Okay, let's get started. So welcome everyone to the metal plus production. Oh, no, it started the mysterious disappearance of the makeup and lesson and paradoxes and fallacies and probability and statistics. I'll let the gears now take it from here. Welcome, everyone. So first of all, let's start off by saying congratulations to all the 2023 math anger winners, all of you. Yeah, I'll introduce myself. So hi, I'm bigger. I earned my bachelor's and master's in electrical engineering in computer science, all in four years last year from Stanford. I'm one of the co founders of meta plus. In the past I've interned at large companies like Amazon and Microsoft. And currently I'm working as a software engineer at a Silicon Valley startup, not too far from Stanford. And like many of you have been participating, I have participated in math kangaroo for several years. And so in the well, I was still a student I placed first place internationally twice and top 10 nationally several times. Hi, everyone. I'm happy. Yeah, I'm bigger. It's older sister. I did my bachelor's and master's in electrical engineering computer science in four years from MIT. I'm also co founder of meta plus. I currently work as a software engineer at Microsoft. And when I was a student, I placed top 10 nationally math kangaroo. So today, our presentation is going to be focusing on probability and statistics. And while I'm sure most of you are familiar with the basics, I think it's important to have a shared vocabulary. So let's just start talking about events. So a lot of the field of probability and statistics is concerned with random events. An event is basically the set of the events or a set of outcomes of a random experiment. So there's different types of events. So for example, you could have independent events where the result of the second event is not affected by the result of the first event. For example, if I roll a pair of die, the number rolled on the second die isn't influenced by the first die. And so I can keep on rolling dice as many times I want. And all of the probabilities of rolling a six, for example, each time is completely independent from the probabilities of rolling dice previously, it's not dependent on those results. However, there are also dependent events where the result of a second event is affected by the result of the first event. So for example, you may be scoring well on a test because you studied for it. So the probability that you study for a test and the probability that you score well on that test aren't completely independent, they are dependent. Then there's also mutually exclusive events. So these are two events that cannot occur at the same time. So for example, a car turning left and right, that cannot happen at the same time. A dice, rolling a pair of dice and getting a pair of sixes and a pair of ones, that's something that can occur at the same time. Now, mathematical paradoxes are specifically, these are mathematical conclusions that are so unexpected that it's difficult to accept exactly what the reasoning is, even though every step of the reasoning is valid. So this is not, these aren't incorrect, they're just very counterintuitive. And we'll see examples of those later. But I'm sure you've faced or seen examples of such logic in your own life. Right now I'm in the right now I'm in the workforce. So something that they always tell us while we're in college or high school is that there's there's this sort of like paradox, a catch 22, where you need experience to get a job, and you need a job to get experience. And it makes sense, both both parts of that seem to be logically consistent in the sense that, yeah, you need a job to get experience, of course, and who's going to hire you for a job if you don't have experience, so you need experience to get a job as well. But as you can see, then it's very hard to enter the loop, which one do you get first experience or a job? Then there's also mathematical fallacies. So this is improper reasoning that leads to an unexpected result that is patently false or absurd. So you may have seen a proof that has some sort of flawed reasoning, maybe, for example, you divide by zero in order to get a completely absurd result, like one equals two. And we'll see some examples of like flawed reasoning and logical reasoning later on as well. I'm so sorry, guys. I'm so sorry. Hello? Oh, I'm so sorry. It's my friend, Dr. Roo. Dr. Roo, I'm in the middle of a presentation. Oh, oh, okay. I see. Dr. Roo wants to say something to all of you. So this is my friend, Dr. Roo. He's a famous world renowned mathematician. I'm Dr. Roo and I was invited to a party at a haunted house. All my friends were invited. Dr. Fisher the Kingfisher, Miss Paws, Nostradamus the No-Fork Terrier, Anthony the Fox, Mr. Whiskers, Florence Nightingale, and of course, I, Dr. Roo, the greatest mathematician in the world. And we were all competing for the famed MacGuffin Macaroon. Anthony the Fox, Mr. Whiskers, and Florence Nightingale were all particularly exemplary players who had at least won one coin during the night. Anthony the Fox had been winning all his games. We were enjoying our game when suddenly the MacGuffin Macaroon, the legendary sweet we were competing for, disappeared from its screw shut glass box. Someone in the room must have stolen it. But who? I need your help. I'm Dr. Roo. So Dr. Roo needs our help. So I'm sorry, I know we were going to talk about probability and statistics, but we might have to put a pin in that for a second. Let's let's talk to Dr. Roo, figure out the facts, learn about who was at his party. So these are the facts that Dr. Roo sent over to me. He said that he and his friends are present at a hundred person party where they were playing card games. And at the end of the night, the individual with the highest number of games won would have received the MacGuffin Macaroon. Anthony the Fox, Mr. Whiskers, and Florence Nightingale were all showing themselves to be great players. And Anthony the Fox has won every single game he's participated in. But now it turns out the MacGuffin Macaroon is stolen. So are you guys willing to help out help Dr. Roo figure out what happened to the Macaroon? Anyone in the chat? I'm going to assume that's a yes. Because silence gives consent, right? I mean, okay, great. So thanks for climbing on board. Dr. Roo really appreciates your support. So this is just a diagram of Dr. Roo and his friends. So here you can see Ms. Paz, Dr. Fisher, Florence Nightingale, Mr. Whiskers, Anthony the Fox, Nostradamus, and of course, Dr. Roo at the center. Okay, but first, Dr. Roo needs us to prove our probability and statistical thinking ability. So he's given a question to start us off with. Given a fair coin and four coin flips, which sequence of events is more likely to be an actual sequence of coin flips? A heads, heads, heads, heads, B heads, heads, heads, tails, C heads, tails, heads, tails, D heads, heads, tails, tails, or E any of them are equally likely. And you might win some prizes if you get the highest number of questions right. So there's an added incentive. So everyone should be able to vote. My sister just put up a poll. But if you can't vote, please let us know in the chat. And we are happy to look over answers that come in the chat. I think that's true if you're, for example, accessing zoom through an internet browser. And then we'll just keep track manually of everyone's score who's participating in the chat. So okay, is that a time? Okay, great. And so let's discuss the answer. It's E, any of them, because any sequence of four coin flips is equally likely. We might assign special significance to certain patterns such as like having four heads in a row that seems unlikely. But the thing is, each of these sequences has exactly one in 16 chance of occurring one half times one half times one half times one half is a probability for getting each of this these events, each of these four events. And so you know, it's certainly more likely to roll a combination of two heads and two tails. But that's because there's more permutations available. So you know, there's there's six ways that you could have two heads and two tails. So that's why you might think Oh, C or D seem more likely. But the thing is, in order to the probability of getting this specific permutation is the same for all of them. Does that make sense? If not, let's please let us know in the chat. Otherwise, we'll move on. Okay, so Dr. Ruz is ready to tell us the next part of the fortune teller and Nostradamus can roll my tea leaves to divine who's the culprit. You have been guessing who will win these games from the start and have not had one correct answer. If we were to approach this as a scientific experiment, it would suggest that you have no magical powers of prediction. I admit that I've been wrong a lot today. But this is exactly why I'm due for the correct answer. Okay, as a fortune teller, I know. So probabilistically speaking, if Nostradamus his predictions have been wrong so far, is he due for a correct answer? Please let us know in the chat. What do you all think? I'll wait until I see someone responds. No. Okay. Alex, that's that's Tony. No. Okay. It looks like the consensus from the audience is no. And I would agree with you. So let's talk about the gambler's fallacy. So the gambler's fallacy, some of you may have