We have, I think a lot of people, so Mr. Bagheerz, if we want to get started. Sure, let's get started. I want to wish all of our Math Kangaroo 2025 winners a round of applause and congratulations. Great job. Some of you may know that we do this META Plus math theatrical production every year. So this year, our presentation is called Do My Genes Fit? A Lesson in Genetics and Math. So a little bit about us. I'm Bagheerz Mehta. I'm one of your presenters for today. I received my bachelor's and master's in electrical engineering and computer science in 2022 from Stanford. I'm one of the co-founders of META Plus. I currently work as a software engineer at Microsoft as well as a part-time researcher at the Stanford Genome Technology Center. And like all of you, I'm a Math Kangaroo alum. I've placed first place internationally twice and I was top 10 nationally many times. Hi, everyone. I'm Ms. Haripriya. I did my bachelor's and master's in electrical engineering and computer science in four years at MIT. I'm the other co-founder of META Plus. I'm also a software engineer at Microsoft. And I too was a Math Kangaroo alum. I placed top 10 nationally. And yeah, congratulations again. So as part of our presentation today, we'll be asking you some math questions based off of the topics we're teaching. And we'll be trying to do these questions through Zoom's polling feature. So let's create a practice poll question just so we can test out if the poll works on your side. And if you don't see the poll, the poll should have popped up now. But if you don't see it, then please make sure you're connected to Zoom from the application. I don't think it works in the browser. And if you don't have the Zoom application downloaded, download it now, please, or join on a device that has it. So you should be able to see the question. It's kind of a doozy. What color is typically associated with oranges? What's hilarious is that someone did have a sense of humor and chose green. OK. So if all of you can see the poll, that's great, because then we'll be able to collect your answers, collect them, and then award a prize to the winners at the end. Yeah, I guess that's true. That is true. OK. So as I mentioned earlier, our topic today is genetics. So what is genetics? Genetics is the branch of biology that studies genes, heredity, and variation in living organisms. It explains how traits and characteristics are passed from parents to offspring through DNA. So how is genetics used? Well, primarily, it's used to understand how traits are inherited, and it's often talked about in the context of diseases that are congenital, meaning ones you can inherit. This is why an eye doctor might ask you about your family history of cataracts, for example, to understand if you are also at risk. It's especially important for preventive care, for example, genetic screening for someone to understand what diseases they may be at risk of. Another application of genetics is medical advancements in treating diseases and forming personalized medicine. And this is somewhat related to the work I do at the Stanford Genome Technology Center. So genetics can play a key role in diagnosing, treating, and preventing diseases. It allows us to design better treatment options because we can then give people exactly the medicine they need or recommend them the vitamins, nutrients, or diets that is best suited for them. More recently, CRISPR is a tool that was developed to edit DNA, and it can be used to genetically engineer bacteria that attack cancer cells, for example. And there's many other applications of how technology is used in combination with genetics in order to get people the best care that they need. So this is one of the most interesting engineering questions of the 21st century. Genetics can also be used to breed desirable crops. So here you can see rice, and then you can see golden rice. Golden rice is a strain of rice that's been genetically engineered to have higher levels of beta carotene, which can be used to create vitamin A in the human body. And it's very helpful for areas where people are lacking in vitamin A. Genetics can also help with breeding plants that have drought resistance or higher yield, improving food security around the world. Genetics can be used to understand evolutionary biology. It provides insight into how species evolved over time and how all life is connected through common ancestry. I can also use it to create something like this phylogenetic tree to see how closely my sister is related to apes. And the last application I'll be talking about is forensics. So DNA evidence is a powerful tool in criminal investigations. There have been many famous cases over the past decade in California and Sweden that were solved with genetics. Certain genes are unique, and the composition of DNA or existence of nearby gene pairs can reveal how closely related people are and how many generations back they share an ancestor. And you can use this to construct a family tree and figure out exactly how a person or how the victim or perpetrator might be related to the genetic subjects that you have at hand. Can you hear that? There's a phone ringing on my side. I think that might be Dr. Roo, the mascot of Math Kangaroo. Let me see if we can pick it up and see what he needs help with. Let me share my screen so you have a chance to hear what we'll be discussing with him. Hi, Dr. Roo. What's going on? Mr. Whiskers, I urgently need your help. I was doing a study on gerbils, but now all my gerbils are all jumbled