 Ding right there questions from the homework. What would you like me to go over now is your chance to ask first from page 440? Whatever this is 443 and then I'll do a couple for the other one as well. There was some nasties. I know oh, but some cool ones Which ones oh? Come on. Yes, you did raise your hand. I was gonna say the homework excellent. I'd love to do number eight Okay Hmm one of the cards is red and the other is black how many cards no no how many cards we picking To treat more than two we'll learn other stuff three I'll sometimes still do a tree, but we did say a tree gets pretty unwieldy when we have three or four or five So and here's my treat. I figure Black on the first one. You know what instead of doing not black. Why don't I just do R for red? Red on the first one second card could be black again red again black again red again and You know what I'll even put a one there for the first and the two there for the second how many black cards in the deck so this is gonna be 26 out of 52 26 out of 52. Oh down this branch. You picked a black card. How many are left in the deck 25? out of 51 26 reds out of 51 Here you picked a red. So you got 26 blacks out of 51 and 25 reds out of 51 Double check to the add to one to the add to one to the add to go back to an error check One card is red and the other is black. Here's what I think. I think it could be this branch or That branch what does or me? So it's gonna be this times this plus this times this come in Howdy, come on in. I'm just gonna hit pause here quickly Sorry about the interruption for those of you following along at home. You're good. It's that makes sense. Okay? By the way, could you now that you see this could you have visualized the tree without drawing it? Sometimes for two events if they're fairly simple events, I'll do that in my homework When in doubt though the tree makes it so clear Okay, and yeah, the shortcut is this times this times two Apparently any others from page 445 and calling. Okay, then I also assigned some questions from page 450 ish and These ones are tougher, but in my mind also, I got to be honest gen these ones are nerdily cooler in My mind they're nerdically delicious or something like that Any I just made that up that good good. I'll use that for the remainder of my teaching career. Okay Any of these you want me to go over Because I know all of you did the what's the probability of a student waking up on time for class question. Yes Some of you did it three or four times and we're just make yeah, okay any of these you want me to go over Yes, sorry. I didn't see you there Five see I would love to do five see Okay How many people are there in this question breath three? Boy, I'll do a tree. This is about the limit of how big a tree I'll do. It's gonna be a fairly complicated tree it's Andy can solve it or Not Berry can solve it or not Berry can solve it or not Curtis can solve it or not Curtis can solve it. Don't put a Or not Curtis can solve it or not Curtis can solve it or not and I'm gonna cheat Because it's such a long tree to draw I'm only gonna fill in These branches. I'm already gonna put a check mark under only one here all three solved it You know what only one guy solved it on this branch right here only a solved it What's another one that has only one a here? Right didn't yes didn't and yes. Yes. Sorry. No. No. Yes there Okay, let's fill in just those branches as a time saver and this is one way you can kind of be in a cut corners a little bit They're independent so it's gonna be one-third two-thirds It's gonna be one half one half Oh, I'm filling in everything for some dumb reason after I told you I was only gonna fill in part of the branches I got carried away with the mathiness of it. Sorry, and it's independent. So it's gonna be three-fifths two-fifths three-fifths Not three. Mr. Do it Two-fifths, and you can see how cluttered it's getting and this is also why a three-level tree is a bit much but it works Multiply down at a cross so this or that or that it's gonna be One-third do any keep going or you take the rest of the way yourself, right? So I keep going. Okay. Love to times one half times Two-fifths, that's That branch or what does or me? I love it Two-thirds, you know what mr. Do it cleverly highlight since you've got this fancy technology two-thirds one half two-fifths or Two-thirds one half three-fifths Now you can pull out your graph and calculate if you need to I think all of these is gonna be over six times five All of these fractions are gonna be over 30. There's my common denominator, and it's gonna be two plus four Plus six which I think I can do in my head two plus four is six Oh Well about a 30 which in lowest terms divide by six divide by six is is it two out of five? Is that what says in the back? Woo-hoo There you go. Okay That's about as much as I'll tackle with a tree three different Events and three levels. That's a lot of work There is a way to do this coming down the side with combinatorics isn't tight any others. They're good So we're gonna move on to What to me is the coolest or one of the funnest lessons of the year and also one of the most counter-intuitive So let me press pause while I'm handing stuff out. I