 Page 434 questions about the homework now is your chance to ask yes Five B and C love to Okay, five B. Oh five B. I was gonna say five A. Why does it say is it okay zoom good banished? Um? Let's see Probability that a student at random walks is point three five has blonde hair is point four five or Is sorry it's point two or is point four five So I'm gonna I guess use the formula here because they gave me all this data It looks like I'm gonna use a w for walks if that's okay, and how about B for blonde? So it looks like it's saying this probability of W is point three five probability of Blonde is point two probability of W or Blonde is point four five and They want us to find and Now we're going to be finding and differently today But you can find it from the or formula for a single event because we said this the probability of W or B is the probability of W plus the probability of B minus and Is that okay so far? Oh, you know what I forgot to put the B and how about putting the B there mister do it? Lea what I'm gonna do is I'm gonna plus this over to this side, and I'm gonna minus this over to that side To get the and by itself I'm gonna get a new equation. It's gonna look like this and Is gonna be the first one Plus the second one Minus the or It's gonna be that plus that minus that oh, I can do this in my head actually point three five plus point two is point five five Minus point four five. I think the and must have been point one. I hope that's what it does in the back Okay, I'm gonna use the formula for C as well now in C They want me to find or so I'm gonna see if I can go straight to the equation Let's see how good I am here. Lea. I'm gonna go the probability of heart or Face card it's gonna be the probability of heart Point three right Plus the probability of face card point two Minus and Point one. Is that all right? Yeah, first day Next 8 a love to eat the one. I'm oh, yeah, cool Is there a part a and a part B here? I'm gonna go Venn diagram, especially because I noticed there's two different Categories and they've given me both. I think it's gonna be easier to solve this with a Venn diagram I'm gonna draw my Venn diagram here in the middle Probably do this with the formula, but I'm gonna say to myself self. Here's the first car Here's the second car Both cars started 40% of the time as a decimal that's point four His first car started 20 times You know what? I've got a problem because this is 40% and this is in Numbers not percent so I'm gonna pause now and I'm gonna instead of putting a point for there I'm gonna say how many days of the month were there. What does the question say read carefully? 30 you know what? 40% of 30 I'm gonna find 40% of 30 and you may remember from your math eight days Jesse of means multiple 40 percent of means really times point four Times 30. You know how many days 40% of the time is this is 12 days? That's the number. I'm gonna put there 12 How many times did his first car start? I'm not gonna put a 20 there though. What am I gonna put here? eight Because there's my 20 Right. Oh by the way, how many times did only his first car start eight times? Second car started 18 times. There's a 12 there. This must be a six How many days of the month did you say there were? How many am I gonna have to put out here then to make it out up to 30 yep Sit with authority Jesse for Now I could answer any question they want to at least one that means that or that or that and Or got a means add 20 sit. You know what 26 out of 30 neither 4 out of 30 he's falling for a ride. Yeah Any others? Yep Number nine I think Yep, what does this question want me to find only oatmeal I Think Leah even though there isn't a part a and a part b I think a Venn diagram is gonna work better because you know what they want me to find this one here Not the overlap. I I find I prefer Venn diagrams I've used the formula with you guys But my first approach is always can I draw this as a Venn diagram? Have they told me the overlap and if not then I'll bail and try using the formula Especially because here they've given me numbers. I'm gonna go like this We have corn flakes an oatmeal. I always want to start in the middle Did they tell me both did they tell me the overlap? They did what good? How many kids liked corn flakes? 48 so I'm gonna put a 48 here, right? Oh, no, no, I'm gonna put 26 How many kids like oatmeal don't know oh? What's this 20 neither where's this 20 gonna go? Right here and Jesse what do all these have to add to how many kids ah with authority 80? Oh So what's gotta go right there? Uh 80 minus 20 minus 26 minus 22 12 is that yes no am I wrong? Yeah, you have to do the arithmetic but 12 now I've got every possible outcome. What do they want me to find only oatmeal? Oh, there it is only oatmeal 12 out of 80 now you can probably get there Using the formula I'm not quite Sure, it would be all that tidy Is that all right? First yes. Yeah, good anymore. Okay Now we're gonna start to ask ourselves basically up until now we've been picking one thing One card one kid Liking oatmeal or not now we're gonna start to look at multiple events What if you're doing something twice? What if you're picking two cards? What if you're rolling two dice? What if you're today? We're gonna ask what if you're doing one thing and another thing and you want to find the probability