up. What do you mean? I wanted to compare two generations of gerbils for an experiment. But they chewed through the labels on their containers. I can't tell who is who. Let me see what I can do. Can you send me some information about them? I need their DNA sequences. Will do, thanks. Okay, let's split like a nucleic acid and get to work. Okay, so before we can help Dr. Roo with his question, let's start learning about the basics of genetics. So the fundamental unit through which genes are carried is DNA. So DNA stands for deoxyribose nucleic acid. It's known for its double helix structure, which looks a little bit like in the graphic on the top right. And it's made up of deoxyribose, which is essentially some kind of sugar, phosphates, and nucleotides, i.e. bases. So we refer to these green, red, yellow, and purple things that you see in this diagram as bases. And so DNA is used to encode a series of bases with one of four in any given location. We have four types of bases, adenine, cytosine, guanine, and thymine. And we typically abbreviate these as A, C, G, and T. A sequence of bases at a given location that share a common purpose is called a gene. So genes are the fundamental unit of DNA. And genes code for certain traits and can be used to produce proteins or regulate behavior of other genes. Basically, this is how we inherit different traits from our parents and pass them on to offspring. An allele is a specific version of a gene. For example, for a gene with six bases, if we know it's six bases long, then CAT, CAT is a possible allele. So that brings us to our first question. If a gene consists of five bases, so it's five bases long, and there are four possibilities for each base, how many possible variations of the gene can exist? So the question is, how many possible alleles can exist for a given gene if that gene is five bases long? And the poll question should have popped up. Please don't put your answer in the chat. Please just answer the quiz. That way if it's right, you get it right. And if it's wrong, we don't lead others astray. It is a competition, a friendly competition, but there are some prizes at the end. How are you doing on responses? We have, we're waiting on like 15 more people. So I will cut it off at the one minute. Right now we have 15 more seconds. So make sure to put something right. You have 25% chance of getting it right. Three, two, one, zero. I'm cutting it off. Okay. So the answer is C. So why is it C? Well, with four options for each base and five total bases, there are four times four times four times four times four. So that's four to the fifth power equal to 1,024 possible variation of a single gene. Now that brings us to question two. In reality, genes can be a hundred, hundreds or hundreds of thousands of bases large. That means a gene with 1,000 bases has four to the 1,000 possibilities. That's a number in between 10 to the power of X and 10 to the power of X plus one, where X is some integer. What is the sum of the digits in X? So the question here is you have to figure out what is X, like what is the order of magnitude essentially that four to the 1,000th power is, and then add up all the digits that are in X. So here's some info that can help you. Let's define a number A such that 10 to the power of A is equal to two. And for those of you who are familiar with logarithms, you can maybe say that in the chat. If you are, A is a log of two base 10, but even if you're not, that's fine. A is approximately equal to 0.30102. And sorry, the font is not rendering properly, but that question mark is supposed to be an approximately symbol. So, what is the sum of the digits in X? Think about all your exponent rules, if you're familiar with logarithms, that might help, but that's not necessary. So those of you who just came in, basically answer the question and answer the future questions that we have through the quiz functionality. Okay, so we have 55% participants have submitted their answer. This one is a little bit harder, so I'm going to give a few more seconds. Okay, I'm waiting for 10 more responses, so I'm going to cut it off at two minutes, so we have five more seconds. Three, two, one. Okay, if you want to submit an answer, just go ahead, randomly choose, because I am ending the poll. Okay, so the answer is eight. Okay, and how do we get eight? Well, we know that there's four to the 1,000 possibilities, that's equal to two to the 2,000th power, right, because four is two squared, and two squared to the thousandth power is two to the 2,000th power. Now, we also know that two is equal to 10 to the power of a, which is that number that we just defined, so that's also equivalent to 10 to the log of two base 10 power, or base 10 if that's something you're familiar with. So two to the 2,000th power, that's equal to 10 to the a, because we know two is equivalent to 10 to the eighth power, raised to the 2,000th power, so that we can then multiply the powers, right, so 2,000 times a, right, that becomes the new power of 10, and just simplify that statement, and we know a is equal to approximately 0.30102, so if we multiply that out, that means that two to the 2,000th power is approximately equal to 10 to the 602 power, 0.04, so that means two to the 2,000th power is roughly between 10 to the 602 power and 10 to the 603 power, that means x is 602, and then the sum of the digits in x is eight, so the answer is eight. That was a bit of a trickier question, don't worry, we have some easier and harder questions alternating throughout the