hit pause lesson five Conditional probability and it correct me if I'm wrong on your notes on the back page the right part of the page is blank Yes, so we're gonna do just to kind of jog our memory here a Generic kind of a tree so if you can go to where the blank section is down here and As a little heading you can write generic probability Tree and all I'm gonna do is I'm gonna fill in a tree Completely with symbols so that if you even get a question with symbols on it, you can oh, they're talking about this branch Okay The cameras way and closer to me because I need my caffeine So we're gonna have two events events A and B generic events. So we would go like this a What am I gonna write over here? What's our symbol for not? Not a and then we're gonna have event B whatever it is And not be B and Not be and then what we introduced last day Jen was the idea of waiting the branches We put numbers here in terms of symbols The number that we put here was the probability of a occurring So in case they give it to you that way, you'll know that's that number What would I call this as a symbol the probability of take a guess? not a okay, and Then we introduced the notion of conditional probability or given that we called this The probability of B occurring given that you're down the a branch Given that a has occurred. What do you think we call this the probability of? B not occurring Given that you're down the a branch. All right, let's see if we can use our math nerd brains What am I gonna call this one here? And I'm gonna have to kind of write small wait a minute, mr. Do it there's an advantage to the software mr. Do it Where's your lasso right there? Move that over a little bit Boom boom ha It's an ugly tree, but I want it to be big enough so you can read things. What do you think I'm gonna call this one here probability of B given that a did not occur and David since you're on a roll. What am I gonna call this one the probability of? B didn't occur and I already know that a didn't occur. Oh And we said you know they're independent if these two branches match those two branches That was the easy definition of independence as opposed to thinking to yourself does it affect or the yucky math definition? What we're gonna look at today instead of saying what's the probability of B given that a have occurred filling in this one What we're gonna say is this if we know this occurred Which could be this branch or this branch? What are the odds that I came down this one, but not that one we're gonna go backwards up the tree Okay, that's our goal. So here's our generic tree and today is going backwards That's about as ugly a tree as I could have drawn, but I'm gonna leave it because I'm an imperfect teacher I'm not embarrassed by it So a Pot is rent that we're gonna start out first and we'll get used a little more complicated Then we're gonna move backwards up the tree here. We have two separate little pots We're gonna pick a pot first and then we're gonna pick a bill and It wants to know what's the probability that we get a $10 bill, but you know what let's fill in our tree So the first first event here is we have to pick a pot. We have to pick pot one or Pot two how many pots are there? What are the odds of picking pot one? Yeah, now sometimes what they'll do to make it more interesting is they'll have you roll a dice And if you get a one or a two you'll pick pot one and a three four five or six You'll pick pot two in which case it's not one or two, but this one. I think is straight 50-50 Then what are the possibilities from each pot? I can get a zero dollar bill and I know we taught but a zero dollar bill a Ten dollar bill and a twenty dollar bill. So I think I'm gonna have zero ten or 20 zero ten or 20 right Let's fill in the weights if we're in pot one down this branch What are the odds of getting a zero? One out of three, what are the odds of getting a ten? One out of three, what are the odds of getting a 20? One out of three double-check is that to one? Yeah, got it, right? And that's why Brett I rarely don't do the whole tree that built an error check to me is so handy That's why I started doing it instinctively when you asked me that question earlier. Oh Now we're in pot two What are the odds of getting a zero? One out of four, what are the odds of getting a ten? Yeah, what are the odds of getting a 20? One out of four. Okay, we've got our treat and you'll find you get pretty quick at drawing these Now let's answer the questions What's the probability of a ten dollar bill is chosen if it's known that pot one is the selected pot? Okay, here's what this is saying if is another word for given that it's actually saying probability of ten Given we're down one. That's why I did that generic tree earlier. Can you see which number they're talking about? Which number is sitting in the ten given one branch? one-third It's not multiplied down. They didn't say what are the odds of getting pot one and a ten dollar bill They said look, you know, you're in pot one. What are the odds of getting a ten? two What are the odds of a ten dollar bill if you know you're in pot two in terms of our notation? It's a ten given that two has occurred sorry two out of four What's the