heard about it, or may not know the name, but know what it is. There's a it's the mistaken belief about sequences of random events. So the incorrect belief that if a particular event occurs more frequently than normal in the past, it is less likely to happen in the future. So for example, if you, you know, roll five sixes in a row, you might think that rolling a six six in a row is unlikely. However, the probability that you will roll a six six, given that you've already rolling those five sixes, it's still one in six, right? It's all about just the next dice roll. And all of these systems are stateless or memoryless, right? There's no, it's not like the the dice or the coin flip or any of these things, you know, they don't remember anything about the previous dice or coin flip, whatever happened on those those rolls or flips, right? So it's completely independent of previous events. And I think that that there's a cool example of this in like 1913 in Monte Carlo, when if you guys are familiar with the game of roulette, there's like almost, almost 50% chance that a ball might land on either red or black. And it landed on the same color, I think black 26 times in a row. And after about like, you know, seven times people thought, okay, now it's due to have a long streak in red, it's had a long enough streak on black, but each time it just kept landing on black. And that caused millions of pounds to be lost as people kept on betting on the incorrect color thinking that, oh, you know, our luck is due to switch, there's no such thing as being due to switch anything. And so Nostradamus said his tea leaves could divine what's going on here. tea leaves are a stateless memoryless system, there's no reason why you know, it's going to work all of a sudden, just because it wasn't working before. And there's no reason also why a person you know, a striving of random value to the output of a random process should give you the truth about anything. Okay, so hopefully, we understand the gambler's fallacy. And we understand why Nostradamus was wrong. So let's try applying this now. So this is quiz number two, if you roll 610 times in a row, what is the probability that you will roll six again on your next role? Given that you're using a fair six sided dice? Is it a still one six? B is a greater than one sixth. See, it's less than one six, and D it's impossible to roll a 610 times in a row, let alone 11. So my sister will put the poll up. And again, if for some reason you can't participate in a poll, you're free to put your answers in the chat. But make sure you put it before we end the poll. Okay, so that's time. See yeah, so the answer is a still one six. Reason why is like we said earlier, you know, there's nothing that the previous 10 rolls don't matter. They don't factor into it at all. And you know, on a fair six sided dice, what's the probability that you roll a six, one and six? Okay. Well, earlier in the evening, when I was using today's horoscope to make predictions on the game winners, I was able to predict Anthony one every single time, I can use the horoscope to make a prediction about the culprit since it's accurate. Okay, so now here we have Nostradamus giving us another idea for who he thinks or how he thinks he can figure out what the culprit is. Now, what do we think about Nostradamus's claim on accuracy in horoscopes? I mean, he was able to predict Anthony one every single time through his horoscopes. And Anthony did in fact, win every single one of his games. So the horoscope is a pretty accurate method of detecting anything, right? It has 100% accuracy. Would everyone agree with Nostradamus's logic here? I want to see some movement in the chat. No, Tony, Kaden, they both think no, Alex says no. Okay. Why not? Okay, well, this is something called the accuracy paradigm. So accuracy is not a good metric for predictive models when classifying. And this may be surprising, but let's talk about why. So in this specific example, every single time Anthony won all of his games. So anyone just betting on Anthony randomly for whatever reason it might be, they would always be right. That doesn't mean they have any more idea about what's going on than you or I do. But they would have 100% accuracy. And horoscopes are, quite frankly, meaningless. There's no scientific basis to them. There's nothing about their predictive power. So it's just the simple fact that it worked to predict that Anthony was winning all of his games. It's just a mere coincidence. It doesn't mean anything. And it's important to also think about what are valid predictions that you can think about when you're trying to measure the accuracy of something. So for example, if I predict that tomorrow I will not meet a unicorn, there's probably a pretty good chance of that, 100% chance. That doesn't make me a wizard. I don't have any special magical predictive powers. On the other hand, if I'm a very good sports analyst, and I predict that a certain player is going to have a very good season, there might be something there. Because I'm looking at certain metrics. I'm looking at certain statistics and history in order to make that prediction. So even if I'm maybe making very safe, easy predictions, then the accuracy of my predictions probably still is not the best metric to take into account. But at least you can say that my predictions are worth thinking about, and they're valid. They're not something completely bogus like a horoscope. Okay, so quiz number three. So, oh yeah, my sister put in the chat. So the accuracy paradox is a very important concept in many fields, including AI and machine learning. You might have heard of using precision or recall in AI instead of accuracy. Yeah. So for example, you could have a data set where you have 80 people who are healthy, and then like 8,000 who have COVID. And let's say you have a magical algorithm that predicts who will have COVID, and you run all these patients through it. Now, you might be accurate most of the time. But the thing is, if you're always predicting that the patient had COVID, then just because my worst fear is the fact that most of the patients in this data set had COVID. And you're not actually doing something different or having an algorithm that actually thinks about each patient differently, you will be getting a pretty accurate answer in general for everyone. So hopefully that makes sense. If anyone has any questions, please, as always, please feel free to put it in the chat. But let's move on to quiz number three. We want to make sure we're showing Dr. Roo that we are really grasping these concepts of probability. So I've put four different predictions. So one is I predicted that it would snow yesterday in Colorado Springs. So this is just based on the meteorological forecast and my powers of being able to understand like weather patterns and storm fronts. Two, I predicted that it would snow yesterday somewhere in the world due to my magical divining ability. Three, I predicted the sun will rise in the east. And four, I predicted that I would finish my homework. So each prediction has a score next to it, one, two, four, eight, and add up the scores of all predictions that are valid, actually take some actual skill to forecast in advance and then answer with whatever is the final sum. So if you think two, three, and four are all accurate, then that means this final sum is two plus four plus eight, which is 14. So the answer would be D in that case. Okay. I think we can call time on that one. Waiting for one more person. If you'd like to respond. I'll wait till a minute and then I'll close. You know, you have 25% of probability, you guess anything randomly to get it correct. So, okay, I'm going to end the poll. Okay, so the correct answer is a. I think prediction that is a really a valid prediction is the first one. Saying that it will snow somewhere in the world, not very informative, not very creative it doesn't really take a genius to say that number prediction three, the sun will rise in the east. I mean it does that every day. It's again, not really a prediction, and for I predicted that I would finish my homework. I mean I'm the one who has control over that right so if you can control an event it's not really a prediction, when you're controlling Does that make sense to everyone. So the only possible valid prediction is a. And, you know, if, if you wanted to measure like a weather forecasters predictive ability, like yes you could you could try to look at their results and, you know, match up their predictions to the actual that ends up happening in a certain area but accuracy is probably not the best measure there are other measures like precision or recall, and you take into account things like you know true positives true negatives false positives false negatives, and also account for like the imbalance of the data set right so you know you predict snow every single day in Colorado Springs of an area that tends to have a lot of snow. You know, again, accuracy, not the best metric, but, you know, if you're more or less good at predicting snow on the days it does snow and no snow on the days it doesn't snow, then that's better. Okay, so let's move on. It looks like there's a breakthrough in the case someone's discovered a clue. I can narrow down the culprit with my legendary skills of deduction. You can observe coconut flakes