presentation. In reality, of course, the genes can be hundreds to thousands of bases large, as I said before, but typically only a few dozen or a thousand variations are found for most genes across most humans, that's because evolution tends to filter out harmful variants, and neutral mutations may accumulate slowly over time, but there's also many variations that tend to have the same result in the genetic code, because the genetic code has redundancies, so humans end up being overall fine, despite the overwhelming number of possibilities, we tend to have only a few number of alleles, and usually the good ones survive, only the good ones survive. Okay, so now chromosomes are the packages that an organism's DNA is in. Chromosomes and mammals are typically found in a pack of chromosomes, chromosomes and mammals are typically found in pairs with one chromosome coming from each parent, this means every mammal gene typically has two alleles, not necessarily distinct, because we have for every type of chromosome, we have two, right, we have pairs of chromosomes, and that means for every gene in that chromosome, we have two copies, so there's two alleles, right, and you know, those alleles may be the same, right, they may be exactly the same base for base, or they might be different. Humans have 23 pairs of chromosomes, so that means we have 46 in total, and you can see somewhat what the chromosomes look like in the top right hand corner. Some animals, like cats, have fewer chromosomes, they have 19 pairs, and some animals, like dogs, have more, they have 38 pairs of chromosomes. So the combinations of alleles that an organism has is called its genotypes. The actual traits exhibited by that combination of alleles is the phenotype. So genotype is basically what you see in the DNA, and the phenotype is the observable characteristic. So in the top right hand corner, this is a way that you can use to draw out chromosomes, and you can see that these are the same chromosome, but this is like the same type, but they're a pair, so we, in the same spot of the chromosome, we have the same gene, but on the left, it makes the flower color white. That's what that allele represents. The phenotype would be white that corresponds to this allele, and then the phenotype corresponding to the allele on the right is purple flowers. So the gene can have two alleles, if there's two copies of a chromosome, and then each allele could be different, it could be the same. So let's say, for example, that the sequence ACGT is an allele for a gene that influences flower color, and let's call this gene B, and if a flower has two copies of this gene, its genotype is capital B, capital B. If this causes the flower's color to be white, a white flower color is the flower's phenotype. So hopefully that makes sense, how we use the words genotype and phenotype. Okay, so since humans have two copies of most genes, and we might have two different alleles, and the fancy biology word for that is heterozygous, right, so if we have two different alleles as a result of having two copies of most genes, how do these different alleles influence traits, right? How do you know, for example, like in the flower example, right, how do we know if the end result is that the flowers will have purple petals or white petals? Well, it depends. Some alleles are dominant, while others are recessive, meaning that if even one copy of a dominant allele exists in a human, that allele will determine the phenotype no matter what the other allele is. So in this example, we have a black cat, and we have this orange tabby cat, and the black cat passes its alleles, which encode for the black phenotype, to its child, while the tabby cat passes the color orange to its child. Now, since black is the dominant allele, the offspring ends up being black, despite having inherited an allele for the orange color from its mother. Yes, someone said in the chat that this is like eye colors. Yes, exactly, yes. So some alleles are co-dominant, which means both phenotypes are observed. So in this case, we have both the black and the orange are observed, right? Part of the cat is black, some part of it is orange, right? And some alleles are incompletely dominant, which means neither phenotype is fully observed, but rather something in between. So in this case, it's not quite black, it's not quite orange, it's halfway in between. So these are all types of behavior that you might see in different genes. Usually, one gene has a fixed behavior, so it'll either be dominant recessive, it'll be co-dominant, it'll be incomplete dominant. But I tried to use the fur color as an example of what it would look like if the allele was dominant, co-dominant, or incompletely dominant. Now, because this can get a little complicated, we'll only focus on dominant and recessive alleles for now. There are a lot more complexities, such as mutations, that might lead to someone having blue eyes or blonde hair, linkage, this is how some genes tend to be related, and epigenetics, this is how environmental factors can influence whether or not your genes turn on and off. And this could mean that our statistical models don't end up being quite correct, especially when we have only small sample sizes, but we're going to ignore all of these complexities for today. So for now, just focus on dominant and recessive alleles and pretend that all the genes I'm talking about follow this dominant recessive pattern. Okay, so in order to better understand what offspring will, what results they'll