 Let's start. Is it recording? I think it is. Good. Lesson one says experimental and theoretical probability. It lists some objectives, but it's easier just to jump on in. It says visualize rolling a die. By the way, grammatically, one die, two dice. However, you're often going to hear me call this a dice because it's sort of become an idiomatic It's become proper in English, but technically a dice is wrong. It's like saying a car's or a truck's. It's one die. So most of the notes will be grammatically correct. Visualize rolling a die. What's the theoretical probability of getting a five? What are the odds of getting a five? And all of us have a very, very basic, intuitive understanding of the odds. Probability. All of us have a horrible anything beyond basic understanding of probability. And if you don't believe me, think about how superstitious, how horoscopes, how we try and read things into everything, we have a bad understanding of randomness. But the very basics we're pretty good at. What are the odds? Okay, I don't want to write out the theoretical probability of getting a five is one out. So here's our notation. First of all, for probability dill in its traditional, we use a capital letter p for the probability. Makes sense. And then we put a bracket and in the bracket, we write the condition that we're talking about. In this case, it's rolling a five. Close bracket is one out of six. I would read that out loud, Matt, not p bracket five, bracket equals one slash six. I would read this as probability of getting a five is one out of six. It's a good notation. Now, this is the theoretical probability. So theoretically, if I rolled the dice 300 times, how many fives should I get? Theoretically. Do the arithmetic? Try. 60, 50, 50. Okay. Theoretically, if I roll the die 300 times, theoretically I would get 50 out of 300. Do you think if I actually did the experiment, I would get exactly 50? In fact, I'd be very suspicious if somebody told me that they got exactly 50. I'd be suspicious if they told me they got one. I'd probably, Miguel, take their word for it if they said they got anywhere between 40 and 60. I probably wouldn't but you know what, even if they said 25, I'd say prove it but it's possible. So theoretically, 50 out of 300, which reduces to one out of six versus experimental. Now, they want us to simulate rolling it and on your TI graphing calculators, some of you will have a probability simulator, we're not going to. We're just going to understand that the experimental would be different, probably. But here's what, oh, come on, mathematician begins with the letter L. Not fair, Matt. Is it? Maybe it is. One of the guys who initially wrote the first drafts of the mathematics behind probability and by the way, the earliest probability mathematics was done because a wealthy noble gambler had been interrupted in the middle of a game and he wanted to know what was the best mathematical way to split the pot because he was ahead but he hadn't won yet. So he was saying, should I get it all because I was ahead? So he wrote a letter to a mathematician named Fairmont and Fairmont in this letter back to him, basically developed most of probability mathematics. Anyhow, here's what Fairmont said. Now, this is our theoretical probability, 1 out of 6. He said, if you rolled 300 times, you might not get 50 out of 300. 1 out of 6. If you rolled 3,000 times, you would get closer to 1 out of 6. If you rolled 3 million times, you'd get even closer to 1 out of 6. In fact, he said that the more you do the experiment, the closer it approaches the theory. Good morning, sir. Again, we have a pretty good understanding. Example 2. Suppose one card is drawn from a deck of 52 cards and we drew a little 52 card list for you here so you can kind of visualize. You have four suits. You have how many cards in each suit? 13, 52 cards altogether. Okay, what's the probability that it is a face card? And I would write it as probability of f for face card. Well, the first rule of probability, Justine, and it's so obvious we're not going to write it down, the first rule is if you can count it, you can solve it. Face cards. I'm going to circle them with my pencil. Those are face cards. Those are face cards because those are the queens. And the other face cards are the jacks. The jacks. I can count it. How many face cards have I circled, Justine? 12 out of how many cards are there altogether? 52. Now, by the way, you're going to find most of the time the fractions will reduce. I never will. And I'm going to tell you not to, although they will reduce the fractions in the multiple choice answers, but we'll be using the fraction button very quickly on our graphing calculators because we're lazy. We want to be efficient. I prefer them not reduced because I'm going to tell you the odds are very good. If I wanted to do any kind of math with this, I'd want a common denominator of 52 anytime I'm dealing with cards anyways. And I just have to find a common denominator. I'll leave it like that. A red ace. Okay. Probability of red A. I'll circle those in red. What do you get? Four out of 52. Oh, Mr. Dewick, you did it wrong. Good gosh. You're right because it's the red ones. It's only these two, isn't it? How about two out of 52? Good gosh, Mr. Dewick. That was terrible. By the way, so again, visualize our deck of cards. You know, I should have a deck of