on the clothes of Miss paws macaroons are made from coconuts, ergo, given that the thief has eaten the macaroon. They should be expected to be dusted in coconuts. So we must expect Miss paws to be the culprit. Okay, so we've cracked the case. What do you guys think is Dr. Fisher right. Do we agree with Dr. Fishers claim that Miss pauses the culprits and she's dusted in coconut flakes and the macaroon was also a coconut flavor macaroon. Okay. I see a no from Caden, we agree with Caden. Anyone else. Alex also says no. Okay, so there seems to be agreement in the chat that Dr. Fishers claim doesn't make sense. But let's try to explain this probabilistically, why exactly. So let's talk about conditional probability. The, so this is also known as Bayesian probability. And it's the idea that you take into account the probability that something like event A will occur, given another event event B. And so Bayes' law is something that allows you to determine the conditional probability of an event. So it's the probability of A given B is equal to the probability of B given A times the probability of A divided by the probability of B. And so this can get a little complicated. Let me just type that into the chat, just so you all have it for your reference. And we can determine the probability that Ms. Paz has eaten the macaroon, given the coconut flakes were found on her. So let's take into account, you know, the, the probability of A given B, so the probability that Ms. Paz is the culprit, given that she has coconut flakes on her. So that's equal to the probability that she has coconut flakes on her, given that she is the culprit, times the probability of her being the culprit, divided by the probability of having coconut flakes on her altogether. Now, these seem like hard numbers, you know, to calculate exactly, but we don't need to have the exact numbers. We can just have a general idea. So what's the probability that Ms. Paz is going to have coconut flakes on her? Are there other food items that may have coconut flakes on them? Anyone, anyone know of anything that, you know, may have coconut flakes? Yeah. So Kaden says yes. Yeah. For example, coconut muffins have coconut flakes in them, right? There's other types of cookies besides macaroons that could have coconut flakes. You know, a lot of, maybe she just likes eating coconut flakes with her cereal, who knows, right? So the probability that she has coconut flakes on her could be very high, right? And the probability of her having coconut flakes just because she's the culprit who sold the macaroon, that's relatively small. Or sorry, no, sorry. The probability of having coconut flakes, given that she's the culprit is probably 100%, right? If you know, if you're eating a cookie, maybe you'll have some crumbs all over you. So if you multiply that by the probability that she's the culprit, we don't really suspect Ms. Paz. We like cats here. Ms. Paz is a friend of Dr. Rue. So the numerator is going to be a small number because the probability she's the culprit is small. And the probability that she might have coconut flakes on her is really high. And thus, we have a small number divided by a large number. And that gives us a very small number. So the probability that Ms. Paz is the culprit just because she has coconut flakes found on her, we're going to say is a very, very low one. And in fact, Dr. Rue is telling me right now, I'm just checking my text. It looks like Ms. Paz was eating coconut muffins earlier, and there are other witnesses there who can corroborate that. So it looks like she's in the clear. Nice job, everyone. Okay. And let's talk about also the overgeneralization fallacy. So this is the mistaken belief that statistics about a particular population is asserted to hold among members of a group for which the original population is not a representative sample. So you don't want to extend a conclusion too far. And certainly, you don't want to overgeneralize the way that Dr. Fisher did, where he basically assumed that since one thing was true, where the probability of Ms. Paz having coconut flakes on her just because she's the culprit is true, that doesn't mean the flip side is true. Ms. Paz is not the culprit just because she has coconut flakes on her. Does that make sense to everyone? And so the overgeneralization fallacy plays into Bayes' law a lot because there'll be a lot of court cases where, for example, like there's a famous court case from the 20th century where a brother and sister in certain clothes were found near a crime scene, where the criminals were reported to be wearing similar clothes and also appeared to be brother and sister. So, you know, out of all the population in the world, what's the chance that you would find such a brother and sister? It's pretty small. However, that really isn't the question to ask here. It's, you know, what's the probability that given that we have found such a brother and sister, that they are the criminals? And actually, that was pretty small. So, like, basically, sorry, I think I said it backwards, but the chance that the people would be wearing certain clothes and, you know, a random brother or sister would be found so near the crime scene, that seemed pretty unlikely. So that seemed to mean that they were the criminals. But actually, like the fact that, you know, given that we have found such a brother and sister, the fact that the probability that they're criminals is actually unlikely. Does that make sense? So essentially, because of overgeneralization, initially, like innocent brother and sister were convicted because it looked like, hey, you know, the probability that they're just here for innocent reasons is low, they're probably criminals. But after using Bayes' law, people realize, oh, actually, no, the probability that they're criminals is pretty low. Okay, so yes, Miss Spada was seen eating coconut muffins earlier in the party. So she has been exonerated. Thank God. Okay, so let's move on. We have maybe another I had a test that is 99% effective at detecting if someone has recently eaten a raspberry jam. I administered it to all hundred party guests and wrote down each result on a piece of paper and folded the paper using a mouthful. Only one person, Mr. Whiskers, tested positive. We have found our culprit. Okay, so let's establish the facts. We have a test that is 99% accurate. Okay, so that means that 1% of the time, it'll identify someone who's not the culprit as a culprit. And similarly, 1% of the time, if there was a culprit, we may identify that person as not being the culprit. So do we agree with Dr. Fisher's claim that Mr. Whiskers is the culprit since he tested positive on a 99% accurate raspberry jam test? No, no, no. Okay, so we have some no's in the chat. Okay, great. Now let's see why that might be. So okay, what first let's define the term false positive and false negative. So a false positive is the yes, thank you. There's a there are 100 people in the party. That's an important reminder. So this is an outcome where the positive class is incorrectly predicted. So for example, if you're trying to diagnose a person who has eaten a raspberry jam, it's a false positive. If you're trying to diagnose a disease, you have some kind of tests like testing someone maybe if they have COVID or not. So someone may not have COVID, would you diagnose them with COVID? That is a false positive. So in this case, we're hoping that for Mr. Whiskers' sake, this is a false positive. We have identified him as the culprit despite him not being the culprit. Now, the false positive paradox is something where, you know, this might, you know, in our case, maybe causing us to convict an innocent person. So this is a situation where there are more false positives than true positives. So how could that be? Well, let's think about this for a second. How many culprits are there? There's just one, right? So, you know, what's the probability that we'll identify that culprit? So, okay, actually, let me take a step back. So we have one culprit. What's the probability that the test is accurate? It's 99%. So what is our expected number of like, you know, people will identify with this test as being positive, you know, you know, actually being the culprit when they are the culprit, it's 0.99. Does that make sense to everyone? So it's, you know, there's one person who is the culprit, and there's a 99% chance we're accurate about them, we accurately identify them as the culprit. So we would expect to find, you know, 0.99 culprits that way. However, we have 99 people who aren't culprits. Okay. And again, our test is 99% accurate, but that means 1% of the time, it's inaccurate. So for those 99% people, sorry, for those 99 non-culprits, 1% of the time, we will have, we will be identifying them as culprits. Again, the expected number of culprits we detect from there is also 0.99. Right? So does that make sense? So 0.99 culprits, or basically about one, are the number of true positives we might expect. And one is also about the number of false positives we may expect. So we have basically exactly the same number of true positives and false positives, which means that just by virtue of the fact that Mr. Whiskers tested positive, all it tells us is that there's a 50% chance he's guilty, right? Because there's a 50% chance he's a true positive, but there's a 50% chance he's