Today boys and girls Sesame Street is brought to you by the word and oh I bet you there's like a grover little love Sesame Street and I should find that from YouTube. That'd be a great segue into this wouldn't it today lesson four the event a and B Well, there's kind of a hint or means what ad if you look at the first objective What do you think and means and means multiply and Basically, that's the whole lesson except I'm just gonna show you various ways to use that Dylan's feeling a little short. You can help yourself to a taller chair No problem. So Today, I'm going to introduce to you Very firmly the notion the idea of a probability tree probably my favorite probability tool Every get the sheet. Let's see. No Dylan you put a check mark here somewhere You didn't you need to Here you go. So Here's situation number one. Yes or no, it's a simple thing. You're not sure Or did you put it on somebody else's name is not here? No, you got everybody. Okay, here's the situation We have a pot and the pot has two white marbles and one black marble Well, almost always start with a real simple one that we can wrap our brains around and then go more complex So a ball is randomly selected a marble is randomly selected and it's not replaced and Then we're gonna draw a second Marble and this is what I said today. We're gonna be asking what happens if there's two of it to two experiments It says define the following events event a is the first ball is white the event B is the second ball is white Says find the probability of a okay What are the odds of getting a white ball right now? How many white balls are there in the container? Chew how many marbles are there in the container grand total two out of three and then we're gonna answer Some further questions, but first I'm gonna walk over here if I was doing this I would quickly sketch a tree and the tree looks like this You could get a white ball or not on the first draw You can get a white ball on the second or not white ball in the second or not and then what we're going to do is we're Going to make what we call a weighted probability tree. What did you tell me the odds were of getting a white ball on the first draw? I'd like you to put a little two out of three right there By the way, what are the odds of not getting a white ball on the first draw? one Out of three now we're gonna walk down this branch now down this branch. We picked a white ball Event B is that you get a second white ball if we picked a white ball. How many white balls are left in this box? one out of how many This is gonna be one half the odds of that occurring is one half Down this branch. We picked a white ball. What are the odds of getting a black ball on your second draw? How many black balls are left in the container one out of? Two Victoria down this branch you picked a black ball How many white balls are now left in the container out of yes? I know it's one, but I never reduce fractions Not tell me in what are the odds of getting a black ball on the second draw if you've already got a black ball in your hand? Zero out of two we call this a weighted probability tree and justine You'll get quick enough that this takes you all of one second to draw There is a built-in error check by the way. What do those two add to? one What do these two add to? One what do these two add to if you've done your tree right each sub branch see each group of sub branches better add to One if not you've counted wrongly miss something so nice little built-in error check and it's very very visual because says this What's the probability? Be that the second ball is white Given that the first ball drawn is white. It's saying look if you know the first ball was white What's the probability of the second ball is white? Now I'm going to give you some notation. This is way too much to write So we write instead of writing out the phrase given that the following has occurred we write this probability of B The abbreviation for given that is a long vertical line a has occurred and I read that as the probability of B given a the probability that the second ball was white Given that the first ball was white if I want to fill in what a and b are and it's One half What's the probability of a and B both occurring at the same time? What's the probability of getting a white ball and a white ball? it's two out of three times one out of two and means multiply and You don't need to pull out a calculator because multiplying fractions is the easiest operations How do I multiply fractions top times? top bottom times, so Two out of six. Yes, and I know it's one of three. Okay Last day. We gave you the term mutually exclusive. No overlap Today, we're going to be talking about Dependent and independent, but I'm going to come back to that. I want to look at this situation to first Situation two is exactly the same as situation one, but instead of holding on to the first ball We're going to put it back in shake the jar up and start over. We're going to do it with replacement so Let's fill in our tree first. What are the odds of getting a white ball two out of three? What are the odds of getting a black ball? One out of three Okay, you put the ball back in you shake it up What are the odds of getting a white ball on your second draw? How many white balls are in the jar still two out of three? How many black balls