cards in my hand for this. So suppose I shuffle this. What's the probability that if I reach in for a card, that it's a face card and a red ace at the same time? It's a trick question. Zero. And how can you tell from my little diagram there because nothing is overlapping in the circles? We're going to talk about what that means as well. We're going to be very important for us to understand whether something can happen at the same time or not. We're going to give that a special name. So here's the official mathematical definition of probability. It says, if there are n equally likely outcomes, 52 cards, and our outcomes are favorable to your event A, two red aces if that's your event, or 12 face cards if that's your event, then the theoretical probability of event A is it's the number of successful outcomes divided by the total number of outcomes. And that was how it was first defined. One of the first, one of the best tools I'm going to give you is a tree diagram. So here's a great question that very quickly tells you how bad our intuitive understanding of even basic probability is actually. What's the probability that in a family of three children, there are exactly two girls? In a family of three kids, there are exactly two girls. Now we are going to assume there's a 50-50 chance of when a child is born that it's a boy or a girl. I've been told that actually it's not quite, but it's something like 49.9999 and 50.00s. Anyways, we're going to assume it's 50-50. I need to double check that. Here's what I'd like you to do. So here's the question. What are the odds? What's the probability that if you know a family has three kids, exactly two of them are girls, I'd like you in the margin to write down what you think the answer might be right now. Best guess. What's your gut instinct? Write something down. What out of what? And then we're going to find the actual answer. And Stacey, the reason I'm doing this is I want you to get suspicious of your own gut instincts. If you don't think we have bad gut instincts, you've never been to Las Vegas. The people there are unbelievably stupid when it comes to their understanding of probabilities. Terrible. So here's how I would solve this. I would look at the fact that you can have your first child, your second child, and your third child. Don't draw those in just yet. I'm going to put those right now. You'll draw those in afterwards. I'm going to use a leveled tree. I'm going to say for the first child, you could have a boy or a girl. Then you could have a boy or a girl, a boy or a girl, and then you could have a boy or a girl, a boy or a girl, a boy or a girl, a boy or a girl. Write that. And you don't necessarily have to put the levels there, but you know what? In our notes we will, so we know what the heck we did. We said here there are three events. First kid, second kid, third kid. So solution steps, it says draw tree diagram, represent the possible arrangements of boys and girls in a family of three. Each branch is equally likely, it says. Why? What are our assumptions? We're assuming the odds of having a boy or a girl are the same. Reasonable assumption. How many branches, once you've drawn this and you're going to get good enough at trees, they'll take you about one second, but they're so useful. How many branches, once you've drawn this out, contain exactly two girls? Let's see. This branch here has three boys. This branch here has one girl. Oh, here is a two girl branch. Are there any other branches that have exactly two girls? Here, here. Now you can use a highlighter if you want to, or if you don't have a highlighter, another thing that works great is putting check marks underneath. That one has three girls. Sorry, two girls. That one has two girls. That one has two girls. The probability of two girls is how many branches have I highlighted? How many branches have a check mark? Three out of how many branches were there? How many outcomes were there when we listed them all using a tree? Eight. Did anybody say that the odds were three out of eight? It's that eight you did? He's done some probability before. This guy, we've got to be careful. The rest of you though, if you're trying to do this intuitively, I defy you when you're told three girls, two girls, three kids, oh, and they're going to get a pop out of there. Yeah, it's two times, two times, two. It's the fundamental accounting principle. It's going to come back eventually. We're going to find better ways to do it, but very basic questions, surprising answer. In fact, most people might say, I suspect the most common answer, well, maybe two out of three or something like that. Trying to desperately do something with the numbers. Turn the page. Example four. When the pointer is spun twice, find the probability that you win a total of $10. How many times are we spinning the pointer? Twice. Generally, if there are two events, I'll use a tree. This one is going to be a bit of a bigger tree. It's going to have four possible outcomes. My tree is going to look like this. On the first spin, I can have a five, a 10, oh, hang on. Let's be organized, Mr. Dewick. A zero, a five, a zero, a five, a 10, or 20, and then I can have zero, five, 10, 20, zero, five, 10, 20, zero, five, 10, 20, zero, five, 10, 20. By the way, how many outcomes are there? How many possible outcomes are there? 