a false positive. So is that a very good test, do you guys think? Is that strong enough to convict him? No. Okay. And does that make sense to everyone, how that logic works? You know, 0.99 true positives, 0.99 true positives, 0.99 false positives. So if you do the math, 0.99 divided by 1.98 is half. So 50% chance Mr. Whiskers is the true culprit. Is it true positive? Okay, great. So yeah, let's not convict Mr. Whiskers just yet. But we can take a second to test ourselves. Do we understand what's going on here? So this is a bit tricky. So I would suggest writing it out with pencil and paper. We have a test that is 98% accurate on 100 people. How many additional people would we have to successfully test meaning we test them and we get the exact right answer for them? How many additional people we have to successfully test to bring our accuracy up to 99% without any additional false positives or negatives? And consider writing it out. So, you know, think about how many errors you have so far, if it's 98% accurate, think about how many total people you would need to test. And if you're not sure there is a 25% chance of guessing correctly. It is one of the four answers I promise you. Yeah, and I'll close the poll at two minutes right now we're at one minute 30 seconds. If you need more time just let us know in the chat. Anyone need more time? Feel free to guess randomly cause I'm ending the poll in five, four, three, two, one. Okay. So the correct answer is D. Yeah. And I'm not sure if that makes sense to everyone right off the bat. So let me flip it for you guys. So we have a test that is 2% inaccurate on a hundred people. We want to shrink that accuracy to being 1%. Okay. So if we aren't increasing the number of failures if we have still the same number of failures how does that 2% become 1% of the overall population? Well, we have to double our overall population, right? So if we tested this on a hundred people earlier now we have to test this on 200 people overall make sure that the last hundred that we test is a hundred percent accurate. And then that way those failures that we had that represent 2% of the population they'll shrink to just representing 1% of the population. So that means if we, yeah, we want to represent or if you want to test 200 people overall that means we want to test an additional a hundred people. So the answer is D. Okay. We'll move on unless anyone has any questions. Now this, ooh, this is a tricky question, but don't get fooled. It's a bit of a trick. So Houdini the Hare says he has a magical die that will roll a six each time he rolls it. You've not inspected the die. So you don't know anything about it. He rolls a six 15 times in a row and he says he can roll another six. Do you believe him? So the four answer choices are A, no because he is due to roll another number. B, no, because there's no such thing as magic. C, yes, he may roll another six. We can assume with more than a one in six chance or D, yes, he may roll another six but we can assume it's with a one in six chance only. I'm closing the poll in five, four, three, two, one. Okay, so the answer is C. Now, I don't know how many of you got that. I don't know if that's surprising to everyone, but let me explain why. So, yeah, there's no such thing as magic. Magic's not real, let's say, but it is certainly possible to create illusions with a fair amount of preparation and deception. So rolling 15 sixes in a row and being so sure about rolling a 16th six is unlikely. And we've never stipulated that the die is fair, right? We never said that anywhere in the problem. And Houdini, the heir himself, said it's a magical die. So at this point, it's rational to assume the die is not fair, meaning that he likely has not just a one in six chance to roll a six, but more than a one in six chance. And of course, A is wrong because he's not due to roll another number. That's the gambler's fallacy. And so, yeah. So let's actually maybe think about this using Bayes' rule. And so if we think about the probability of the die being fair, given that he rolled 15 sixes in a row, right? This equals, using Bayes' law, the probability that he rolled 15 sixes in a row, given that the die is fair, times the probability of the die being fair, divided by the probability of rolling 15 sixes in a row. Can people still hear me? Okay. Yeah, no, yeah, we lost you for a second there, but now we can hear you. Yeah, sorry, I think my Zoom chat is freezing. Sorry, let me fix that typo. So hopefully this calculation makes sense based off of Bayes' law. Hopefully you guys wrote down that slide or copied it down from when I put it in the chat above. So what's the probability of rolling 15 sixes in a row given that the dice is fair? Well, it's not a very large number. I calculated this out and it's about 2.12 times 10 to the negative 12. Okay, so that first probability is a very small number. Now, the second and third probabilities, the probability of a die being fair and the probability of rolling 15 sixes in a row for any dice that we may have, that's a little trickier to calculate, but we can make some assumptions here and just fill in some numbers. So let's say we think that, hey, maybe we have a very high faith in Houdini and we think that probably that the dice is fair is 99%. So we're going to be very, very cautious. We don't want to assume that the dice is unfair. We're going to say that there's only a 99%, sorry, that there's a 99% chance that the dice is fair and there's only a 1% chance that maybe the dice is unfair and it's maybe weighted or loaded in some way. So let's just fill that in. Okay. And now if the die is unfair, we don't know anything about it, but let's say that, you know, if the die is unfair, we're going to pretend that instead of rolling a six with a one in six probability, it rolled a six with a one in three probability. So it's not super unfair, you know, like, like it just doubles our chances of rolling a six. Okay. And so if you do the math, if you work out what's the probability of rolling 15 sixes in a row, given that 99% of the time it's a one in six chance of rolling a six and 1% of the time, there's a one in three chance of rolling a six. Then basically you can plug in the value for the denominator. We know the values for both of the probabilities in the numerator, and we get that the overall probability that the dice is unfair, or sorry, we're solving for the dice being fair. The overall probability of the dice being fair is 0.3%. Meaning that the probability that the dice is unfair is 99.7%. Even though we started off by being very, very cautious. And we assume that, okay, even if the dice is unfair, it's not super unfair. And there's a 99% chance that the dice is fair. So just by virtue of the fact that rolling 15 sixes in a row, so yeah, let me write it out. So we have, so one over six is the probability of rolling a single six, right? And so if we, you know, take that to the 15th power, that gives us the probability of rolling 15 sixes in a row, 15 sixes in a row using a fair dice. So that's what this is, this number that I'm, let me put that in. Okay. Sorry, that should be an equal sign. I can fix. Yeah, thank you. And then, and then the, oh, you can't see it. Ah, geez. Okay. Oh, oh, oh, oh my gosh. I'm so sorry. This is, oh my gosh. I thought I was typing to everyone. I think I started typing to hosts and panelists only. Okay. Let me write out everything again. Thank you. Thank you for letting me know. Okay. So, sorry, this is our assumption. And then, so in order to calculate the numerator, this is a bit tricky, but basically we're saying P of rolling 15 sixes in a row, you have to take a weighted average. So 0.99 of the time we have a fair dice. So then that gives us a one in six to the 15th power of rolling 15 sixes in a row and 0.01 times, or like 1% of the time we roll a six with one in three chance, let's say. So that's how I get this number. And basically now we have the probability of 15 sixes in a row, given that the dice is being fair, we have the probability of the die being fair, and we have the probability of 15 sixes in a row. So once you plug all of those numbers in now, you know, you, you'll need a calculator maybe for, for some of this to do it efficiently, but I went ahead and calculated it for you. So the probability of the dice being fair, given that we rolled 15 sixes in a row is only 0.3%. The probability that the dice is unfair is 99.7%. Now, if we assume that the, there, we have an unfair dice that rolls a six 50% of the time, not just, you know, one in three, but one in two times, the probability that the dice is unfair would be 99.93%, just based off of our priors. And I had a probability teacher who actually had a loaded dice that would roll a six 99% of the time. So, and he would like bet with students, Hey, like, you know, I bet I can roll this many sixes. So you can imagine, you know, the, the, the numbers would be even more skewed in favor of assuming that the dice is unfair. Now we were being very nice here and assuming the dice was fair with a 99% chance, right? We could have assumed even lower prior probability. We might actually think that there's only a 10% chance, Harry, that, that, that Houdini the hare is using a fair dice. So, you know, this depends on the priors that we're giving, but even giving the most generous priors we