exhibit, once you know that they have some, once you know what the dominant recessive alleles are, right, so if you know the genotypes of both parents, you can draw out a diagram to predict the probability of each genotype and phenotype of their offspring. So we can start by labeling a dominant gene. We usually use a capital letter by convention, so we can use the letter capital B. Then we'll label the recessive gene with some letter, and we can use the same letter but lowercase, so let's use B, lowercase B. So this is what a Punnett square is. You can see, since each parent has two alleles, we can list the alleles for one parent vertically and the other parent horizontally across the square. So one of the parents has a capital B and lowercase B, and the other parent also has a capital B and lowercase B as their two alleles, right, so they each have one dominant and one recessive allele, right. So we can set this up across the squares and divide it up into quadrants, and we can use this to analyze what the offspring's chances of getting each genotype and phenotype is. So let's say both parents have a genotype of BB, as I said, then the Punnett square will look something like this. Now note that we order dominant alleles before recessive ones just to make it easier to read through, so we have the capital B before the lowercase B. Now what's the probability then that an offspring will have one dominant gene and one recessive gene, i.e. capital B, lowercase B? Well, the Punnett square makes it easy to see that the probability is two quarters or half, right, because two of these four cells have the capital B, lowercase B combination for the genotype, right, and there's four total cells. So the probability of having capital B, lowercase B is a half. Okay, now what's the probability of exhibiting the dominant phenotype? Well, remember we said that if there's at least one dominant allele present, then the dominant phenotype will be exhibited. That means that if you have capital B, capital B, or capital B, lowercase B, in either case you will exhibit the dominant phenotype. So what's the probability, and by the way, this is just fancy probability syntax, so what's the probability of BB, right, the probability of the event of having BB plus the probability of event of having capital B, lowercase B, right, that's equal to one quarter plus two quarters equals to three quarters, right, so three of our four cells have dominant phenotypes, the ones that are in red. Does that make sense so far? Okay, great. So that brings us to question three. You don't necessarily need a Punnett square for this question, but it might help you out. So we have some data on a family with a congenital disease. Remember, that means the disease is inherited through genes. Based on this family tree, what do we know about this disease? And you can see that there's a legend down here, so the triangle means male, circle means female, and gray means we know that this person has no disease, the blue means we know this person has the disease, and yellow means we're not sure. So here are the four answer choices. A, we know that this disease is a dominant phenotype because the percentage of people with the disease goes up across generations. B, it is a dominant phenotype because daughter one is healthy. C, it is a recessive phenotype because the father has it despite not having a parent with the disease. And D, it is a recessive phenotype because the mother has it despite not having a parent with the disease. Think about what statements you know to be true, what you know, what statements you don't know to be true, what can you prove. How are we doing on time? So we have three fourths of the participant respond. I can give maybe. Let me see. We have so we have time. Less than 10. So I'll go to the two minute 15 seconds. And we're at two minute five. Three, two, one, I'm going to end the poll. Okay, so this is a bit of a trickier question, because we have this family tree with paternal grandparents maternal grandparents, a father, mother, a daughter, one son one and daughter to. But the answer should be pretty simple. So, a, it's a dominant phenotype because the percentage of people with the disease grows up across generations. This is never the right answer right dominant phenotype, just means it will be exhibited. If the dominant allele is present right that doesn't mean that it will dominate and be the only phenotype that exists. It doesn't mean that it will go up across generations. Be. It's a dominant phenotype because daughter one is healthy. That's just a meaningless statement, it doesn't really mean anything. It's a recessive phenotype because the father has it. If you look at the legend, we don't know what the father status is we don't know if the father has the disease or not. So, that leaves by process of elimination D, it is a recessive phenotype because the mother has it, despite not having a parent with the disease. It is a recessive phenotype, because the mother has been having a parent with disease. If the disease were the dominant phenotype for a person to not have a disease, they would need to have two copies of the recessive gene. In that case, both maternal grandparents would have had two copies of the recessive gene. So the mother would have inherited two copies of recessive gene and not have the disease. Therefore, the disease must be recessive right it must be the recessive phenotype to have the disease, and each material maternal grandparent must have had one recessive gene. We can see this