16. Probability of $10 is. Now, I noticed for part B, I'm going to be reusing the same tree, so I'm not going to use a highlighter because that's really going to muck this up. You can, but I want to try and see if I can use this twice without redrawing it. I have a bit of a natural laziness. In this case, I'm going to go to checkmarks. You can't. It'll be fine. You might airy find it easier though just to quickly redraw the tree down here. Anyways, exactly $10, and I'll use green since I've used green over here. This one, this one, this one, oh, nothing over there. What are the odds of winning exactly $10? Three out of 16. Then we have probability of same number. Okay. That can be zero, zero, five, five, 10, 10, 20, 20, oh, four out of 16, four out of 16. We're going to also eventually find ways to do this without a tree. Specifically, when you have three or four events, it gets a bit ugly. We had three events here and even a three-level tree with two options already started to get a bit unwieldy by that third row. If we were spinning this pointer three times, how many outcomes would we have? Each one of these would have four more branches. I think 64, you know what? There comes the point when a tree breaks down, but we're going to do a lot of our initial probabilities with two or three nice events, and then we'll eventually try and generalize. Example five, two bills are randomly selected from the pot without replacement. You know what? That's a keyword. We're going to underline the word without replacement. Find the probability that you win a total of $10. Let's do our tree. On the first draw, we can get a zero, a five, a 10, or a 20, but now you're not replacing the bill. On this second draw, there's only three options. I've already picked a zero. What's left in this particular pot? Five, 10, and 20. Down this branch, there's also only three options. What's left? Zero, 10, and 20. Zero, five, and 20. Zero, five, 10. Right? How many outcomes are there this time? Not 16, yet four times three if you're trying to bring the fundamental counter-principle. Yes. Probability of $10 is, is it just two? Two out of 12. So there is lesson part one. So lesson two, part two, a bit more terminology, related events, and it gives us some objectives. So definition, the list of all the possible outcomes of an experiment is called the sample space. For example, when I flip a coin, what's the sample space? More specific, not two. What are they? When I flip a coin, what are the possible outcomes? What's the sample space? Heads or tails? Okay. When I roll a dice, what's the sample space? One, or two, or three, or four, or five, or six. Can I get a 1.3 when I roll a die? Can I get a negative four when I roll a die? So part of the first thing you have to be able to do is figure out what can and what cannot happen. One bill, example one, one bill is randomly selected from this pot. What's the sample space? Traditionally, we use a capital letter S for sample space, equals. And traditionally, because it's a set, we use the fancy curvy set brackets, those ones. I won't take marks off if you don't, but I'll be proper. And the sample space here would be zero, five, and 10. And what is the probability that a $5 bill is selected? I don't think we need to do a tree for this one. We can do it intuitively. What is the probability that a $5 bill is selected? One out of three. Let's do a more interesting question. Example two, one bill is randomly selected from pot A and one is selected from pot B. So now we're picking from each, it says use a table to construct a sample space and use a tree diagram to construct a sample space. Which one is easier? I'm going to draw a little line down the middle. Generally, I use a table if there are two main samples. And there's more than three. Specifically, I use a table when I'm rolling two dice because it's six and six. If I did a tree, my tree would have 36 branches which is a bit of overkill. But both are acceptable. So if I was going to do a table, I would go like this. A and I would write a little zero, five, and 10 right there. And then I would write a little B. And just to the left of the first zero and below it, I would write a five, 10, and 20. And then I would draw a line like this so that I can sort of build my table. Can you see how this is going to look or going to work? It's going to go like this. I guess I can get a zero and a five, a five, and a five, a 10, and a five. This is another way to list all of the outcomes in an organized manner. Oh, I can get a zero and a 10, a five, and a 10, a 10, and a 10. I can get a zero and a 20, a five, and a 20, a 10, and a 20. How many outcomes? Nine. I could also have done a tree. And this would be row A, zero, five, 10, row B, five, 10, 20, five, 10, 20, five, 10, 20. Which of those is easier to draw? I'm better at trees, but I'm going to tell you right now, if you're doing two dice and you will be doing two dice on your test as a classic question, table works way better because a 36-branch tree is overkill. Basically, I quickly use the fundamental counting principle to make a guess as to how many branches I'm going to have. It's less than 12, 12 or less. I'll probably go with a tree. What's the probability that some of the two bills is $10? So there's two ways I can get that. From my table, I could say, now they want the sum of the two bills. That means that it adds to 10. I could circle probability that