are in the jar still? one out of three Okay, this down this branch you've got a black ball you looked at it Ah, but this one says with replacement so you put it back in the jar you shook it all up What are the odds of getting a white ball? two out of three and One out of three Quick built an error check adds to one adds to one adds one good. I've probably got everything right So what's the probability of event a? Two out of three What's the probability of event B given that event a has it? Oh, let's write our shorthand probability of B given a is Also Two out of three What's the probability of a and B? Two out of three times two out of three Four out of nine So we want to now define the term dependent and independent There's a mathematical definition, and then there's a much easier tree definition Dependent means the odds change depending on whether or not the first one occurred or not This event here is dependent because if you get a white ball the first time White ball the second time the odds are one-half if you don't the odds are two out of two This event here is independent because if you get a white ball What were the odds of getting a white ball the second drop two out of three? What if you didn't get a white ball, but were the odds getting the white ball the second draw two out of three? so the mathematical definition of The probability of dependence says this, the probability that B occurs if you know that A occurs is not the same as the probability of B occurring if A didn't occur. Hold on, write it down and then patience. Oh, and independence, the probability of B occurring if you know that A occurred is the same as B occurring if A didn't occur. The odds of B occurring don't change whether A occurs or not. The odds of B occurring do change whether A occurs or not. That's the mathematical definition. The easier tree definition is this. Victoria, you know the, oh, sorry, people are still writing, oh wait. You know they are dependent if these two branches are different from those two branches. It depends which branch you go down, you get different odds. You know they're independent if these two branches are the, they are. That's what I use, that's my, because I can quickly glance, oh, independent. Well, in some of its common sense as well. If I said to you, what are the odds of rolling a six and on the second, what are the odds of picking a jack, because whether you roll a six or not have any effect on whether you're going to pick a jack or not, they're independent. What are the odds of picking a black card and on the second card picking a jack? That depends because the first card could have been a black jack and that'll change the odds of whether this, it'll change how many jacks are in the deck. They're dependent. Easiest way though is you spot it in the tree. So the multiplication law says this, the probability of A and B is the probability of A times and means multiply. The probability of B, make sure you're in the A branch given that A occurred. Make sure you're on the same branch as the A. Oh, and for what it's worth Dylan, if they are independent, it doesn't matter whether A occurred or not. So you can drop the little given that A occurs. This is the special sub case. I don't memorize that. In fact, I don't memorize this one, but this one is the one that's on your formula sheet. Although sometimes they'll divide by probability of A and they'll get the given by itself. In fact, this one we're going to spend some time with later. It really becomes much clearer because I've told you not a big fan of the formulas. You're going to see only one time am I going to give you a formula. And even then I'm going to cheat and ditch the letters. We're going to memorize a little acronym, a little mantra, a little chant to use. That's next class. I'm going to do a tree. Example one. Example one says two cards. How many cards? Two. So a Venn diagram not going to help me much because Venn diagram is only if you're looking at one single card. Or what we did last day when we were doing or and we were saying, well, one card, we're not going to do that. We're doing two. And the first event is the first card is a face card. The second card is a face card. We're going to do a tree. It's going to look like this. Event A, event not A. How many face cards are there in the deck? Got to think a little bit. 12 out of 52. How many non-face cards are there in the deck? Don't count. Use the compliment, please. 40, right? I hope you didn't go count. I hope you went 52 minus 12. Compliment is also where it works really well. 40 out of 52. And then we could have had a second face card or not. A second face card or not. Now, down this branch, we already picked a face card and we're holding it in our hand. So, Miguel, how many face cards are now left in the deck? 11 out of, not 52, but 51. By the way, that's the probability of being given A. Oh, how many non-face cards are there in the deck? Which one? 40 out of. How could you double check to make sure you've done it right? Adds to one. That's your trigger. 