 If you want a pretty good idea of what your test is going to be, combine quiz one and quiz two. All right? I think you'll find very few surprises. Number one. Oh, let's do the old because Doug's here and he likes to laugh. Okay. There you go. Number one. When you toss a fair dye three times, what's the probability that you will get a five on the first toss, a six on the second toss, and any number except two on the third toss. How many times are we tossing this dice, Amrit? Three. I think I can, for a dice, I think I can still use a tree for three events because I think this is really saying comma and comma and and what does and mean? Multiply. What are the odds of getting a five on the first toss, one out of six, and what are the odds of getting a six on the second toss, one out of six, and any number except a two, what are the odds of not two, five out of six, five out of 260, one more. By the way, this would be fair game on the provincial as a non-calc question. I think they'd be comfortable expecting you to go one times one times five and six times six times six. But I ask you to memorize some exponents at the very beginning of the year, way back when, when I still like to, I mean, I mean, when, anyways, yeah, right, okay. That came out wrong. Number two, how many cards? Two. Tree, or maybe I can just visualize it. Let's see. First card is a heart, second card is a heart. Are those both the same event, Madison, heart and heart? I think I can visualize this without doing a tree. The probability of heart one followed by heart two, I think it's going to be 13 out of 52 and 12 out of 51. If they had said a heart followed by not a heart, or if they had said instead of giving me the specific order, if they had said one heart and one non-heart, I'd probably have to draw the tree to look at all the different possibilities. But here, since they gave me an order, and by the way, that's technically a permutation, but you may have noticed when I did the combinations with you guys as a lesson, it was a video lesson. I really don't use permutations. I do a tree. I use combinations for card questions, but if it's like they're giving me the specific order, even if it's five cards in a row, I can follow that branch all five in a row, I think. 13 times 12 is, well 12 times 12 is 144, so 156 over 26, 52. What's that in lowest terms? If you reduced it fine, multiple choice, it would be reduced as your answer, so make sure you know how to reduce this, and I'm pragmatic, use your calculator. Okay, suppose you throw a fair six-sided dice. One is white, and the other is black. Let T be the total showing of both dice, and let B be the number showing on the black die. Let the probability that the total is eight, given that conditional, black is two. How many dice? I need to use a chart. I didn't leave you much room to do a chart, sorry, the trees, the trees. I'm going to try and do a chart. Where am I going to do my chart? So I'm going to write down here, use chart, and I'm going to wimp out and just open up a brand new page here. So I'm going to have black dice, one, two, three, four, five, six. What was the other color dice? White, one, two, three, four, five, six, and although this sounds like a lot of work, when you get good, this takes less than 60 seconds, well less than 60 seconds, one, one, one, two. You know what? Maybe I can just kind of visualize what's, well no, this is a quiz, you know, Jasmine in my homework, I might try and just see if I could visualize which ones I'd be circling just from this pattern already, but on a test or a quiz, I'll write it out. One, three, one, four, one, five, one, six, two, one, two, two, three, two, four, two, five, two, six, three, one, three, I'm not going to call these out because I'm having a hard time speaking and doing this at the same time. So on your test, five, five, let's try, what's that Miguel's height, oh sorry, no, five, four, okay, five, one, five, two, five, three. On your test, I'm going to give you Jordan a two dice question, but I'm probably going to make it that four sided tetrahedral dice that we did on the quiz because it's a much smaller chart to do, but I'm game, I'm keep going, I'm almost done. Six, one, six, two, six, three, six, four, six, five, six, six, okay, there's my chart. This is going to be, the probability of that given that is both t equals eight and black equals two divided by black equals two. Now how many of these is the black dice a two and the total is eight? I think only one out of 36, is it not? One out of 36 divided by, how many of these is the black equal starting with a two, one, two, three, four, five, six out of 36, oh wrong one, and the answer is one out of six. Now that I see that, I say, oh yeah, one circled out of six, but I got there with the formula just in case, is that okay, nope, not for two dice, I don't think, I haven't, well yeah, you could do a tree, but you know how many branches your tree would have, 36 