can, we see that there's a very low chance of the die being fair. Now, just, just to clarify though, this, this wouldn't apply to like a scenario where you roll like, you know, 10 sixes in a row using an actual fair dice, because in that case, the probability of the die being fair is one. And so then the, the overall, you know, answer would end up being one just because by virtue of the fact that we say there's a 1% chance that the die might be unfair. That's why we get these giant numbers that suggest that the dice is probably unfair given 15 sixes in a row. Does that make sense? And yeah, thanks. Thanks for letting me know. Okay. So we have a, another experiment on the way. It looks like Dr. Fisher has another method proposed another test to find the culprit. Examine the scene near the light switch for clues and found a strand of hair. You know what this means? The culprit has been eating spicy salsa. Look at the experiment I conducted with various animals eating different levels of salsa. As you can see, the animals that ate the most salsa also shed the most. The ones that ate the least shed the least. Oh, I think that's talked a little early. Find out who was eating the most salsa. All we need to do is find out who was eating the most salsa at the party. And we have our list of suspects. Interesting. So look at the graph. You can see the x-axis teaspoons of salsa consumed. And the y-axis is the percentage of hair shed, you know, as a percentage of body weight. So you can see that there's a upwards trend, right? Do you guys all see that how it's going in that direction overall? So do you think it makes sense to use Dr. Fisher's salsa test to predict who was near the light switch? So remember we saw a lot of hair near the light switch. So based off the fact that there was a lot of hair being shed near the light switch, we're assuming that that person must have consumed a lot of salsa. Does that logic seem to make sense to everyone? No. No. Okay. No from Alex and Caden, who have not been big fans of Mr. Fisher today. And I would agree, actually. Shrey, also no. Yeah. Why not? That's an important question to ask. Saying no is easy, but why? Correlation and causation. That's an interesting point. That's a good point, yeah. Yes, so I think that's a good point, and we'll talk about that in a second. And there's also something else, something more interesting that we can talk about first, which is, actually, I think Alex may have preempted my question that I have later on, but we'll see. Yeah, let's take a closer look at the chart, look at the animals. So if you look at each of these groups, right, the green group, the orange group, the blue group, which animals shed very little body hair just in general, which animals shed, you know, a medium amount of body hair and which animals shed a lot of body hair in general. You can see that within those three groups, there's actually a declining trend. Do you all see that? So even though overall, in the overall data, we saw like a positive correlation, here we're seeing a negative correlation when we separate based on the groups of how much these animals tend to shed. Does that make sense? How there's like this kind of flip side, you know, reverse correlation? Hopefully that makes sense to everyone. You can see the graph. It's a little tricky when you see it for the first time. It's always, you know, tricky when you see this with numbers. But yeah, basically what we can see from the chart is actually capsaicin and salsa. Well, so this is just based on scientific studies. Capsaicin and salsa actually helps with hair growth. So that means hair loss should decrease. And that's actually what our chart shows, right? Like the more salsa that an animal within one of these groups ate, the more percentage of hair they shed. Now, this isn't the best experiment just because there aren't a lot of controls or stuff like that. But even if you just take this one confounding, I'll explain what that is in a second, but with this one confounding variable about how much hair they tend to shed, these animals, you can see that, yeah, definitely there's basically the opposite conclusion of what we initially drew. And in fact, that more salsa leads to less hair being shed. Okay, so animals that shed the most hair were selected to eat the most salsa. And thus, it's also a flawed experiment. And so we should be looking for animals who ate the least amount of salsa. So yeah, so this is called Simpson's paradox. So it's a statistical phenomenon where an association between two variables in a population suddenly emerges or disappears or reverses when the population is divided into subpopulations. So basically, you take another variable, something that we can call like a confounding variable in this case, where, you know, if you don't know about the existence of it and you don't consider the effect of it on your experiment, you may be drawing incorrect or just partial conclusions. You know, the overall conclusion here may not necessarily be wrong, but it's definitely not telling us the full story. And certainly when we take this confounding variable into play, which is the amount of hair that's being shed just generally, then we see a very different conclusion. So a confounding variable, the definition is an extraneous variable that is correlated with the independent variable and causally relates to the dependent variable. So this is, you know, a really powerful paradox and you'll see very interesting results from it. So, for example, like, you know, you could survey like a bunch of men and a bunch of women and ask them, you know, which flavor of ice cream they like the best. So like chocolate or vanilla. And, you know, just because of like maybe skews that are present in the data based on how many men were interviewed and how many women were interviewed and like the relative ratios, you might end up seeing, you know, like overall as a population, if you look at the results, then, you know, maybe people like vanilla more. But if you look by subgroup and you look among men specifically and among women specifically, then each group may like chocolate more. In a similar note, like, for example, UC Berkeley, they were investigating their admissions and they thought they were discriminating against women because it looked like women were getting accepted at lower rates. Now, it actually turned out that women were getting accepted at basically the same rates as men. But it just so happened that they happened to be. You had to break down the data by major. And women had happened to be applying to more competitive majors more commonly. And thus, it looked like the admission rate for women was lower when, in fact, it was basically based on major. You would see that there's there wasn't discrimination between men and women. Yes. Thank you. Exactly. Yes. So there's an interesting phenomenon in baseball, maybe any sport. But like, for example, Derek Jeter, David Justice, famous all-star baseball players. David Justice had a better bat batting average every year for a period of like five years. But Derek Jeter had a better batting average overall. And the reason for that is basically just a skew in the number of bat at bats. So, for example, Derek Jeter or sorry. Yeah. David Justice just played, you know, like a few games in a few years. But in those two games, he got a very good batting average. And so those skewed his batting averages those years upwards. The remaining year, he got only even though he played all the games, he got only a slightly better batting average than Derek Jeter. Derek Jeter mainly. But on the other hand, because of overall like was doing well, you know, he had an overall better batting average. So basically you'll see this in baseball where like you can find this correlation every single year or for every like single, you know, period of like three years or five years or whatever that there will be one baseball player has a better batting average every single year when you're looking per year. But if you look at the overall time period, another baseball player has a better batting average. So you can do this with comparisons between two baseball players and like there's always like some pair of baseball players you can find in any group of years, which is pretty cool. Does it make sense to everyone how that works? I am going to move on if I. Yes. OK, great. So quiz number six. Were there any other fallacies present in this reasoning? So we said, hey, because there was a bunch of hair shed on the floor, thus the person must have eaten a bunch of salsa. Is that so? Sorry, let me let me read this out. So no, because we saw a bunch of hair. Let me not. I don't want to give away the answer. But basically, because there was a bunch of hair on the floor, we thought we must have had a culprit who ate a lot of salsa. And why did we think this? It's because Dr. Fisher thought that eating a lot of salsa causes a lot of hair loss. So we know the probability of hair loss given that someone's eating salsa is high, at least that's what Dr. Fisher thought. So given that. Does it make sense, the conclusion he was drawing, or are there some other fallacies like the errors with Bayesian reasoning? Gambler's fallacy or generalization, false positive paradox. Oh, yes. OK. Certainly some of these errors can tend to overlap and there may be like. You