probability that a ten dollar bill is chosen now? They have think probability tree diagram We already did ours here. What's the probability that a ten dollar bill is chosen? It really means this branch or This branch and here is where we're going to go multiply down At a cross the probability that a ten dollar bill is chosen. What we're really saying is in Our notation it could be pot one and a ten dollar bill or Pot two and a ten dollar bill and I'm not a big fan of the notation I like the tree, but Madison I want to make sure if they use it you're kind of not terrified by it Comma means and what does and means Times what does or mean at it's gonna be this times that it's gonna be one-half times one-third or One-half times two quarters Is there a built-in common denominator here? Then you know what I'm gonna wimp out and go to my graphing calculator And it's not that I don't want you to know how to do fractions. It is that I'm interested in saving time So I'm gonna go one-half times one-third plus one-half times Two quarters and then how do you get that? Oh, yeah math and or and or five out of 12 piece of cake Okay, here is the question. We're gonna try and ask today Suppose David has just played this game, but I didn't watch him, but he's holding a ten dollar bill What's the probability it came from pot one? Can you see what they've done in this part D? They've told you the bottom outcome and they're asking about the top outcome Going backwards Okay Says begin by translating this as a conditional probability probability of Let's see What do I know here? What's the support by the way if that's the given? Given that we have a ten dollar bill What do they want us to find? One so the first step to doing conditional probability And this is the only time that I'm gonna do this compute completely with a formula is Translating it into probability speak Yeah Sure Although if you were really able to concentrate and you were a good student you should be able to handle the district shut up Mr. Duke, okay fine my bad Do me a favor this is because this is the only time I'm using the formula Can all of you either find your formula sheets or flip to the inside cover of your workbook because it's there too and What I'd like you to find is the formula that says this The probability of B given a it's on there Which by the way is really what we have here with different stuff Be given a is on there. Is it not is a given B? Okay, sorry. Sorry. Sorry my bad. I couldn't remember is it a given B and What does it say? It's the probability of a and B divided by the probability of B That's the formula that we're gonna use but I got to say that's fairly ugly Because I always get confused by the letters So I'd like you to write this down and then I'm gonna give you mr. Dukes handy-dandy truncated version of this Okay, so there's the formula I Don't remember it that way. I Remember it as when they give me a I got to translate this statement I look for words like if or given or supposed as another word that they use and I say it's the probability of both Over the probability of the given I Remember both of you It's much easier for me to keep track of the letters because Brandon often The a is where the B is and the B is where the a is and again can be forget it I'm not gonna wrap my brain around it. I know that for conditional probability It's gonna be the probability of both over the given one. So let's go to this question that we have here The one that we're actually trying to answer if David won ten bucks What are the odds that it came from pot one? Given that ten dollars has occurred. What's the probability that it came from pot one? That's going to be the probability of Another word for both is and One and ten Divided by the probability of what ten now. Why is that so helpful go look at your tree? Do we have a branch that has one and ten? Walk down that branch because and means what? Multiply down what numbers are gonna go on top of the fraction here two fractions. What are they? one out of two and And you know, it's the one-third because this says both. I know I'm going down the one column not the two call over Ten Now ten would be that or That what does or mean? Oh, wait a minute. Wait a minute. Didn't we just figure that out right here? Conveniently they snuck in this is the probability of ten Right, we did look at both possibilities Multiply down add across in fact the number that's going to go here is just five out of twelve Don't reach for your calculators yet the top Multiplying fractions is the easiest operation top times top bottom times one one out of six Divided by five out of twelve boys and girls. How do you divide by a fraction? I couldn't a pinch do this whole thing by hand. It's gonna be one out of six times twelve out of five so 12 over 30 or in lowest terms, which is what they would put by the way two out of five Strangely enough the same answer as Brett's question earlier today. That's a fluke But conditional probability going backwards if they tell you the bottom has occurred And they want you to figure out the odds that one of the top things has occurred It's both over the given one example two One of the places this is used is an industry to track