branches, you really want to do that? We can, so rule of thumb, Brett, dice, two of them, chart, okay, number four, Dan, how many people are we selecting in number four, five, tree, no, are we selecting without replacement, bucket, because it's dependent. I'm going to go boys, girls, four boys, five girls, it says we want three girls, how many boys do we have to choose then if we're picking five people, two, my equation's going to look like this, four boys choose two, and five girls choose three, now that's the number of ways to pick three girls to make it a probability divided by the number of ways to pick five people. Oh yeah, nine, choose five, I got my built-in error check there, plus, first, plus, first equals first, last, plus, last equals last, okay, now I have to go to my calculator, I think, here we go, four math back, choose two, and five math back, choose three, divided by nine math back arrow, choose five, double check if I typed it in right, four choose two, five choose three, nine choose five, and lo and behold, I get an answer of .4762, how many decimal places, if they don't say, I usually go to four, so .4762, by the way, if you wrote .4761 in grade 12, we'll take a half mark off if you don't know how to round math eight, so don't be sloppy, Kelvin's still awake, I got you before, right, okay. Number five, two dart players, how many dart players, two, I'm already thinking maybe three, they throw independently one dart at a target, the probability of each player hitting a bullseye is .3 and .4 respectively, what does respectively mean in that order, what's the probability that at least one of them, you know what, player one hits or misses, player two hits or misses, hits or misses, what are the odds that player one hits a bullseye, .3, what are the odds that he misses a bullseye, .7, what are the odds that player two hits a bullseye, .4, .6, .4, .6, by the way, how can you tell these are independent because these two branches are the same, Amrit, what's this question, why don't we define the probability that what, now at least one is, it seems to me, this branch is at least one, this branch is at least one, this branch because two is at least one, you know what an easier, now you could get, what does or mean by the way, so you could go plus, plus, plus, I'm going to be clever and I'm going to go one minus the probability of none because it's less typing, you could use compliment here, I'm going to go one minus .7 times .6, also because I can do this completely in my head, I know that 7 times 6 is 42, so .7 times .6 is .42 and I can go one minus .42 in my head and get, is the answer 0.58, yes, pretty cool, huh, or Jordan you could have gone plus, plus, and plus, multiply down at a cross, but it's very handy, very useful, keep in the back of your mind the idea of the compliment, great shortcut, I like number six, so bag A contains one red and two white marbles, bag B contains one white and two red, a marble is chosen randomly from bag A and placed in bag B, a marble is then randomly chosen from bag B, determine the probability that the marble selected from bag B is white, huh, how many marbles are we picking, two, three, so here's what we have, it seems to me that starting in bag A, we could have been red or white, and you know what, I'll go red from A, white from A, how many red marbles are there in bag A, one out of, how many white marbles out of, okay, then we can have red from bag B, white from bag B, red from bag B, white from bag B, now down this branch we picked a red marble, and what did we do with that red marble, we put it in bag B, so how many marbles are there in bag B grand total now, not three, but, and how many are red down this branch, three out of four, one out of four, continue, you see where we're going Dan, we can actually, when I first saw this question, I thought holy smokes, changing marbles back and forth, no if it's two marbles, I can come up with every possibility, because Jordan down here, I picked a white marble, what do I do with that white marble, put it into bag B, so how many marbles are there in bag B, four, how many of them are white down this branch, two out of four, two out of four, this is a very neat dependent tree, because I'm actually changing the condition of the second bag each time, does that make sense, there's my tree, now I'm going to answer the question, they want the probability that we have that, it seems to me that it's this branch or that branch, is it not, it seems to me that it's going to be one third times one quarter or two thirds times two quarters, and conveniently I've even got a built-in common denominator, I think the answer is going to be five out of 12, how would I give out marks here, if you did the tree that would get you one mark, if you did it correctly, and then I would give you one mark for the answer, yeah it's worth two, that's actually, if you guys found that easy great, because in previous years my kids have found that really a challenge, but I'm trying to let you know we got some pretty flexible to it, specifically for two events Amy, we can solve almost anything, if this had been like replacing, picking five marbles going back and forth, that would be much tougher if I didn't do the tree, Tally find it, were you here last class, nope continue, next page, okay now multiple choice test, Jen this is a classic binomial, binome PDF or CDF question, because the odds here never change, on each question it's going to be one out of four, one out of four, so it's not like cards tree, we're going to use our binome thing, so how many questions? 