know, there's there's different ways of explaining maybe the same statistical error, but I think there's a one very clear answer that really explains what's going on in this particular case. Again, it is advantageous for you to randomly guess because you have a 25% chance of getting it right. I will, we're almost at the two minute mark. I will wait for two minutes and 30 seconds before I close the poll. If you have any questions or if you need reminders for what any of these are, then feel free to ask in the chat. Otherwise, I will be closing the poll in 20 seconds. and five, four, three, two, one. Okay, I will close the poll because I did say I would close it in 30 seconds, but maybe we can now discuss what each of these means. Okay, sure. So the correct answer is A. Let's start at the bottom. So false positive paradox, what does that mean? Well, we talked about how just because of the fact that you have someone who has tested positive for something doesn't mean that they are likely, in fact, positive for that thing. And in fact, when you have something where the chance of actually being positive is very low, so for example, if only 1% of the population has the disease or only 1% of the population is the culprit in this case, if your test is 1% inaccurate, as low as that inaccuracy may seem, just because of the fact that there's so many negative examples and you're not 100% accurate on those negatives, you'll have roughly the same number of true positives and false positives, right? You'll have the actual number of people who are supposed to test positive balance out roughly with the number of people who are negative who tested positive, or you may even have more false positives than true positives. So there's a famous example of a case where police officers were using breathalyzers on a lot of people and the breathalyzers were pretty, they were 100% accurate on people who had been drinking. They were 95% accurate on people who hadn't been drinking, meaning that only 5% of the time they were false positives. But because the percentage of population who was drinking and driving was so small, the only 2% of the drivers that failed the breathalyzer test were, in fact, drunk. So does that make sense? There's a very, because of the imbalance in the actual true, in the reality of how many positives and negatives there are in a population, even a small inaccuracy in a test that results in false positives or false negatives can cause extreme imbalance in terms of, if you compare the true positives to the total number of people who tested positive, right? So basically the chance that someone is actually positive just because they tested positive ends up being very low, just because there'll be so many false positives, just because the negatives in the population outweigh the positives. So hopefully that's clear. I know we kind of explained an example of it, but I didn't generalize it. So hopefully that makes sense. Overgeneralization. So just, as you know, colloquially, it's just anytime you're generalizing and trying to draw conclusions that are too broad. But basically it can be, for example, when you're trying to generalize from a small population and drawing some conclusions about the larger population. In some sense, this is happening, right? Like you know something about populations that eat salsa and that this can cause hair loss, and you're using that to generalize about anyone that has hair loss that they must have eaten salsa. But I don't think this is the best answer. And I'll explain why I'd say. So gambler's fallacy, that's basically when you're due, when you think that someone is due for a change in outcome, that's fallacious reasoning, because basically we're pretending that like, you know, there's some, you know, like the dice remembers that, hey, like, you know, it's not really likely for me to roll this many sixes in a row. So I'm just going to roll something else next time, right? There's no such thing. The probability that a dice will roll six, as long as it's fair, is always consistent. So that's not applicable here. But I would say the correct answer is, the best choice is A. So errors with Bayesian logic. So basically, let me write this out. The probability of hair loss given salsa, given eating lots of salsa is high. Let's pretend that Dr. Fisher was right about this. We know he wasn't, but let's just pretend he was. So what Dr. Fisher was trying to tell us is that the probability that given that we have seen that there was hair loss, the culprit, sorry, given that we've seen a lot of hair loss from the culprit, that culprit must have eaten lots of salsa, right? So during, you know, we saw like the lights go out when the macaroon was stolen. So, you know, the culprit must have been near the light switch, near the light switch, we found all this hair. So we know for sure that the hair belongs to the culprit. Okay. Well, we don't know that the first statement is true. We don't know that they've eaten lots of salsa. And that's why, you know, they've lost all this hair. We only know the second statement, which is that they have lost hair. So given that, what's the probability that they've eaten lots of salsa? We don't really know, right? Like, because these two are not equivalent statements, right? The probability of having hair loss, given that you've eaten lots of salsa is not equal, equivalent to the probability that you've eaten lots of salsa, given that you have hair loss. There's, there could be lots of reasons for hair loss, right? Including like just, you know, some animals shed more hair than others, right? Maybe stress causes hair to fall out. You know, we don't know exactly what it is, but unless we were to actually solve the full Bayes law, you know, plug in everything. So the equation would be the probability of hair loss, given you've eaten lots of salsa times the probability of having eaten lots of salsa divided by the probability of hair loss. So we have a bunch of furry animals at this party, right? The probability of someone losing a lot of hair, it's pretty high. So we don't need to know any of the terms in here. We just know the denominator here is pretty high. And thus that means the overall fraction is going to be something that's small, right? So the probability of having eaten lots of salsa, given that you have hair loss, it's a small number, right? Does that make sense? Suhani and everyone else? Yes. Okay. Excellent. I will move on. Let's go to the next slide. We have, it seems, another clue. After examining the crime scene for clues, I found screws lying here alongside a gold coin. The thief must have used this gold coin to unscrew the box and then left it here. Well, I guess there's an equal chance that Anthony or Florence is the culprit. After all, both had at least one gold coin. Okay. Interesting. So let's see if that reasoning makes sense. Okay. So Anthony had 100 gold coins. If you remember our facts sheet at the start, Florence had one gold coin and 99 silver coins. So do we agree with Dr. Fisher's claim that just because we found a gold coin lying around the glass box, the macaroon was in, right? So we know the gold coin came from the culprit. Do we agree with Dr. Fisher's claim that Anthony and Florence can be considered culprits with equal probability? So let's think about what the culprit must've done. They probably, they needed like a coin to unscrew the screws of the box and remove it. They pulled a random coin out and it was dark, right? Because the light switch was off. So we know that there wasn't any planning into like exactly what coin was pulled out. So this was a random coin that was pulled out and it happened to be gold. Now, we know Florence has one gold coin and 99 silver coins. We know Anthony has 100 gold coins. So who do you think, did you think that it's equal probability or do you think it's more likely that it's one or the other? What do you guys think? More likely that's Anthony, Caden says. You know, Caden, I agree with you. Let's see why. So this is, yeah, exactly. It's unequal. And so this is something that I don't want to actually give the quiz. Let me actually explain a bit first. Basically what you can do is think about what are the general overall possibilities that Anthony would have pulled out a gold coin. There's a hundred percent chance, right? That if he had pulled out a random coin, it would be gold. But for Florence, there was only a 1% chance that she would have pulled out a gold coin. So the fact is that even if we think, you know, Anthony or Florence, either one could be equivalently the culprit, just the probability that it's Anthony's is far more likely just because of the fact that, you know, the gold coin is much more likely for him to have pulled it out than her, because she would have been much more likely to pull out a silver coin. So, okay. Hopefully that reasoning makes sense. You can, oh, am I frozen? Can you, okay, you can hear me. Yeah. So let's see if you guys can get this question. Let's say you have three boxes, box A has two gold coins, box B with one gold coin and one silver coin, and box C has two silver coins. You pick a box at random and then pick a coin out of it without looking inside the box. You see that you have picked a gold coin. What is the probability you pick box A, B or C respectively? And so our answers are half, half and zero, one