30, no. Okay. Nice little built-in error check. Really, if you get the first half of each branch right, you're probably not going to get the second half wrong as long as you think. Oh, Miguel, down this branch, we got a non-face card. So, how many face cards are left in the deck? Still. Out of 51. How many non-face cards are left in the deck? 39 out of 51. By the way, Miguel, are these two branches different from those two branches? Oh, they're dependent. What it's really saying is the probability of being given A is different from the probability of being given not A. Whatever. See, now that I have my tree, I can answer every question. Almost with no work. Well, with a minimal amount of work. What does and mean, multiply? Which branch has and, A and B? 12 out of 52 and 11 out of 51. Now, I'm not going to reach for my calculator because, Patrick, I know that that times that. I've done so often. 52 times 51. I've just memorized. It's 26, 52. 26 is half of 50. Really, all I need to do is 12 times 11. In other words, it's actually possible to do some of these in your head. 12 times 11 is 132. You guys can double check me, but top times top, bottom times bottom. You can reduce the fractions if you want to. They will be reduced in lowest terms in the answers. In the answers in your homework or in the answers on the multiple choice, they would have gone 12 out of 52 times 11 out of 51. Remember, how do we turn this into a lowest terms fraction? Math, enter, enter. I guess it's 11 out of 221. That's fine. I'll take that too, but I bought it. The reason I like the tree so much is, what's the probability of not getting a face card on the first draw and getting a face card on the second draw? Can you see which branch that is? Boom, boom. Alex, what times what? Yep. 40 out of 52 times 12 out of 51. Sometimes you'll see them just put the two numbers in brackets. Sometimes they'll put a time sign between whatever. Hopefully you've seen enough math now that you're flexible. Oh, I can do this in my head because I know that 52 times 51 is 26-52. Don't believe me, try it, but it is. And I can go 40 times 12 because 4 times 12 is 48. So just seeing 40 times 12 is 480. I don't need my calculator. By the way, what if we were picking three cards? I'd have two more, two more, two more, two more. How many branches would I have grand total? Two more, two more, two more, two more. How many branches would I have grand total? Eight. What if I was picking four cards? 16. What if I was picking five? Okay, you know what? Three is great for two or at the most three draws. After that, it gets unwieldy. But we're going to start to notice today that maybe we can also pull out, later on, some fundamental counting principle shortcuts. For example, you may notice this is something times one less. Maybe there's some room to bring in some factorial type of notation. And there is. But we always start out with trees. And much like last unit, when I told you my fallback was always the fundamental counting principle. And then see if the shortcut works. My fallback is almost always a tree. If it's two or three cards, two or three events, three. More than that, sometimes even still I'll do a tree if it's a fairly easy branch to visualize. Example two. Two cards are drawn from a well-shuffled deck of 52 cards. What's the probability that? Now I looked quickly at A, B, and C. What event are A, B, and C all concerned with? Hearts. You know what? I can use the same tree for all of those. D is going to be different. I'm going to see if by the time we get to D, I'm good enough that I can visualize the specific branch that they want without actually drawing it out. And that's the next step is I draw a tree on a test all the time. But in my homework, sometimes I'll try and visualize the branch and just see if I can get there. So here, though, because it's A, B, C all involving hearts, it's well worth drawing the tree. So it's going to look like this. Hearts one, not hearts one. What will I use for hearts two, do you think? H2? One thing I don't like about these notes is they always use A and B. I think that's dumb. Like here, I would have used F1 and F2, face card one, face card two, because those are letters that make sense to me and I can keep track of them. So I'm going to go hearts two, not hearts two, hearts two, not hearts two. All right, let's see if we can fill in the branches. Kyle, how many hearts are there in the deck? I should use red for hearts. How many hearts are there in the deck? Sorry, out of. How many non-hearts are there in the deck? Please don't count. Use the compliment. 39 out of 52. Okay, down this branch, Kyle, we picked a heart and it does say without replacement. If it was with replacement, by the way, these branches would all be exactly the same as those ones and it would be independent and kind of boring. Without replacement, how many hearts are left in the deck, Kyle? Out of. Yep. How many non-hearts? Out of. Good. Down this branch, we didn't get a heart. How many hearts are left in the deck? 13 out of 51 and 38 out of 51. Double check. These set of branches adds to one. Yep. Dependent or independent? I think the odds depend. By the way, that's how I remember the term dependent. I really say to myself, yeah, it depends which branch I went to. Okay. Now let's answer the questions. Both cards are hearts. 13 out of 52 and 12 out of 51. It's 26, 52, and I have no idea what 13 times 12 is. Both cards heart, right? This one, right? What is 13 times 12? 