with a Punnett square right so what if the Punnett, what if being healthy was a recessive phenotype. Right, then what's the Punnett square look like, in that case it would mean that being healthy would require to recessive alleles. This means that the mother would have 100% chance of also having to recessive alleles and being healthy, but we see that the mother is not healthy she has the disease. So what if the Punnett square of not having a disease was a dominant phenotype. So what is the Punnett square if not having a disease was a dominant phenotype. In that case, it would mean that being healthy would require at least one dominant allele. We know that the maternal grandparents, then each have at least one dominant allele so let's represent that with capital B. We don't know what the other allele is for either parent so we can represent them with x and y. So, our four cells have three of them have a capital B which means those three cells represent phenotypes that without the disease, right, but x y could be anything right so x y could both be lowercase B, meaning that both of them represent having disease, which means x y is BB. So this means that the phenotype for this that corresponds to this would be having the disease. And this is how the mother would have the disease. This means the mother could have two recessive alleles and thus have the recessive phenotype, ie exhibit the disease. So if the diseases are recessive phenotypes, this means that they're only exhibited if a person does not have a single dominant allele. That's a pretty good thing because that means you have to have, if you have even one dominant allele, you won't have the disease. Okay, now question four, it's a slightly different family tree. Right. So it's a new family. So if the disease described in this family tree is recessive. What is the probability that daughter to has the disease. That is, and that's question one and if question two is if daughter to is healthy. What is the probability that she has at least one recessive allele. So someone's asking the chat to recessive phenotypes make a dominant phenotype. So if you have two parents with recessive phenotypes, the child that they have will have, by virtue of the Punnett square we saw earlier, it must also have two recessive alleles, which means they will also have the recessive phenotype. I assume that's what your question was asking. Please let me know if I understood. Okay, so John seems to have anticipated this question for in the chat. So again, the questions, the disease described as recessive, what's the probability that daughter two has the disease? That's part one. Part two, if daughter two is healthy, what is the probability that she has at least one recessive allele? And here's a hint, just because this family tree doesn't show you the information about what the phenotype or genotype is for the father doesn't mean that you can't perhaps figure it out. Three-fourths of the participants have answered, so I'll give you maybe until three minutes. So we are at 2.25. And remember, we're assuming things like mutations don't exist. We're following the strict rules of what's called Mendelian genetics. It's dominant recessive alleles. Okay, three, two, one, zero. I'm going to end the poll. Okay. I'll show the answer. It is B. So how do we get B? The second part of the question is a little easier to answer, so let's start with that. So the mother has two recessive genes, meaning that all of her children will inherit one recessive gene from her. Therefore, the probability that daughter two has at least one recessive gene is one. And we can see that from a Punnett square here. So we don't know necessarily what the father is at this point, but we know that the mother has lowercase b, lowercase b as her phenotype, or sorry, as her genotype, because she's exhibiting the recessive phenotype, right? So we can see that all of these cells have at least one B in them. So there's 100% chance of inheriting a recessive gene. Okay, now the next part of the question, let's try to first reverse engineer what genotype and phenotype the father must have. So for daughter one to not have the disease, a dominant gene must have come from the father, right? Because if a recessive gene came from the father, then that means that a recessive gene also came from the mother, so the daughter one would have exhibited the recessive phenotype, but she didn't. That means the father must have given the daughter a dominant allele, right? So that means the father must have a dominant allele, if that's what the daughter inherited from him. Now, for son one to have the disease, then he must have two copies of the recessive gene, meaning that a second copy of the recessive gene must have come from the father. Therefore the father has one dominant gene and one recessive gene, right, because he gave one of the dominant genes to his daughter and one recessive gene to his son. Therefore the probability that daughter two will inherit a second recessive gene from her father is half, right? Because we know that the probability she already has one recessive gene from her mom is one, right? So what's the probability that daughter two has the disease? Well, it's essentially equivalent to the probability that the father will give his daughter the recessive allele. He has only two alleles, right? And one of them is recessive, so there's a 50% chance. And we can see the Punnett square for this as well. So we reverse engineer the father as one capital B and one lowercase b. And we see that two of these four cells have two lowercase b's. So the probability the daughter two will exhibit the recessive phenotype, right, and inherit a second recessive gene from her father is half. Okay. So hopefully all those questions made sense. This is a bit easier. Two parent flowers are cross-pollinated, producing 100 seeds. All seeds are planted, resulting in 100 offspring flowers. 