12, that's also how you know this is not a tree, because there's no way I want you to do a 12 level tree now, what are the odds that a student gets none of the questions correct, now I'm going to do this both the long way and the short way, the long way was from 12 questions choose none correct, the odds of getting it correct are one and four, none, the odds of getting it wrong are three and four, 12 of those, and you can totally, yep you can totally do it that way, or if you're going to use your calculator and it's a written question Kelvin, you must write this out, show me the function and show me what went here, otherwise I can't give you full marks, and you would go 12, one quarter, zero, and I'm going to use my calculator because I'm a techie nerd, 12 comma 0.25 which is also one quarter, zero, the odds of getting them all wrong, you know what, Brett you'll probably get one right, or more, 0.0317, 0.0317, 0.032 I take that as well, three questions correct, so 12 questions choose three, and remember Jordan this is the formula that's on your formula sheet for this particular topic, it's n choose x, p to the x, q to the n minus x or something like that, that's there, but you got it built in, 12 choose three, one quarter, three right, three quarters, nine wrong, or binome PDF, 12 comma one quarter comma three, second function enter, hey change the zero to a three, I think this will be reasonable odds, yeah 0.2581, 0.26 sure, about a 26% chance of getting 25%, what does at most three questions mean, David, okay this means three or less, and this is where the binome CDF shines, this means three or two or one or zero, and you could do each of those and just add up your answers and that's perfectly valid, although it was silly, binome CDF of 12 comma one quarter comma three, that will calculate three or two or one or zero and add them up, nope can't do that Mr. Good, CDF of 12 comma 0.25 comma three, you know what you've got a pretty good chance of getting 25% or less on this test, 0.6488, what you really want to know is at least seven because seven or more means you passed, six or more technically but let's say we're shooting for more than bare minimum, so at least seven, the problem is this means seven or more, my calculator cannot do or more, it can only do or less, so I have to use the complement, I have to go one minus six or less, I have to go one minus binome CDF 12 comma 0.25 comma six, one minus binome CDF of 12 comma 0.25 comma six, Brett what are the odds of getting seven or more, not great, 0.0143, 0.0143 and of course the moral of this evidence study, which most of you do, I think the sports applications to me are the more interesting ones because I'm sure Vegas uses a more complicated version of this, but I'm sure they use some version of this, that I was stunned to find out that Kavisi does sports action or whatever it's called, I saw a documentary on the news and I was stunned to find out none of them have a math degree and they don't actually use math to set the odds, they do gut instinct, I'm sure that could be taken advantage of, I'd be a little worried. For me, I think I told you guys about two months ago a mathematician in Ontario cracked the barcodes on the scratch and wins so that about 80% of the time he could pick a winner, okay and he taught it to his 10-year-old kid, so his 10-year-old kid would go into the store with his daddy and say that one, that one, that one, that one and more often than not, they were winners. So he wrote a blog and he told Ontario Lottery and Ontario Lottery denied it, so he wrote, as it turns out in North America there are only two companies that make all of the scratch and win cards in North America, so one of them has a flaw in their algorithm and at one point was denying it except he was saying, look I'm winning far more than chance should ever allow he did it as a math exercise, actually. You know what, okay, really all you're doing though is you're ripping off the hospital charities and things, right, come on, let's think about it. If you're bored, Google, Google scratch ticket pattern or mathematician finds pattern in scratch lottery tickets, you'll find the blog in the article, it came about two months ago. For me, I think in the factory they produce all of the winners at the same time, which meant that the