member C. So this right over here is true. Now, can we write that C is a subset? Can we write that C is a subset of A? Can we write C is a subset of A? Let's see. Every element of C needs to be in A. So A has an 18, it has a 7, it has a 1. But it does not have a 19. So once again, this right over here is false. Now we could have also added-- we could write B is a subset of C. Or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this. And then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B, and this is just another way of saying that B is a subset of A. But the way you could think about this is, A contains every element that is in B. And it might contain more. It might contain exactly every element. So you can kind of view this as you kind of have the equals symbol there. If you were to view this as greater than or equal. They're note quite exactly the same thing. But we know already that we could also write that A is a strict superset of B, which means that A contains everything B has and then some. A is not equivalent to B. So hopefully this familiarizes you with the notions of subsets and supersets and strict subsets.
Sal moves onto more challenging set ideas and notation like the universal set and absolute complement.
What I want to do in this video is introduce the idea of a universal set, or the universe that we care about, and also the idea of a complement, or an absolute complement. If we're for doing it as a Venn diagram, the universe is usually depicted as some type of a rectangle right over here. And it itself is a set. And it usually is denoted with the capital U-- U for universe-- not to be confused with the union set notation. And you could say that the universe is all possible things that could be in a set, including farm animals and kitchen utensils and emotions and types of Italian food or even types of food. But then that just becomes somewhat crazy, because you're thinking of all possible things. Normally when people talk about a universal set, they're talking about a universe of things that they care about. So the set of all people or the set of all real numbers or the set of all countries, whatever the discussion is being focused on. But we'll talk about in abstract terms right now. Now, let's say you have a subset of that universal set, set A. And so set A literally contains everything that I have just shaded in. What we're going to talk about now is the idea of a complement, or the absolute complement of A. And the way you could think about this is this is the set of all things in the universe that aren't in A. And we've already looked at ways of expressing this. The set of all things in the universe that aren't in A, we could also write as a universal set minus A. Once again, this is a capital U. This is not the union symbol right over here. Or we could literally write this as U, and then we write that little slash-looking thing, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U that is not in A. One way to think about it, you could think about it as the relative complement of A that is in U. But when you're taking the relative complement of something that is in the universal set, you're really talking about the absolute complement. Or when people just talk about the complement, that's what they're saying. What's the set of all the things in my universe that are not in A. Now, let's make things a little bit more concrete by talking about sets of numbers. Once again, our sets-- we could have been talking about sets of TV personalities or sets of animals or whatever it might be. But numbers are a nice, simple thing to deal with. And let's say that our universe that we care about right over here is the set of integers. So our universe is the set of integers. So I'll just write U-- capital U-- is equal to the set of integers. And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. And it's Z for Zahlen, from German, for apparently integer. And the bold is this kind of weird looking- they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers. In fact, I'll do a little aside here to do that. So for example, they'll write R like this for the set of real numbers. They'll write a Q in that blackboard bold font, so it looks something like this. They'll write the Q; it might look something like this. This would be the set of rational numbers. And you might say, why Q for a rational? Well, there's a couple of reasons. One, the R is already taken up. And Q for quotient. A rational number can be expressed as a quotient of two integers. And we just saw you have your Z for Zahlen, or integers, the set of all integers. So our universal set-- the universe that we care about right now-- is integers. And let's define a subset of it. Let's call that subset-- I don't know. Let me use a letter that I haven't been using a lot. Let's call it C, the set C. Let's say it's equal to negative 5, 0, and positive 7. And I'm obviously not drawing it to scale. The set of all integers is infinite, while the set C is a finite set. But I'll just kind of just to draw it, that's our set C right over there. And let's think about what is a member of C, and what is not a member of C. So we know that negative 5 is a member of our set C. This little symbol right here, this denotes membership. It looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set. We know that 0 is a member of our set. We know that 7 is a member of our set. Now, we also know some other things. We know that the number negative 8 is not a member of our set. We know that the number 53 is not a member of our set. And 53 is sitting someplace out here. We know the number 42 is not a member of our set. 42 might be sitting someplace out there. Now let's think about C complement, or the complement of C. C complement, which is the same thing as our universe minus C, which is the same thing as universe, or you could say the relative complement of C in our universe. These are all equivalent notation. What is this, first of all, in our Venn diagram? Well, it's all this stuff outside of our set C right over here. And now, all of a sudden, we know that negative 5 is a member of C, so it can't be a member of C complement. So negative 5 is not a member of C complement. 0 is not a member of C complement. 0 sits in C, not in C complement. I could say 53-- 53 is a member of C complement. It's outside of C. It's in the universe, but outside of C. 42 is a member of C complement. So anyway, hopefully that helps clear things up a little bit.