156? I should have known that. 12 times 12 is 144 plus 12 is 13 times. I can't do that. Okay. Neither is a heart. Which branch is neither a heart? H1, H2. Sorry, not H1, not H2. Is that that wrong? It's going to be 39 out of 52 and 38 out of 51. It's going to be 26, 52. I have no idea what 39 times 38 is. 1482. See, exactly one is a heart. Listen very closely. Ready? It seems to me that when it says exactly one of the two cards is a heart, that could be this branch or this branch. What does or mean? Add. You know what? The basic rule for trees is multiply down, add across. It's going to be the first card a heart, second card not. That's this column right here. Or plus 39 out of 52, 13 out of 51. And you know what? I'm also sensing there's going to be a common denominator here of 26, 52. And so I'm just going to go 13 times 39 plus 39 times 13. Or what's an even shorter way that I could do that? I think times two, right? Because 13 times 39 plus 13 times 39. 10, 14. What we're really saying Dylan is, now that you're awake again, the first card or the second card could have been a heart. First card a heart, second one not. Or first card not, second one was. All right. D, both cards are aces. Okay, we're going to try and visualize the tree. I think both aces, I think what that's saying is, find the probability of ace one. My abbreviation for and is a comma, because strange enough, that's what it is in English too. Ace two. Ian, are you asking about D or about C? I don't think so. Let's find out. Ready? So visualize a tree looking like this, but with no numbers. How many aces are in the deck? How many aces are it? Oh, I want to gamble with you. Please come play poker at my house. Please. How many aces are in the deck? Four out of 52, right? So that's the four out of 52. We picked an ace. How many aces are now left in the deck? Three out of 51. By the way, what if I wanted to find the odds of getting three aces in a row? How many aces are now left in the deck? Two out of 50. What if I want to find the odds of getting all four aces back to back to back to back? I would then multiply by one out of 40. We're not going to do that. You can do a tree, a three level or a four level tree in your head if it's the same event over and over and over and over and over. Because that's easy enough to keep track of. If it's different events like this one where one is and one is not a heart, then there's always more than one branch. And to try and keep track of all of those, I'm going to tell you right now, you're going to miss a branch. Draw the tree. Oh, Ian, what's four times three? Yes. And you know what? 52 times 51 has been for every one of these questions. 26-52. By the way, what are the odds of getting two aces in a row? Not very good. It's a good poker hand. What are the odds of getting no hearts? Pretty good. Almost 50-50, but not quite. Okay, so far. Like I said, you can do a tree of more than two levels in your head as long as it's all the same event. Example three says, what's the probability if we draw three cards without replacement that all three are spades? What they're really saying is find the probability of spade one comma spade two comma spade three comma. Erie, how many spades are there in the deck? How many spades are there in the deck? Are you saying 13? Because here's what I'm hearing. Teen. Teen. Okay. Out of. And now how many spades are there in the deck? We've picked one. How many are left? Out of. And now how many spades are left in the deck? 11. Out of. Oh, by the way, if you're playing five card poker, five spades, a spade flush would be 10 out of 49, 9 out of 48. And then if you multiply that by four, because there's four different suits that tells you the odds of getting a flush. Anyways, what's this one? This one, you know what? Let's wimp out. 13 out of 52 times 12 out of 51 times 11 out of 50. Math, enter, enter. Apparently 11 out of 850. By the way, Jesse, do you notice that I can go down by ones here? Probably I can bring in some factorial notation, times by one less, times by one, and same with on the bottom. And that's going to be a couple of days from now. We're going to say, okay, let's find some shortcuts. But when in doubt, I think that I shall never see a thing as lovely as a probability tree. Dude, paraphrase a famous poem. Example four. Randomly select one bill from pot A and one bill from pot B. Okay. From pot A, you can get a zero. You know what? I need to make this probably a little wider, don't I? You can get a zero, a 10, or a 20. And then it looks like from pot B, you can get a zero, a 10, or a 20. A zero, a 10, or a 20. A zero, a 10, or a 20. What are the odds of getting a zero from pot A? One out of three. What are the odds of getting a 10 from pot A? One out of three. What are the odds of getting a 20 from pot A? One out of three. Double check, by the way. Do those add to one? Yep. I haven't done pot B yet. Relax. Ready, Troy? What are the odds of getting a zero from pot B? How many zeros are there in pot B? One. How many cards are there in pot B? Ah. So Troy, my friend, what are the odds of getting a 10 from pot B? Do I need to draw two 10s, or can I just weight my branch with a what? Don't reduce fractions. What can I put here, sir? Okay. Now, down this branch, I got a 10 from pot A. Will that change the odds of pot B? Oh, dependent or independent? Because I'm noticing that these branches are all going to be the same, regardless of which first branch I went down. They're