defects as a matter of fact There was a mathematician back in the 1950s who was doing of his doctorate on probabilistic sampling methods and so he went to the big three automakers for GM in American Motors. He said look I Can guarantee that if you let me go through your factory I can teach you how to set up your factories using probabilistic sampling methods so that you'll catch more defects before they go to the lot in your cars You'll catch more mistakes before they leave your planet Ford GM and American Motors were not interested. No, thank you So okay So he went across the sea to Japan and he pitched the idea to Honda and Nissan which was Datsun at that time and Toyota and they said this is great You mean we don't have to tell our workers to do any better. We don't have to change anything We'll just catch more stuff mathematically. Yes, it won't cost us anything. Nope and That's why in the 60s and in the 70s and in the 80s Japanese cars had a reputation as being better made. It wasn't that they were coming off the assembly line better They were catching more mistakes Mathematically they were able to if they found a mistake go backwards up the tree and find where it come from something like this The American Motor companies didn't start applying a lot of this until the early 90s and they're still trying to catch up So here's one place that it certainly changed several countries economies a Company has two factories that make computer chips cars if you want to go to the 70% of the chips come from factory one 30% of the chips come from factory two in Factory one 25% of the chips are defective in factory two 10% of the chips are defective Suppose you don't know from which factory a chip came. What's the probability that chip is defective? What they're really saying is find the probability of how about the letter D for defective? I think this is the same as the previous question. There's two events pick a factory and Then pick a chip So I'm gonna do a treat. How about F1 for factory one What would be clever to use for factory two? F2 you say What's the probability that something come that a chip comes from factory one read the question? 70% which as a decimal is 0.7. What's the probability that comes from factory to them? 0.3 then from factory one we can be defective or not defective or not What's the probability that a chip is defective given that it's from factory one? Point two five what about not defective? Use the compliment right this this is where the compliment also comes in really handy Okay, given that you're in factory two. What are the odds that the chip was defective? Point one and Point nine So this first question is simply saying what are the odds that it's defective? Well, look how many branches end in defective. I Think it's gonna be this one or This one What does or mean? Multiply down at a cross in fact now in my probability speak This is really the probability that we're defective and Factory one given factory one or the probability that we're defective. Sorry not my bad. Mr. Do it not given and because we're multiplying down we're multiplying down probability that you're defective and Factory two now there's the probability notation, which I'm not a fan of I can see from the tree It's gonna be point seven Times point two five or Point three times point one and yes, you're allowed to use your calculator this time I Will tolerate going to a calculator for basic decimal if you have a stack of chips You don't know What's the odds that it's defective one you pick one at random Jordan what do you get? Okay, sorry Amrit what'd you get since I see you racing for your calculator? Blinding speed off the blur blur. Oh my hair is getting blown back. I feel the wind. I feel the wind What do you get folks? Point two oh five or if they wanted you to go to a percent Twenty point five percent by the way, I'll take both answers But if it's multiple choice you better be able to if they want a percent be able to recognize the percent answer, right? The more interesting question to me is B. Oh by the way, I like this question. I like this question. I like this question. I like this question. I like this question Why do you think I'd say that Dylan? I think there's why would I say that I like this question? Sorry, what? You didn't hear me say it was gonna be on the test. You're just a good student who notices those things Okay There enough Brainiac that's absolutely when I think of you that the first thing that pops into my eye. I almost got it straight I almost got it out. Sorry my friend. I came close though Always Read B to me kiddo Stop That's also an especially for my ESL students. That's also another synonym or trigger word for given The three words they use most often are suppose that or if or Ideally given that's my preferred trigger word. I usually will use that but Start over kiddo. We need to translate this into a probability statement What's the given? What do we know? What's the suppose? No, what's the suppose? What's the given? The given is that it's defective. They've told me the bottom outcome That's also how you can recognize that we have to use this. What do they want us to find? Okay, we're going backwards up this tree and how are