In this Precalculus Course's video, we will learn the Relative Complement Or Difference Between Sets. Sal shows an example finding the relative complement or difference of two sets A and B.
What we're now going to think about is finding the differences between sets. And the first way that we will denote this is we'll start with set A. I've already defined set A. Let me do it in that same shade of green. I've already defined set A here. And in both cases, I've defined these sets with numbers. Instead of having numbers as being the objects in the set, I could have had farm animals there or famous presidents, but numbers hopefully keep things fairly simple. So I'm going to start with set A. And from set A, I'm going to subtract set B. So this is one way of thinking about the difference between set A and set B. And when I've written it this way, this essentially says give me the set of all of the objects that are in A with the things that are in B taken out of that set. So let's think about what that means. So what's in set A with the things that are in B taken out? Well, that means-- let's take set A and take out a 17, a 19-- or take out the 17s, the 19s, and the 6s. So we're going to be left with-- we're going to have the 5. We're going to have the 3. We're not going to have the 17 because we subtracted out set B. 17 is in set B, so take out anything that is in set B. So you get the 5, the 3. See, the 12 is not in set B, so we can keep that in there. And then the 19 is in set B, so we're going to take out the 19 as well. And so that is this right over here is-- you could view it as set B subtracted from set A. So one way of thinking about it, like we just said, these are all of the elements that are in set A that are not in set B. Another way you could think about it is, these are all of the elements that are not in set B, but also in set A. So let me make it clear. You could view this as B subtracted from A. Or you could view this as the relative complement-- I always have trouble spelling things-- relative complement of set B in A. And we're going to talk a lot more about complements in the future. But the complement is the things that are not in B. And so this is saying, look, what are all of the things that are not in B-- so you could say what are all of the things not in B but are in A? So once again, if you said all of the things that aren't in B, then you're thinking about all of the numbers in the whole universe that aren't 17, 19, or 6. And actually, you could even think broader. You're not even just thinking about numbers. It could even be the color orange is not in set B, so that would be in the absolute complement of B. I don't see a zebra here in set B, so that would be its complement. But we're saying, what are the things that are not in B but are in A? And that would be the numbers 5, 3, and 12. Now, when we visualized this as B subtracted from A, you might be saying, hey, wait, look, look. OK. I could imagine you took the 17 out. You took the 19 out. But what about taking the 6 out? Shouldn't you have taken a 6 out? Or in traditional subtraction, maybe we would end up with a negative number or something. And when you subtract a set, if the set you're subtracting from does not have that element, then taking that element out of it doesn't change it. If I start with set A, and if I take all the 6s out of set A, it doesn't change it. There was no 6 to begin with. I could take all the zebras out of set A; it will not change it. Now, another way to denote the relative complement of set B in A or B subtracted from A, is the notation that I'm about to write. We could have written it this way. A and then we would have had this little figure like this. That looks eerily like a division sign, but this also means the difference between set A and B where we're talking about-- when we write it this way, we're talking about all the things in set A that are not in set B. Or the things in set B taken out of set A. Or the relative complement of B in A. Now, with that out of the way, let's think about things the other way around. What would B slash-- I'll just call it a slash right over here. What would B minus A be? So what would be B minus A? Which we could also write it as B minus A. What would this be equal to? Well, just going back, we could view this as all of the things in B with all of the things in A taken out of it. Or all of the things-- the complement of A that happens to be in B. So let's think of it as the set B with all of the things in A taken out of it. So if we start with set B, we have a 17. But a 17 is in set A, so we have to take the 17 out. Then we have a 19. But there's a 19 in set A, so we have to take the 19 out. Then we have a 6. Oh, well, we don't have to take a 6 out of B because the 6 is not in set A. So we're left with just the 6. So this would be just the set with a single element in it, set 6. Now let me ask another question. What would the relative complement of A in A be? Well, this is the same thing as A minus A. And this is literally saying, let's take set A and then take all of the things that are in set A out of it. Well, I start with the 5. Oh, but there's already a 5. There's a 5 in set A. So I have to take the 5 out. Well, there's a 3, but there's a 3 in set A, so I have to take a 3 out. So I'm going to take all of these things out. And so I'm just going to be left with the empty set, often called the null set. And sometimes the notation for that will look like this, the null set, the empty set. There's a set that has absolutely no objects in it.