50 have purple petals, which is the dominant phenotype, and 50 have pink petals, the recessive phenotype. Based on this outcome, what is the most likely genotype of each parent flower? So this is kind of using a Punnett square, but maybe you're trying to reverse engineer what are the parents' genotypes. So the four answer choices, you have two capital B's, capital B and lowercase b. B is capital B, lowercase b, and capital B, lowercase b. C is capital B, lowercase b, and lowercase b, lowercase b. And D is capital B, capital B, and lowercase b, lowercase b. So someone asked, wouldn't the answer to part two of question four be greater than one? The question was asking about the probability. Probabilities range only between zero and one. So the chance that she'll have at least one recessive gene is 100%. Now if you're asking what's the expected number of recessive genes she'll have, you can say that's 1.5, because she definitely has one from her mom, and there's a 50% chance she'll get one from her dad. I'm going to close the poll in 15 seconds. So what is the most likely genotype of each parent flower? That would result in half of them exhibiting the dominant phenotype and half of them exhibiting the recessive phenotype. And yeah, we are past the time, so I am going to end the poll in 3, 2, 1. Okay, so the answer is C, capital B, lowercase b, and lowercase b, lowercase b. And we can draw the Punnett square for each of these. So for A, if we were to draw the Punnett square out, you can see that all four of these cells have a capital B, which means 100% of the offspring will exhibit the dominant phenotype. For answer choice B, we'll see that three of these four cells have a capital B, meaning 75% of the offspring will exhibit the dominant phenotype. And by the way, this is not exact, right? This is an approximation, but when you have 100 offspring, there's a pretty good chance that if you have a three-quarter chance of exhibiting the dominant phenotype for each one, then you'll get about 75 offspring exhibiting the dominant phenotype. Now here, 50% of the offspring will exhibit the dominant phenotype because we have 50% with a capital B, so two of the four cells, and then 50% of the offspring will exhibit the recessive phenotype, so that's a lowercase b. So C must be the correct answer, but let's just see the Punnett square for D. In this case, 100% of the offspring will exhibit the dominant phenotype, so all four of these have a capital B. And what's interesting is that if you look at the distribution of letters, sometimes you'll have two capital Bs and two lowercases b, or capital B, lowercase b, capital B, lowercase b, right? If you look at that, the same four alleles are used from the parents, but depending on how it's inherited, which one it's inherited from, you'll have different Punnett squares. So B and D have the same alleles coming from the parents, it's just that the alleles are distributed across the parents in a different order. So we get a different Punnett square. Okay. Someone's asking, are we going to do 4x4 Punnett squares? Yeah, you could extend this concept to 3x3, 4x4, you could keep extending this answer, and especially it helps when you do something like 4x4 Punnett squares if you had, for example, two genes influencing a trait. That's called polygenic traits. And then you would have maybe like, or yeah, yeah, you would have four cells because there are four combinations of genotypes that can be inherited for the two genes. Someone asked why B instead of any other letter, is it just chance for this webinar or is B the common letter used in genetics? Yeah, it's just by chance for this webinar, you could use different letters. Our question six, we have the letters R, S, and T. So this is not a 4x4 Punnett square. This is, well, maybe depends on how you want to solve this problem. I won't tell you how to do it, but it might be a little bit more complicated than your normal 2x2 Punnett square. So here's the question. A plant's flower color is determined by three genes, R, S, and T. Purple flowers require at least one dominant allele for each gene. So that means if you have at least one capital R, it doesn't matter what the other one is, one capital S, one capital T, right? It doesn't matter what the other three alleles are, right? Then we see that the flower will be purple. Otherwise, flowers are pink. So two plants with the genotypes R, R, S, S, T, T, right? Where one of the R's, S's, and T's are capitalized and one is not, right? Are cross-pollinated. What is the probability of producing a purple flowered offspring? So the answer choices are 1 64th, 1 8th, 3 4th, and 27 64th. Yeah, and so for the person who was asking why B instead of any other letter, I actually went through all the letters in the alphabet and felt that the easiest one to say twice in a row was BB. Some examples that you'll see will, if they're describing like, you know, maybe yellow fur or green fur, they might use like YY or GG, right? Green fur isn't a thing, I guess, but yeah, so it's, it's often the letters related to like maybe the trait that you're examining. So maybe for fur, it would be