all going to be one quarter, two quarters, one quarter. And again, Troy, the reason I yelled that you don't reduce the fractions, isn't it easier to do your check to see that it adds to one like that rather than having a one half there? Yes? Right, adds to one, adds to one. There's my tree. Oh, and as we said already, Jesse, the fact that these three, these three, and these three are all identical, independent. Okay. Let's answer the question. A, what's the probability of getting a $10 bill on each draw? Now, this one is complicated enough. I think I'm going to put check marks underneath the correct branches. So a $10 bill on each draw. Actually, that one, there's only one option. $10 bill on each draw. That's this one here, isn't it? One out of two. Sorry, one out of two, Mr. Dewick. One out of three. And two out of four. Two out of 12. One, six. B, what are the odds of getting only one $10 bill? Here is where I'm going to use check marks, I think. Did I get exactly one $10 bill down this branch? Did I get exactly one $10 bill down this branch? Yeah, a 10 out of zero. Did I get exactly one $10 bill down this branch? Did I get exactly one $10 bill down this branch? Yep. Down here. No, I got two down here. Yeah, I got exactly one 10. I also got a 20. But the question is just asking, getting exactly one 10. That's exactly one 10 and a 20. Oh, you know what? This is exactly one 10. Those are the branches there. There's four of them. Listen close. This branch, or this branch, or this branch, or this branch. Amy, what does or mean? Add. Or if you want a quick way, multiply down, add across. Oh, before I forget, who has the solution keys for the workbook? Okay. The workbook does their trees sideways. I do my trees vertically. They do their trees sideways. I can't draw them. I learned this way, and I tried switching one year, and I just couldn't visualize and draw them properly. So I guess for them, the rule would be multiply down the tree this way, add across. Just letting you know. So anyhow, let's get the answer. One out of three times two out of four, and one out of three times one out of four, and one out of three times one out of four. And Pat, are you going to your calculator for the one times table? Are you serious? Are you serious, my friend? Put that away, my friend. What's the denominator going to be? Please tell me. You can see the denominator is going to be up 12. And Pat, my friend, in your head with no calculator, please God, what's one times two plus one times one plus one times one plus one times. Oops. I think this last one should be a two here. I was too angry when I saw you reaching for your calculator. Really? It's the one times table. I'm pretty sure I get two plus one plus one plus two. I get what? Six? Which, without a calculator, Pat, what could you reduce that to in your head? Oh, thank you for redeeming my faith in you. By the way, someone always does every year, and I always jump on them. You were the first one that I saw. It's the one jump. Come on. These things here should be crutches, but they should not be stretchers, right? You should lean on it, but you shouldn't be collapsing all your weight on it. You should not have your summer arithmetic in your head. Now, let's go to the question that many of you have wondered. In particular, Steven. In particular, Kyle recently. In particular, oh, who's been late a lot for the last, let's look. Dylan's been doing pretty good. Karen, late occasionally. You've often wondered if you're the third person in late. Let's suppose that the previous two have all rolled instead of a five. Let's cross it out and make it a six. Or maybe you're interested. Maybe they both rolled ones. What are the odds of me rolling? Whichever. It's going to be the same no matter what. What's the probability of Mr. Dewick having three students show up late and having all three of them roll back to back to back sixes? Or, in probability speak, what's the probability of six and six on the second and six on the third? What does and mean? Multiply. By the way, when you roll a dice, is it dependent or independent? Does what you roll previously have any effect on what you're rolling next? So you know what? I'm just going to visualize this single branch of the tree. What are the odds of rolling a six? Karen, my angel. What are the odds of rolling a six? On a dice? How many sixes are there on the dice? How many numbers are there on the dice? So what are the odds of rolling a six? Absolutely. And Dylan, what are the odds of rolling a second six? And Kyle, what are the odds of rolling a third six? What are the odds of getting three sixes? Or have I had three ones in a row in this class? I know I have once this year and I've vented because you know what the odds are? They're stinking small. The odds of getting those three ones in a row. I was yelling at the fates saying, what are you doing to me? It's one on 216. That shouldn't happen very often. B, what are the odds of getting at least one five in three rows? Okay, this one's a bit trickier. Oh, but let's cross out the five. Let's make it a six. So three students are late. What are the odds that Mr. Dewick feels some sense of justice at least once? Okay. What does at least once mean? Once or, or, what does or mean? Now here's the problem. What I would have to do to solve this would be first one