third, two thirds and zero, two thirds, one third and zero, or impossible to tell since you are an alchemist who can transmit silver into gold. Let me start the poll. Three, two, one, I'm ending the poll. Okay, great. So, what is the correct answer. It is not D, I'm sorry. It is C. So, hopefully it makes sense to everyone why I think Caden gave an interest rate gave good explanations. So, it's the probability that you have picked box C, of course, is zero. Because there's two silver coins in there, you couldn't have picked a gold coin over there. Now, the probability that you picked one gold coin out of box B, it's only 50% chance. And the probability that you've picked two gold coins, or sorry, one gold coin out of box A, that's 100% chance, right? So, it's twice as much more likely that you've picked box A than box B. So, that gives you like a two-thirds probability for picking box A and one-third for box B. Basically, the paradox comes from the fact that, you know, even though they both have at least one gold coin, because of the way the problem is stated, the answer is not A. It's not equivalent probabilities of picking A and B. Hopefully, does that make sense to everyone? And I think one good explanation that I was thinking of is, you know, like one thing you can do is, as always, Bayes is your best friend. So, the probability of it being A, given that we've picked a gold coin, is the probability of picking a gold coin, given that we've picked A, times the probability of A, divided by the probability of picking a gold coin. So, this is equal to... So, probability of gold coin, given that it's A, is equal to one. Probability of picking A, you know, given that any of these are equivalent of picking one of these things, it's one-third. And the probability of picking a gold coin is equal to the probability of picking A, times the probability of picking a gold coin, given that we've picked A, plus the probability of picking B, times the probability of picking a gold coin, given that we've picked B, and then the probability of picking C, given that... at times the probability of picking a gold coin, given that we've picked C. So, we know that the probability of picking a gold coin, given that we've picked C, is zero. So, we don't have to worry about that last term. We only have to worry about the first two terms. So, the probability of picking A is one-third. The probability of picking a gold coin, given that we've picked A, is one. Similarly, the probability of picking B is one-third. But the probability of picking a gold coin, given that we've picked B, is only a half. So, this ends up being... the probability of picking a gold coin is this. So, now if we do the math, if we fill out all the items in our fraction, we have one times one-third, divided by a half, which gives us two-thirds for A. Now, this is all fine and nice. I like another explanation I came up with much better. Let me just write this out for you a little differently. So, this is a representation of each of the boxes. Box A has two gold coins, box B has a gold coin and silver coin, and box C has two silver coins. So, what's the probability that given we pick a random gold coin, it came from box A? Well, there's a two-in-three chance, right? Because two of the three gold coins are in A. So, hopefully that makes sense. So, this is also known as Bertrand's box paradox. So, the problem setup has like two gold coins, two silver coins, one gold coin, one silver coin. This is exactly what it is. Russell Bertrand, famous mathematician, logician, probabilistician, statistician, I don't know. He came up with this. And yeah, after choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, what is the probability that the next coin will also be a gold coin? So, similarly, how we found like a gold coin near the box, and we think that Anthony is more likely to have picked it out. Sorry, Anthony is more likely to have dropped it out than Florence. This is exactly the same problem. It's just framed a little differently, and the odds are a little more skewed in favor of Anthony being the person who has dropped the gold coin. But hopefully, it makes sense how these are the equivalent questions and why the answer is two-thirds and not a half. Now, I'm going to give you something a little trippy. Again, read it very carefully. So, it's different framing. We have three boxes. Again, similar setup in terms of what the boxes contain. But you pick a box at random, and then I take a look at both of the coins inside. So, I tell you that you have picked a box with at least one gold coin in it. What is the probability that you have picked box A, B, or C respectively? So, we'll put a couple of seconds on the clock. So, let me... While you guys are doing that, I saw some interesting stats about Jeter, so... So Jeter was just slightly worse than Justice in both seasons, 1995 and 1996, but because, sorry, I think I flipped it around when I explained it, because he barely played in 1995, his 1996 season, which was by far better than both his and Justice's 1995 season, that made up the majority of his at-bats and so that really scooted his overall average upwards. And so his overall average was better even though his per-season average was worse. But sorry, I don't want to distract you from the question. Make sure you understand what the difference is between this setup and the previous setup. I'm going to close the poll in 3, 2, 1. Okay, so the answer is A. The reason why is because, hopefully it makes sense. I mean, there's two boxes with at least one goal point in it. If I've told you that you've picked a box with at least one goal point in it, then the probability you've picked A is half and the probability you've picked B is half and the probability you've picked C is zero. Okay, great. And now another question, similar lines. So we have a box with two silver coins and one gold coin. Maybe draw a diagram to help you out. You randomly pick up a coin without looking at it and you just hold it in your hand like this. Now, what is the probability that the coin you're holding is silver? Now I say that I'll look inside the box and remove a coin that I know to be silver. Regardless of whatever coin you picked up, we know that there's at least one silver coin in the box, right? So I'm telling you that I will remove a silver coin from the box. So now there's only one coin remaining in the box. What is the probability that the coin remaining in the box is silver? So there's two questions here. What is the probability that the coin remaining in the box is silver? And what's the probability that the coin you're holding is silver? So make sure you're thinking about the answer to both questions. What's the probability of the coin you initially pick up is silver? And what's the probability that the coin remaining in the box is silver? Also just curious in the chat, is anyone familiar with the Monty Hall problem? So honey says yes. Trey says yes. Okay, that's exciting. And let me know if you think you understand why I'm asking you guys about that. Okay. Three, two, one. Okay. So the correct answer is B. So hopefully it makes sense. There's two silver coins to start off with. I don't know why my highlighter is on. But yeah, there's two silver coins to start off with. And there's one gold coin to start off with. So the probability you pick a silver coin randomly from the three coins is two out of three. So it's definitely either B or C at this point. And then inside of the box, we have three options. Either you picked up the... Let's see. Let's see. How do I write this out? Well, okay. So either we have a gold and silver coin left because you picked up the second silver coin. Or we have the gold coin left and the second silver coin because you picked up the first silver coin. Or you picked up the gold coin and there's two silver coins left. So in two out of three cases, if I remove a silver coin, then... Yeah, we can talk about the logistics at the end. Yes. So if we remove the silver coin from two out of the three possibilities, that means we're left with the gold coin inside the box. So in one out of the three possibilities, though, the other remaining coin in the box is still silver. And so that's the probability the remaining coin in the box is silver is one third. So hopefully, that makes sense. The reason why I brought up the Monty Hall problem, if some of you are familiar with that, you basically had a very similar setup except like you have three doors. There's a car behind one of them and like two, like, I don't know, donkeys or something behind the other two. You pick one of the doors at random. The host will open up a door that specifically has a donkey behind it. So that means one of the donkey is taken off the board. And the question is, should you switch? Because you have the option to switch to the other door and say, no, I actually want to take what's behind the other door. So should you switch? And since there's a two in three chance that you've picked up a donkey on your first turn, and there's only a one in three chance, that means that the remaining door has a donkey behind it. That means there's a two in three chance that that other door has a car behind it. And the door that you currently have selected, there's only a one in three chance that you have a car behind it. So you're doubling your odds of getting a