Sal shows examples of intersection and union of sets and introduces some set notation.
What I want to do in this video is familiarize ourselves with the notion of a set and also perform some operations on sets. So a set is really just a collection of distinct objects. So for example, I could have a set-- let's call this set X. And I'll deal with numbers right now. But a set could contain anything. It could contain colors. It could contain people. It could contain other sets. It could contain cars. It could contain farm animals. But the numbers will be easy to deal with just because-- well, they're numbers. So let's say I have a set X, and it has the distinct objects in it, the number 3, the number 12, the number 5, and the number 13. That right there is a set. I could have another set. Let's call that set Y. I didn't have to call it Y. I could have called it A. I could have called it Sal. I could have called it a bunch of different things. But I'll just call it Y. And let's say that set Y-- it's a collection of the distinct objects, the number 14, the number 15, the number 6, and the number 3. So fair enough, those are just two set definitions. The way that we typically do it in mathematics is we put these little curly brackets around the objects that are separated by commas. Now let's do some basic operations on sets. And the first operation that I will do is called intersection. And so we would say X intersect-- the intersection of X and Y-- X intersect Y. And the way that I think about this, this is going to yield another set that contains the elements that are in both X and Y. So I often view this intersection symbol right here as "and." So all of the things that are in X and in Y. So what are those things going to be? Well, let's look at both sets X and Y. So the number 3 is in set X. Is it in set Y as well? Well, sure. It's in both. So it will be in the intersection of X and Y. Now, the number 12, that's in set X but it isn't at Y. So we're not going to include that. The number 5, it's in X, but it's not in Y. And then we have the number 13 is in X, but it's not in Y. And so over here, the intersection of X and Y, is the set that only has one object in it. It only has the number 3 So we are done. The intersection of X and Y is 3. Now, another common operation on sets is union. So you could have the union of X and Y. And the union I often view-- or people often view-- as "or." So we're thinking about all of the elements that are in X or Y. So in some ways you can kind of imagine that we're bringing these two sets together. So this is going to be-- and the key here is that we care-- a set is a collection of distinct objects. And the way we're conceptualizing things right here, this is the number 3. This isn't like somebody's score on a test or the number of apples they have. So there you could have multiple people with the same number of apples. Here we're talking about the object, the number 3, so we can only have a 3 once. But a 3 is in X or Y, so I'll put a 3 there. A 12 is in X or Y. A 5 is in X or Y. The 13 is in X or Y. And just to simplify things, we really don't care about order if we're just talking about a set. I've just put all of the things that are in set X here. And now let's see what we have to add from set Y. So we haven't put a 14 yet. So let's put a 14. We haven't put a 15 yet. We haven't put the 6 yet. And we already have a 3 in our set. So there you go. You have the union of X and Y. And one way to visualize sets and visualize intersections and unions and more complicated things, is using a Venn diagram. So let's say this whole box is-- you could view that as the set of all numbers. So that's all the numbers right over there. We have set X-- I'll just draw as circle right over here. And I could even draw the elements of set X. So you have 3 and 5 and 12 and 13. And then we can draw set Y. And notice, I drew a little overlapping here because they overlap at 3. 3 is an element in both set X and set Y. But set Y also has the numbers 14, 15, and 6. And so when we're talking about X intersect Y, we're talking about where the two sets overlap. So we're talking about this region right over here. And the only place that they overlap the way I've drawn it is at the number 3. So this is X intersect Y. And then X union Y is the combination of these two sets. So X union Y is literally everything right here that we are combining. Let's do one more example, just so that we make sure we understand intersection and union. So let's say that I have set A. And set A has the numbers 11, 4, 12, and 7 in it. And I have set B, and it has the numbers 13, 4, 12, 10, and 3 in it. So first of all, let's think about what A-- let me do that in A's color. Let's think about what A intersect B is going to be equal to. Well, it's the things that are in both sets. So I have 11 here. I don't have an 11 there. So that doesn't make the intersection. I have a 4 here. I also have a 4 here. So 4 is in A and B. It's in A and B. So I'll put a 4 here. The number 12, it's in A and B. So I'll put a 12 here. The number 7 is only in A. And the number, I guess, 13, 10 and 3 is only in B, so we're done. The set of 4 and 12 is the intersection of sets A and B. And we could even, if we want to, we could even label this as a new set. We could say set C is the intersection of A and B, and it's this set right over here. Now let's think about union. Let's think about A-- I want to do that in orange. Let's think about A union B. What are all the elements that are in A or B? Well, we can just literally put all the elements in A, 11, 4, 12, 7. And then put the things in B that aren't already in A. So let's see, 13. We already put the 4 and the 12, a 10 and a 3. And I could write this in any order I want. We don't care about order if we're thinking about a set. So this right here is the union.