sum equals 10 is what out of what? Two out of nine. Or from here, I would go, oh, they got to add to 10. Yay, no, yay, no, no, no, no. And there's my two out of nine as well. Example three. You're going to make it? No, nothing. One card is drawn from a well-shuffled deck of 52 cards. Use the graphical display from the sample space and find the probability of getting, okay. First of all, we want to find the probability of getting a red card. What I'm going to do is I'm going to circle all the red cards. I'm doing this for a reason. So if you have your pencil and you have different colors, great. Even if you don't, we're still going to circle and just keep track of things. Now, all the red cards are those ones and those ones because we're going to try to develop an equation shortly. What's the probability of getting a red card? 26 out of 52, which I know is one half, but I'm not going to reduce. Probability of getting a jack. So probability of a jack. So the jacks are those ones there. What's the probability of getting a jack four out of 52? What's the probability of not getting a jack? Okay, some terminology. Don't write this down. Here's what I don't want to write. That's dumb, waste of time. We have a better notation than that because so often in probability we want to find the odds of something not occurring. We have a special symbol for it. That's the probability of getting a jack. The symbol for not is if you draw a horizontal line above it. Write that down. That is the event not a jack. What is the probability of not getting a jack? Did you count the 48 cards? How did you get the 48? So this is going to give us another tool. I think what you went was 52, 55 Mr. Dewick. 52 out of 52 minus four out of 52. You went 52 take away four, yes? 48 out of 52. We call this using the complement. We say, you know what? Sometimes it's easier to find the odds of the opposite and then subtract from one if you really want to be fussy. What's the probability of a red jack? Can you see it from what we've circled? What we're really looking for here is the overlap. The overlap right there and that's why I went with this circling approach. How many red jacks are there? Two out of 52. Now compare that with the probability of red or jack. Now we need to define what or means in probability. Patrick you were late for class so you can have a detention at lunch or a detention after school. In English you understand in English the word or means one or the other but not both. In probability or means one or the other or both. In other words when it says a red card or a jack it also includes the red jacks and that's something kids often forget because the English meaning and the probability meaning are different. What it's really asking is how many cards are circled once or twice? I don't care. How many cards are red or jacks? Stop. How can I figure it out without counting them all? Could I go 26 plus 4? Why? In fact what I do is I go 26 plus 4 minus the overlap and that's going to give us next class our first probability formula. We're going to say if you ever want to find or it's the first one plus the second one minus any overlap because you've counted it twice. Anyways 26 plus 4 minus 28 yes. Circling things like this is a very good way to visualize stuff. In fact we use something called a Venn diagram. Charles Venn was a mathematician who honestly was not very good but one of the things he did in his notes is he popularized diagrams like this. They're named after him and it's one of the ironies because he really didn't knew much but Venn diagrams even non-MAT people often know what a Venn diagram is. It's a nice visual representation. So here is a Venn diagram. We would call this the event A. What would I call that there with the bar above it? The event not A and there is no overlap. You can't be A and not A at the same time. Sorry you can't. We call the event A not occurring the compliment and this is a word you're going to want to know. I'm even going to highlight it. It's a vocabulary word of event A and it's denoted by not A. It's sunny outside. What's the compliment of its sunny outside? It's not sunny outside. The Canucks won yesterday. What's the compliment of the Canucks winning yesterday? Not in the playoff game. What's the compliment of the Canucks winning yesterday? Canucks losing yesterday. You can't win and lose at the same time and the probability of something happening or something not happening has to add to 100%. Something either happens or it doesn't happen guaranteed every time or we can rewrite this. The probability of something not happening is one minus the probability of something happening. So let's go back to the Canucks yesterday. Okay. Over the regular season, let's suppose their winning percentage was 65%. So the probability of them winning any one game is 65%. What was the probability of them losing yesterday? B. 35%, 0.3%. A and B. So the event that both A and B occur at the same time is denoted by A and B and an event diagram, it's the overlap. That there is how I represent A and B. A or B. The event that A or B occurs or both is denoted by A or B and A or B is this or the overlap or that. This section right here is A only. This section over here is B only. This section over here is both. All of them together are A or B. What would this white section out here be? A or B. No. Was that a phone? No. Was that