six, second two not next to not first one not second one. Yes. Third one. No. First one. No. Second one. No. Third one. Yes. I would have to look at first person, second person, third person. I don't want to do that. That's why I put in brackets think compliment. What's the opposite of at least one six? Think about it. What did you say at least one was what or what or what? One or two or three. What's the opposite of that? I heard it finally. What? None. The opposite of at least one is none. This is going to be much easier to do instead of finding, sorry, let's write it this way. Instead of going the probability of at least one six, it's going to be much easier to go one take away the probability of none because none is the compliment. And we said last day that if you know the compliment, one minus the compliment is the other one. In fact, we've been doing the compliment half the time on our card trees. I'm willing to bet nobody counted 39 or 38. You just subtracted. Why is this so nice? So this is going to be one minus the probability of not a six, not a six, not a six, which is going to be one minus. What are the odds of not rolling a six? How many non-sixes are there on the dice? Five out of six and five out of six and five out of six. So five out of six times five out of six times five out of six, that's the odds of getting no sixes. One minus that is the odds of getting at least one six. What are the odds of Mr. Dewick feeling that there is some sense of justice in the universe? You know what? I'm not even sure I need a calculator for this, Pat, because I know this is one minus 125 over 216 because way back in logarithms Mr. Dewick maybe memorized certain exponents. It's five to the third over six to the third. And you know what one is? It's also 216 over 216. This is really 216 take away 125, which is 116 take away 25, which is 96 take away five. I'm willing to bet the answer is 91 over 216. In my head you guys can fall back on your stretchers and check me if you need to. Am I right? Yeah. And what that means is almost half the time I should feel justice. You know what else that means? I don't think these dice are properly balanced because I'm telling you I don't feel justice almost half the time. I get hoaxed an awful lot of the time I think in my dice of fate rolling. Man, I have some kids that just happily trounce through ones and twos this entire year. I got some kids that are up to double digits on ones and twos. And oh, why do you mock me? Calm blue ocean. Calm blue ocean. I could. I could drill a little hole, put a little weight on the one there, cut that out, blew it back in, and then it'd be more likely to end up that way with the one on the bottom and the six on the top. I could. Can you turn, please, to page 443? Page 443. Well, first you can turn to page 441 because I know some of you like to use your book for studying as well. This is the AND formula. It's on your formula sheet. But, Justine, did we really use it? I showed it to you, but then I said treat, treat, treat, treat, treat. But that's the justification for the treat. Fine, fair enough, fair enough. Okay. Oh, and then it has the multiplication law for independent. You know what? I don't know why they make a big song and dance about, okay, fine, they're independent, so the branches are the same. And stale means multiply. Whatever. Then it walks you through some lovely examples. I'd like you to now go to page 443. And I'm not going to sign every single thing that I said in the back, but I am going to give you some of these. Let's see. I think number one is good. Six is good. Seven, now seven is with replacement. That means your branches won't change on the second level because you're putting the card back in. These are independent. That's okay. Eight is the same question, but without replacement. Oh, look at number nine. The probability that Sarah will pass grade 12 math and grade 12 physics. Oh, I have to look at that one. Now they told me that they're independent. What that means is my second half branches will be both the same either side. Okay, nice. Skip 10, I think. 12 is good. And then if you could turn to page 450. On page 448, they introduce a tree, but you can see I said to you they do their trees sideways, which in some ways I guess is nicer. I have a hard time my math brain and I know I'm weird. I have a hard time reading these and drawing these sideways. If you like them better sideways, you'll get full marks. No problem. And every year I have a couple of kids who look at the solution key and like that approach better. Good. I'm fine with it. Anyways, I said turn to page 450. I got to do some basketball. You're into my heart. So number one, I got enough hockey players that will do number three. Take a look at number seven, please. With Dylan in this class. Karen in this class. I think we have to do with Michelle in this class, not in this class right now. I think we have to do number seven, the probability, especially with her alarm having gone off last class that was at about 915 in class. Okay. So we'll definitely have to do number seven. That's kind of a neat one by the way. I'm not going to worry about Dr. Chang and Dr. Barbara and Dr. Adams. I think we're going to stop there. I'll assign the ones from page 460 next class. I'm about 15 minutes, just a little bit shy of 15 minutes.