car if you flip. Hopefully, that makes sense. Basically, yeah, you want to try to win what's behind the door. And there's a two in three chance that probability that's essentially like locked in at the start of you picking a donkey. And so when the host narrows down the options for you and gives you basically takes off one of the donkeys off the board, that means that there's like a two in three chance that the remaining door has a car behind it. And so you want the car and you'll get that. Okay, great. So, yeah, let's talk about more clues that we have found. Yeah, so there appear to have been claw marks on the glass because the thief wasn't able to open the box. So that's why, as I had presumed, they randomly pulled out the coin out of the pocket. Since Anthony had 100 times as many gold coins as Florence, the probability of Anthony being the culprit is 100 over 101. The probability of Florence being the culprit is 1 over 101. So that means 100 times more chance that Anthony is the culprit. I say that's enough evidence to at least get the police to investigate him. So he had at least 100 gold coins in his possession. And he's an animal that shed a lot but ate little salsa. So he's our most likely thief. Okay, great. So have we solved our mystery? Wait, so Anthony is the culprit. Anthony's horoscope said he would exhibit great skill but also fall prey to temptation. The horoscope predicted that Anthony would win all those games and that he was the bad guy. Why did you say the horoscope did not work? Horoscopes are pretty generic and meaningless. While there is a tenuous link between what his horoscope said and the reality appeared to be, in reality, Anthony won all his games by cheating. See, he had a fake set of playing cards. Did you know he cheated? It would be impossible for anyone to win all those games of go fish in a row. The game largely depends on luck, not skill. Wait, aren't you guilty of committing the gambler's fallacy? And did you suspect that Anthony was the culprit because he was a cheater? So is Dr. Ru guilty of committing the gambler's fallacy? And did he suspect Anthony the fox because he's a cheater? So I know time's running out. So if you guys can quickly put your guesses in the chat or what would you think? Is Dr. Ru guilty of committing the gambler's fallacy? Or is this more like the Houdini the hair problem? At some point, you have to assume that like, hey, the dice isn't fair because there's no way someone could do that so many times in a row. Yeah, Dr. Ru is not guilty of committing the gambler's fallacy. Exactly. And did you suspect Anthony the fox because he's a cheater? No, there's a difference between cheating and thievery. And while we may have a stronger bias that Anthony the fox is likely to commit a crime just because he's committed one already, cheating at card games, that doesn't make him the criminal. And we only evaluated this case based on its own merits. Also, just want to go back a step. When Nostradamus asked, like, why the horoscope hasn't been proven to work, you know, when it actually did say that he would fall prey to great temptation. Horoscopes still don't work. That doesn't mean anything just because it happened to be correct. Again, this is the accuracy paradox. OK, so, yeah, making a future prediction based on previous predictions in a game of chance where games have no related. There's no relation between them. They're not dependent events. And the probability of winning N plus one games, given that a person has won and games is the same as them winning one game. So that's what the gambler's fallacy is. Right. And basically, that's not what's happening here. Right. Because this is this is, you know, it's just unlikely overall, though, that someone could still win N plus one games. The probability was still low overall, even if the probability like the doctor who never said, OK, Anthony's due to have a loss at some point. So, yeah, because this is what the gambler's fallacy would be if we violated this, this thinking. Also, how did Dr. Arun know, you know, that Anthony was cheating? Well, he was counting cards and he figured out exactly what was going on. Yeah. So, well, this brings up, though, like the horoscope on the note of the horoscope, you know, not actually being proof of anything and not actually being a valid method of prediction. Correlation is not causation just because a pair of variables change. It doesn't mean one is causing the other to change. So Nostradamus's horoscope and Anthony's unethical behavior in the card game may have correlated with his actions, but that did not cause him to steal the macaroon. So this is, let's see, perhaps our last quiz question for the day. So which of the following are examples of us committing logical fallacies of assuming correlation implies causation? So, A, people with smart watches tend to be healthier. So smart watches are good for your health. B, Miss Paws is sad every day that it happens to rain. So her sadness causes the rain. LDL cholesterol, that's low-density cholesterol, is associated with low mortality. Therefore, high LDL cholesterol increases mortality. And finally, B, I walk two miles more on days when my car is not working. Thus, not driving leads me to walk more. So, or E is A through C or F is A through D. So which of these logical fallacies are, yeah, go ahead. Yeah, which of these logical fallacies are correct? Yeah, and for people who are asking about a recording of the session, we will be emailing it. Joanna will be emailing the recording of the session once the meeting is done. Yeah, so, sorry, which of these are actually logical fallacies and which of these are correct? So pick the ones that you think are logical. Yeah, very quickly. I'll give you three more seconds. Three, two, one. Mahirat, you can explain very quickly. So the correct answer is E. I mean, you need to have more proof for A through C. And certainly, I think some of these are ridiculous. Like B, Miss Paws does not cause the rain with her sadness. But D definitely makes sense. Definitely, the consequence is a very clear conclusion that not driving would lead me to walk more since I'm not walking to work or something instead of driving. Okay, great. So are we done with our mystery? In reality, he sold the macaroni because he wanted it, but because someone paid him to. Someone who is afraid of losing the competition tonight. So Anthony is not working alone, apparently. Look, Anthony had an uneaten macaroni and a pile of cash folded using the map fold. If Anthony wanted to eat the macaroni, he would have. He just wanted to prevent the competition from taking place. This pile of cash must be from whoever hired him. This is your signature fold, isn't it, Dr. Fisher? I noticed all night that you were leading us away from the true culprit with your poor statistics. Now it makes sense why. So if you remember earlier, Dr. Fisher had his jam test. And when he was compiling the results, he wrote it on a piece of paper and then folded it in half using the map fold. So that seems pretty clear that Dr. Fisher is the culprit. And apparently the police agreed. So Dr. Fisher and Anthony are both of our culprits. Okay, so hopefully you all enjoyed that and you enjoyed figuring out who was behind the mystery. Yes, let's see. One second, I need to... If you guys can quickly write your names, that way I can correlate your answer questions. So what will happen is after we close the meeting, I will get to see your poll results and I can see who scored the highest and there will be a prize. The prize will be one of our camps, Joanna, and I will discuss and we will get back to you. So yeah, if you did super well today, I will be emailing you. Yeah, so please submit your... I'll give you 30 more seconds for the name. But you can show the next slide and next slide while people are writing down their names. So we hope you enjoyed a Method Plus production. Thank you for joining us at this very lovely night at the theater. Our cast of characters, Dr. Rue, of course, and also Ms. Paws and Mr. Whiskers joined us tonight. So, or this morning, I don't know why it's tonight. So if you would like to learn other cool topics, you can actually join us. We have a YouTube channel, youtube.com slash at Ms. Paws and Friends. So our YouTube channel name is Ms. Paws and Friends and we do different educational videos on mathematical topics, scientific topics, history, English, everything you can imagine and fun documentaries, mini documentaries. Okay. And by the way, the whole, the Dr. Rue that you saw, the backdrop, we needed a green screen. So the backdrop, if you can go one slide back, the backdrop was one of our Math Kangaroo shirts. The kangaroo is also award that we received when we were students. That shirt is from 2008. So that was maybe, I don't know, maybe before some of you guys were born. Wow. How I feel. Awesome. Well, thank you all very much. If anyone has questions, feel free to ask in the chat. Otherwise we will get back to you if you did very well in today's competition and congratulations on doing very well in the Math Kangaroo as well. Congrats on finding Dr. Fisher and Anthony the Fox.

probability

statistics

paradoxes

fallacies

interactive quizzes

logical reasoning

statistical fallacies

narrative storytelling

critical thinking

math concepts

logical fallacies

software engineers