we gonna do that? We can use the fancy schmancy complicated Conditional probability formula from our formula sheet or mr. Dook said it's really easier to remember. This is going to be Both over The given and by both I mean and It's gonna be defective and Factory one divided by just plain old defective Now the nice thing here is what did we find in part a the probability of what? We actually found the denominator already They won't do that always but often part a will be find this and then part B will be a conditional and Hey, great defective, which is point two oh five defective and f1 oh point seven times point two five Put your pencils down and look up for a second Once you've written that down Just for what it's worth Good, you may you know I'm big on built-in error checks this point two oh five came from 87 times point two five Plus point three times point one it came from this right First of all, can I cancel have I so can't cancel but if you are doing a conditional Probability one of the built-in error checks is the both will always appear somewhere in the bottom If it doesn't you messed up or miss something Having said that I know this is point two oh five I'll leave this here in the notes. You guys just wrote point two oh five. So the answer is going to be Point seven times point two five point two five point two five just to do it divided by point two oh five What are the odds that it came from factory one? eighty five point three sorry eighty five point four percent point eight five four point eight five four or eighty five point four percent turn the page a few more and then Maybe I'll save your life. What really yeah example three oh We're picking two cards without replacement got you they want the probability that the second card drawn is a king You know what it depends What does it depend on? First card is a king. So let's set up our tree. We could have King on the first Not king on the first king on the second Not king on the second king on the second not king on the second Let's fill in the branches How many kings are there in the deck out of how many non-kings? Please don't count use the compliment 48 out of 52. All right down this branch. We picked a king We picked up We picked up We picked a king Ha ha how many kings are now left in the deck Out of how many non-kings 48 out of 51 now here. I didn't get a king How many kings are left in the deck? four out of 51 How many non-kings? 47 out of 51. Okay, we got our tree and Trust me the trees get pretty quick to draw they want the probability of King to how many branches end in King to two of them multiply down out across four out of 52 three out of 51 King one and King two or First card could have not been a king Good gosh, mr. Do it. Let's fix that make that a proper two and second card could have been a king now We noticed yesterday that 52 times 51 was 26 52 I've got that one memorized because it shows up so often the top is gonna be four times three 12 plus 48 times four 196 12 and 196 is 196 and 12 is to is it 208 someone check my math. I'm doing it all my head. Am I no 204 208 is it 208 208 thought I was wrong, please. I never make mistakes All right. I have a doubter here in the front row or times three plus 48 times four is 204 I Am wrong Mr. Do it David. Did you set me up for that because if you did I say to you well played? Yes Yes, you may hustle that kid. Oh, where did I watch that because it's not a hundred ninety six mr. Do it because 192 plus 12 Now that question Nicole we actually did yesterday last day What's the first word Nicole of part B conditional? Okay, so let's set up the statement Given what's the given that they gave me you right say louder in my notation. What is it? K1 not K1 K2 not K2 which K2 right What's the probability that what? What's the probability that the first card was? Not so what am I gonna write here? Not K1, okay I'm getting a bit lazy Jordan. I'm actually just gonna write that's both over the given Okay Which branch has both of those that's what's going to go on top? This one has both this one. Yes Yes, yes, yes, it's gonna be 48 out of 52 and four out of 51 Yo Yeah Try see if you can finish the rest of this on your own my children We're back. By the way, did you guys actually calculate K2 or did you just use this number from here? You could have just dropped this ugly fraction But in case you hadn't done part a ahead of time you could have just said oh get K2. Oh that's whoop Turn the fan on mr. Dukes That's K2 Or that's K2 you could have excuse me. I got the hiccups of me. You could have gone four out of 52 and three out of 51 or 48 out of 52 and four out of 51 However, since Jen we already figured out what this works out to I'm gonna this time erase it and I'm just gonna go to saying hey, this is 204 divided by 26 52 I like to for these because they're multi-level multi-level fractions I'm a little concerned about how my calculator will actually work So usually what I do here is work out the whole top in lowest terms I go 48 over 52 times 4 over 51 enter math enter enter this is 16 out of 2 21 over 204 out of 26 52 and then I would say I'm gonna go the first fraction in brackets 16 divided by 2 21 divided by the second fraction in brackets