barcodes had some kind of numerical pattern which was, because the barcode has to identify an individual ticket, each barcode is unique, so he figured out the pattern that identified mostly winning tickets. Google it, boy this is going to be an interesting one online for people that are watching this. Yo, yep, below, below, and it only works below, yep. Okay, so at least seven, what does that include, seven or eight or nine? So how can I get instead of seven or eight or nine or 10 or 11 or 12, which I could do individually, what's the opposite of at least seven, six or less? So I'm using the compliment, I'm saying if I want to find seven or more, take 100% minus six or less, because whatever's left over should be seven or more. And how do I do six or less, that's the binome CDF. And in fact, I'm going to be honest with you guys, probably on your test, if I ask you a CDF question, it's going to be an or more question so that you'll have to clue in, I've got to go or less to make this work. Does that make sense? I like number eight, I like number eight, I like number eight, I like number eight, I like number eight, I like number eight. Okay. Number eight, we're doing two things, picking a jar, picking a marble. I think two things Jordan treat. First thing I'm going to do is pick jar one or jar two. Is it 50 50 chance of picking jar one and jar two? It was in some of the homework questions, but I told you I find that kind of boring. So here it looks like we're rolling a die. If a one or two comes up, we go to jar one. So what are the odds that we end up picking from jar one? Two out of six, what are the odds that we end up picking from jar two? Four out of six. Then we have red, white, black, red, white, black. In jar one, it's how many red? Three out of eight, two out of eight, five out of, that's what I said. Ten, can't you read my writing? You think by now, at this point in the year, you would actually be able to read my writing. That just tells me you haven't done the notes very often. He says, trying desperately to recover. All the times I've mocked you guys for doing math on your calculator and I can't add to 10. Really? Here we have four out of 12 on my right out of 12. Four out of 12, five out of 12, three out of 12. I don't think I'm going to get a common denominator here. I'm going to end up using my calculator. Here we go. Nick, what does a want me to find? What does a want me to find? You know what? Here or, here, what does or mean? Multiply down, add across. I think the probability of red is going to be two out of six times three out of 10 or four out of six times four out of 12. Is there going to be a built-in common denominator this time? No. Then you know what? Do yourself a favor. Use the technology. Two out of six and three out of 10 or four out of six and four out of 12. Enter math, enter, enter. 29 out of 90. B. Nicole, what's the first word in B? If give, you know what? Given conditional. And here's also how I know if what? The bottom level, find what? No, find the probability what? Top level. We're going backwards up the tree. This is how I recognize math. This is a conditional probability question. As a statement, it's going to look like this. Given red, find star one. Right? Now, this formula is on your formula sheet, but it sucks. I've given you what I think is a better way to remember this. Do you remember it? It's going to be the probability of what? Both over given. What do I mean by both? It's going to be the probability of one and red divided by the probability of red. Now, one and red is this branch. Right? It's going to be two out of six and three out of 10 divided by. Now, the probability of red, Madison, is both of these branches, and I could calculate it. But conveniently, I think I just did it over here anyways. 29 out of 90 is what I would get if I calculated it. But in fact, I'm going to get six out of 60 over 29 out of 90. Now, if you go to your calculator on this line, you do have to be a bit careful. Amy, you're going to have to put the top fraction in brackets divided by, and then put the bottom fraction in brackets. Otherwise, your calculator won't know this is a four-level fraction, or you could simply say, how do I divide by a fraction? Flip it and multiply, and then if you write it as a multiplication question, your calculator won't freak out. I'll go to my brackets. So, bracket six over 60 over 60, Mr. Doock. Six t closed bracket divided by bracket 29 over 90 closed bracket. Enter math, enter, enter. Is the final answer nine out of 29? Yes. Now, each of these, oh, no, for some reason, I only made this worth a total of two marks. On a test, that's a probably worth four marks, two marks each. For now, I'll give you a half mark for the tree, a half mark for this answer, a half mark if I saw this statement or that statement, and a half mark for the final answer. And if you would be so kind as to give yourself a score out of count them 23. Oh, is the back page, I'm not done? Oh, yeah, there's even more. That seemed a little short. Oh, my favorite. Bend diagrams. Cool. How many dots are there? 