your alarm to wake up normally on days like, oh, we're going to talk. We're going to talk. Example four. So two dice, a black die and a white die. And this is where if I was doing a two dice question, I would quickly go one, two, three, four, five, six. One, two, three, four, five, six. I would draw a line, draw a line, and in 15 seconds or less, I can list all of the outcomes. There's 36 of them. It really doesn't take much time, go systematic. I've done that for you. So we have two events. Event A is the sum of the two dice is at least nine. Event B is the black die is one less than the white die. Because I can do color, I'm going to call this one blue, and I'm going to go with this one red. You guys are fine without color, but if you have color, pick a couple of colors. It says use the sample space in the table to find the following probabilities. Event A, probability of A. What was event A, Ian? Read it to me. What was event A? Read it out loud to me now that you're back with me. What I'm going to do is I'm going to circle any dice that add to nine or more. Here, here, here, here, because nine is at least nine. Oh, in fact, I'm spotting a bit of a pattern. It's this bottom corner, isn't it? Everything in that bottom corner adds to at least nine, which is also one of the nice things about a table. Often you'll find there's convenient patterns like that. How many did we circle? What is the probability? Oh, you know what, it's going to be out of 36. I can tell you that. What's the probability that two dice add to at least nine? Now, event B is the black die is one less than the white die. Let's see. So I think something like this, that one, that one, that one. Oh, I'm spotting a pattern here too, that one and that one. What's the probability of B? Five out of 36. What's the probability of A and B at the same time? Can you see it? What's the overlap? Two out of 36. What's the probability of A or B? How many are one or the other or both? You know what, how many are circled? 10 plus 5 minus 2, 13. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 13, if we count them by hand, the shortcut is add them up, but subtract the overlap because you've counted it twice. 13 out of 52. Out of 52, out of 36. Good gosh, I'm doing card questions. As soon as I wrote that, 13. What's the probability of not A without counting, please? 26 out of 36. In English, what does the event not A represent? Well, Ian, can you read to me the event A again? So in English, what does not A represent? I think it represents the sum of the two dice is, Troy, less than 9. So dealing for two dice, I quickly draw a sample space. You could do a tree, 36 branches is yucky. You could. You'd go 1, 2, 3, 4, 5, 6, 1, 3, 5, 6, 1, 3, 5, 6, forget it. Here is the lovely bend diagram. I don't know what the event is. I really don't care. They put dots to symbolize how many of each appear in each section. So there are 13 total outcomes and they're all equally likely. We've included that phrase in most of these questions. In a couple of days, we're going to ask, what if they're not equally likely? We can tweak it and adjust it, no problem. But for now, we're going to assume every event, every outcome has the same odds of occurring. What's the probability of A? I'll give you a hint. It's out of 13. I'm going to give you another hint. Don't say 3 out of 13 because it's not. What? Yeah, there are four circles in oval A. Four out of 13. What's the probability of B? Five out of 13. What's the probability of not A without counting, preferably? 9 out of 13. If it's 4 out of 13, 8, 9 out of 13, right? You can count, but let's use the complement when we can. It's always nice to use complements. What's the probability of not B? 8 out of 13. Okay. What's the probability of A and B occurring at the same time? You see it? 1, right? There's only one overlap. What's the probability of A or B? Now, there's two ways to get it. I can count 1, 3, 4, 5, 8, or I could go A plus B minus the overlap. Either of those works just fine. Here, I would almost certainly just in count because it's like a lovely diagram. Still, 8 out of 13. What's the probability of not A and B? 12 out of 13. And what that's really saying is what's not overlapping? What's the probability of not A or B? Now, I can get it from the complement here. 3, 5. Oh, those guys are not A or B or both. So far so good. Can you open your workbook please if you would be so kind as to page 400 and 26? I need to look at lesson 1 and see what I technically assigned the end of the lesson. 3 to 4. Sure, 3 is good. 4 is good. I said try number 6. Let's see. Tree diagram. Three people, but it's just so it's going to be a three level tree, but it's flipping a coin. So two outcomes. In fact, it's going to look almost the same as the boy-girl tree that we did at the very, very beginning. So 6 is okay. 7, yep. I'm going to assign 8 because we've done, by the way, if the probability of X is 0.2, what is the probability of not X in your heads right now? What is it? Yeah, the answer is D. Okay, we just did 8 anyways. And 9. So 9, you're going to have a two level tree. Box A, red, blue, yellow. Box B, red, blue, yellow, green. Draw your tree and then which branches have the same color? So number 9 is good. And then what did I assign on lesson 2? Number 8, which I already assigned. Okay, I'm going to add 10 and 11 and I'm going to hit stop here.