I'm a little paranoid with multi-level fractions I need to play around with my calculator to see whether I can lose the brackets or not You get 16 out of 17 Numbers questions see Second card drawn as a face card so we could redo this Except instead of a king what would we have here a face card? What's the probability of a face card how many face cards other than 12 and 40 11 we could do this and that would be face card second card and then First card was a face card given that second card was not a face card. You'd actually be doing Second card not first card was you'd be going this way up the branch I'm going to pass on that for a second instead I'd like you to scroll down here below to where you wrote your generic probability tree A couple of conditional probability questions that the math nerd within me just loves tests for Diseases so write this one down, please Okay, suppose there is a Terrible disease called Vitalitis Just making something up off the top of my head the test for this disease Is 98 percent accurate And there have been numerous studies that have been done in the population and you know that two percent of the population suffers from this horrible Insidious Well, okay, we'll leave the adjectives off because I could fill a page from this disease Okay So you know that 98 percent of the population is healthy by the way I'm doing this humorously on purpose but replace the italitis with cancer with hiv with anything you want to something more serious Suppose you test positive What is the probability that you actually have the disease And the sad thing is Most people if they know that the test was 98 percent accurate And they know that 98 percent of the population doesn't have it They assume if they test positive There's a 98 chance that they have it and sadly some doctors think that too and give advice that way Let's see what the real answer is So there's two events here. First of all, you have the disease or you don't And then you test positive or you don't Let's define D You have the disease And how about p Test positive and remember what I said and a lot of people will think 98 chance that they've got cancer hiv whatever Oh, and by the way the tests I've made up here Medical tests are normally nowhere near this accurate. So I'm going almost worst case scenario or best case scenario Let's set up a tree So the first issue is you can have the disease Or not What percent of the population has the disease Point zero two By the way, the most common dumb mistake I see is kids do this That's not two percent. What percent is that? It happens every year to the point where I've made sure one of the probabilities that I give you is below 10 Just to tempt you to write it wrong telling you that's on your test. Don't do that What's going here then Okay, and then you can test positive or not positive Or not now in medical language If you have the disease and the test doesn't catch it. We call that a false negative If you don't have the disease But the test says you do we call that a false positive If you have the disease should you test positive or not? If you have it, okay 98 of the time you should test positive 2 of the time you'll get a false negative reading Down this branch. You don't have the disease Should you test negative or positive then? 98 of the time you should test negative 2 of the time you should test positive you get a false positive Because no lab tests are perfect ever What's the given in this question? What's the suppose? Okay, we want to find Suppose you test positive What do we want to put here? What's the probability that you? We want to find d given p. Can you see we're going backwards up the tree? And you know what it's going to be it's going to be both over the given one so you're ready This is going to be both over The given one both d and p at the which branches both d and up Boom boom point zero two times point nine eight That's both over Dan, what's the given here letter p which stands for? Okay, you know what that's this branch or This branch What does or mean? that So it's going to be Point zero two times point nine eight or point nine eight Times point zero two How many numbers do I have in the denominator? Put the bottom in brackets and get out your calculators I guarantee the answer is surprisingly less than 98 percent What do you get what? What do you get I got to try this myself? This is one of the most counter-intuitive questions you'll ever come across point zero two times point nine eight plus point zero two times point nine eight I put the bottom in brackets check I've typed it in right check You're telling me I only got a 50 chance of having the disease That's not bad odds And here is the horrible horrible tragedy And I apologize. I'm going to get a little serious here every year People find out they've tested positive for something And they kill themselves Because they don't want to face it when they don't realize the conditional probability map Doesn't mean what they think it means So what do I say to all of you if you are ever going through a medical situation? Get a second test always Oh jasmine if you extend this tree One more level and then you work out the odds of if you've tested positive twice