19, okay. What's the probability that A and B occurs? Two out of 19. What's the probability that A and not B occurs? By the way, another way to say this in English is A only. How many dots are in A only but not in B at the same time? Four. Oh, by the way, we're falling back Jasmine on our very first rule, which was if you can count it, you can solve it, right? Neither A nor B. I think that's these eight, at least one of A or B. That means one or the other or both. One, two, three, four, five, six, seven, eight, nine, ten, eleven. I think 11 out of 19. At most one of A or B occurs but not both. Dylan, what do you think? At most one but not both. It means don't count the overlap. Nine, yes? Okay. I think F and G are conditional probability questions. So I'm going to set them up as a statement. F, probability, what's the given, which is going to be A and B divided by the given one. I think we already did A and B. What was A and B? It's listed there. What was A and B? Two out of 19 divided by, what's the probability of B? One, two, three. This isn't B2, is it not? Seven out of 19. How do I divide by fraction? You know what? Two out of seven. Now that I see that, I say, oh, what they're really saying is, if you know you're in B, how many are also in A? Two out of those seven. Gotcha. B occurs given that A has occurred. I think the answer is going to be two out of six, but let's see if I can prove that with the conditional equation. It's going to be B given A, which is the probability of B and A divided by the probability of A, both over the given one. B and A, oh, that was two out of 19 because AND was the overlap, divided by A, six out of 19, and lo and behold, I end up with two out of six or one third. Conditional, given, if, suppose. Last one, I think, yes, oh yes, the defective question, which I like, I like, I like, I like, I like. I think I have two versions of this test. One, for its conditional probability question, has the marbles one that we did a couple of questions earlier. One has factory defective. This is a nice industry kind of an application. Machine A, Rio produces 60% of the product, while Machine B produces 40% of the product. 3% of the pop production from Machine A is defective, while 2% is from Machine B is defective. Let's do a treat. First, we can be from Machine A or Machine B. Adam, what are the odds that we're from Machine A? 0.6. What are the odds that we're from Machine B? 0.4. And then we can be defective or not, defective or not. As a decimal, what are the odds that were defective if we're from A? Now, I love it because about five of you made the one dumb mistake that will drive me crazy on this test. I heard my friend Dan say it and correct himself. He said, oh, 0.3. Now, what is that as a percentage? That's 30 and everything else is going to be wrong in garbage because you're going to put a 0.7 there. So don't make that mistake. What is it properly, Dan? 0.03 and 0.97. And down B it's 0.02 and 0.98. And then if they want the probability that it's defective, I think I'm pretty sure that's this branch or this branch. Rhea, what does or mean? Plus it's going to be 0.6 times 0.03 or 0.4 times 0.02. I think I could almost do this in my head. 0.6 times 0.03 I think is 0.18 plus 0.4 times 0.02, 0.08. Is the answer 0.26? No? Yes? 0.026? No, 0.26. Isn't it? Oh no, I'm missing a zero. You're right. I'm off here. Extra decimal places, Mr. Dewick. Thank you, David. I can't do this in my head apparently. It would be this. Is that right? Whoo-hoo! Which makes sense because I was getting a 26% chance that it was defective. But wait a minute. It's three and it's two. How are you getting 26% out of that? B, B. Megan, what's the first word of B? If given, it's conditional. Given what? Given defective. Oh, which is the bottom branch. Yeah, we're going up. What's the probability it came from? B, which is going to be both over the given one. It's going to be B and defective divided by defective. B and defective, I'm pretty sure is that branch there. I'm pretty sure it's 0.4 times 0.02 divided by defective, which I think we just figured out it would be both of these branches. But since I already did the arithmetic Madison, I'm just going to put a 0.026 there. And since it's decimals, I'm going to my calculator. 0.4 times 0.02 divided by 0.26 and I get curses, red baron. And I get that or 4 out of 13 if you went as a fraction. I'm guessing most of you went with decimals 0.0377. No, 0.3077. How about learning to read, Mr. Dewick, as well? Wow, am I stopping? How many marks is this worth? Three? I would probably go like this. One mark for the tree, one mark for the answer, and one mark for this answer. Now I think you can give yourself a total score. Correct me if I'm wrong, out of 23. Can you not? And if you need to lawyer with me, now is the time.