In this example, determine the scenarios that are probable given the rolling of two dice.
We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes.
Find the probability of rolling doubles on two six-sided dice numbered from 1 to 6. So when they're talking about rolling doubles, they're just saying, if I roll the two dice, I get the same number on the top of both. So, for example, a 1 and a 1, that's doubles. A 2 and a 2, that is doubles. A 3 and a 3, a 4 and a 4, a 5 and a 5, a 6 and a 6, all of those are instances of doubles. So the event in question is rolling doubles on two six-sided dice numbered from 1 to 6. So let's think about all of the possible outcomes. Or another way to think about it, let's think about the sample space here. So what can we roll on the first die. So let me write this as die number 1. What are the possible rolls? Well, they're numbered from 1 to 6. It's a six-sided die, so I can get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the second die, so die number 2. Well, exact same thing. I could get a 1, a 2, a 3, a 4, a 5, or a 6. Now, given these possible outcomes for each of the die, we can now think of the outcomes for both die. So, for example, in this-- let me draw a grid here just to make it a little bit neater. So let me draw a line there and then a line right over there. Let me draw actually several of these, just so that we could really do this a little bit clearer. So let me draw a full grid. All right. And then let me draw the vertical lines, only a few more left. There we go. Now, all of this top row, these are the outcomes where I roll a 1 on the first die. So I roll a 1 on the first die. These are all of those outcomes. And this would be I run a 1 on the second die, but I'll fill that in later. These are all of the outcomes where I roll a 2 on the first die. This is where I roll a 3 on the first die. 4-- I think you get the idea-- on the first die. And then a 5 on the first to die. And then finally, this last row is all the outcomes where I roll a 6 on the first die. Now, we can go through the columns, and this first column is where we roll a 1 on the second die. This is where we roll a 2 on the second die. So let's draw that out, write it out, and fill in the chart. Here's where we roll a 3 on the second die. This is a comma that I'm doing between the two numbers. Here is where we have a 4. And then here is where we roll a 5 on the second die, just filling this in. This last column is where we roll a 6 on the second die. Now, every one of these represents a possible outcome. This outcome is where we roll a 1 on the first die and a 1 on the second die. This outcome is where we roll a 3 on the first die, a 2 on the second die. This outcome is where we roll a 4 on the first die and a 5 on the second die. And you can see here, there are 36 possible outcomes, 6 times 6 possible outcomes. Now, with this out of the way, how many of these outcomes satisfy our criteria of rolling doubles on two six-sided dice? How many of these outcomes are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling a 1 and 1, that's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6 events satisfy this event, or are the outcomes that are consistent with this event. Now given that, let's answer our question. What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? Well, the probability is going to be equal to the number of outcomes that satisfy our criteria, or the number of outcomes for this event, which are 6-- we just figured that out-- over the total-- I want to do that pink color-- number of outcomes, over the size of our sample space. So this right over here, we have 36 total outcomes. So we have 36 outcomes, and if you simplify this, 6/36 is the same thing as 1/6. So the probability of rolling doubles on two six-sided dice numbered from 1 to 6 is 1/6.