Then it's much closer to 92 or 93. Okay Oh also Being a bit judgmental if you ever hear of let's say an athlete or a celebrity testing positive for a drug Suspend your judgment until you hear the results of the second test Because when so-and-so test positive at the olympics It's actually Might mean there's a 50 50 chance that they didn't Now Realistically probably not it's the olympics after all but suspend your judgment and certainly in medical tests. Okay See this question. We're going to just change it slightly So suppose a lie detector is 98 percent accurate And two percent inaccurate And you know that in the court of law 98 percent of the time people will tell the truth Suppose the lie detector caused it calls david a liar in the court of law. What's the actual probability that he's lying With a 98 percent accurate lie detector It's only a 50 50 chance that he's lying. So polygraphs. I'm sorry And the reason is because when it comes right down to it Most people are going to tell the truth You're going to get a bunch of false positives and that's what skews this result Come on. This was some nerdly cool stuff. Hey, huh? I really I really Pardon me. What was that? Oh, I mean polygraphs. I mean they're sorry By the way, the only way you could get these numbers accurate is to make the polygraph 100 accurate Do you really think lie detector tests are 100 accurate? In fact, let's be clear. Do you really think they're 98 percent accurate? I doubt it I doubt it and medical tests as well are not 98 percent accurate. So So The fact that you have lots of false positives in a conditional probability question can dramatically skew your results Absolutely I would hope so Well, no, the reason they don't use them is Constitutionally, you're not allowed to incriminate yourself. I'm worried with the current climate with 9 11 or not that they might want to And they don't realize Mathematically they're garbage Garbage why because most people will tell the truth Almost everyone will tell the truth in the court of law with lie detector tests and unless it's perfect then that means you're gonna have a Bunch of false positives people are going to be found guilty of lying when they're not And that could really ruin someone's life, right? So what's your homework? Well, first of all, stay healthy and tell the truth Okay Questions to try now. I wrote page 461 to 8 and page 468 I see you guys tomorrow tomorrow with the short. Oh, I don't know. I don't see you guys tomorrow Yeah, I do it's a short friday, and it's a tuesday schedule I may for you guys press pause. So i'm going to give you a bunch of homework But I don't think i'm doing a lesson tomorrow So page 460. Did you learn something new? Hey, huh? Page 460, um I'm going to assign number one. I hate the wording of number one, but it often shows up. It's weird english So i'm going to assign it I said to try number two I'm actually going to give number three instead now number three instead of using a tree You're going to fill in the chart But from the chart you can get all your probabilities by saying how many out of how many how many out of how many And visualize the tree. So that's good practice Four is good Five is a disease question So five yes Six is okay Seven is a bit overkill. It's a long question. Although it's nearly cool Eight is good. So that's on page 460 Then on page 468 so in page 460 on almost all of those you're still going straight down the tree Only a couple of times you're going backwards The fancy term for conditional probability Is bay's law Oh remind me tomorrow. I need to tell you why had marsha clark had a mathematician on their staff OJ simpson would have been found guilty because the defense used a bad conditional probability as one of their main arguments That's tomorrow See how I left you hook. Dangle there Okay Number one Four, uh, no, I'm gonna nuke five. Sorry. Why did I nuke five because I didn't assign it here? Why did I not assign five? Oh, yeah Now let's get five I think the one I'm looking for Yeah, that's nice number seven because it's university and passing exams, which some of you might be interested in But it's a three level tree You know what I'm gonna do instead I'm not gonna do the three level tree I am going to assign number eight. Now number eight requires you to turn back to page 461 Okay Last thing so i'm done the lesson and if you need to go to the washroom now is the time however Have you ever been standing in line like at the supermarket? And you look at the line to your right And you look at the line to your left And you're convinced that the fates are conspiring against you that no matter what happens Your line is traveling slower Is that your imagination as it turns out there's a valid probability argument that suggests That's not see if you could this this math is at your level. See if you can follow it I'm sure this week you've been trapped in a slumbered line in this box I have a gift which shows how to choose stores with the shortest way It's in old-fashioned telephone. Now, of course, if you're using the phone it's important Okay, I have to pause for a second because apparently I can't have all this software open at the same time So i'm gonna go right click