capital F, lowercase f. But I think the capital B, lowercase b is this easiest to distinguish in any font. So that's the letters I went with. Some people, you know, use the letter P for Punnett square, right? Think about why I didn't want to use P twice in a row. Can I unsubmit my answer? I don't think you can, unfortunately. I'll give you guys five more seconds. Three, two, one. Okay, there's like 88%. Let's get a few more answers in the next few seconds, or I'm going to end the poll. Okay, five, four, three, I'm really going to end the poll this time. Two, one, zero. Okay, I'm ending it. Okay. So the answer is D, 27 64th. So how do we get that answer? So let's look at the Punnett square just for R. We have a three quarters chance of having a dominant allele of R, right? Because we have this capital R, lowercase R, capital R, lowercase R. We go through all the cells and we see that three of the four have a capital R, right? And the same will be true for S and T, right? Because we can, you know, use a capital S, lowercase S instead everywhere. And the Punnett square would be exactly the same or distribution wise and capital T, lowercase T, same thing, right? So that means for each of these alleles or each of these genes, R, S and T, we have a three quarters chance of having a dominant allele. For the color purple, all three genes must have a dominant allele present. And since the genes are independent, the probability of this happening is three quarters times three quarters times three quarters equals to 27 64th. So that's how the answer is D. Okay. Just a quick vibe check. How are people doing so far? Can you maybe put in the chat like, hey, does this make sense? Do you understand Punnett square? Do you understand what we've been discussing so far? Maybe just great. Good. Okay. A lot of people are saying great. Good. That's good to hear. Excellent. Okay. I am happy to hear this. Okay. Question seven. So a plant's flower color is determined by three genes. It's that same plant from before. Right. And the same thing is true about requiring at least one dominant allele. Right. For each gene in order for the flower to be purple. Right. So and those two same plants are cross pollinated. But now the question is, if the offspring has purple flowers, what is the probability it is heterozygous for all three genes? Heterozygous means you have different copies of the allele. Right. So you don't have two capital R's. You don't have two lowercase R's. You have one capital R and one lowercase R. Right. So the probability that's heterozygous for all three genes, what we're really asking is what's the probability that the genotype of the offspring will be capital R lowercase R, capital S lowercase S, capital T lowercase T. Since that's the only combination where you have something that's heterozygous for all three genes. Some people are asking what grade you learned this. I think I learned this in second grade, but that might just be my dad. I think normally people learn this in grade nine when they're taking biology. Neither of us took regular biology because we took AP biology directly. So I'm not sure how much this is covered in the regular biology, but it's definitely covered in great depth in AP biology. Some people learned this in middle school. I know some of these are not just biology questions, right? They're mathematical questions as well. I'm curious, when did you all learn about logs? I think some people may have learned in middle school, some people maybe in high school. Yeah, the basics they definitely cover in middle school of dominant and recessive genes. Yeah, and for the, someone asked like what about RS squares or RT squares or ST square? You could make a square with now like four things on the top, four things on the bottom for like all the combinations of RS that a parent could pass on to its child, right? You could also do something with eight things on the top and eight things on the bottom for all the things that could be passed on to the child. But here's the thing, we don't have to actually draw that out, right? Because drawing an eight by eight, you know, square out, that would be 64 cells to fill in. But I drew a four by four, or sorry, two by two square, which gave me just four cells. And I know that the same property applies for RS as it does for Ss and Ts. So just by symmetry, I can calculate the right answer, rather than having to draw out the whole thing. Now for some genes that are more complicated, maybe if it doesn't follow the strict dominant recessive behavior, and we want to examine something more specific, like what's the probability we have exactly this genotype? You might want to do some more complicated Punnett squares, like four by fours, or eight by eights, or something like that, right? You might want to use multiple genes, and each cell would contain like, you know, capital R, capital S, lowercase r, lowercase s, something like that. Okay, I'm gonna end the poll, it's been quite some time, so 3, 2, 1, 0. Okay, so the answer is C, 8 27th. And how do we get this? Well, let's look at the Punnett square just for R. We have three entries that could result in a dominant phenotype. Two of them are heterozygous. So that means the chance of having a heterozygous genotype given a dominant, sorry, heterozygous genotype given a dominant phenotype is two-thirds, right? This is, remember, this is conditional probability. So what's the condition? That you have a dominant phenotype, so we're only looking at these cells in red, and then out of them, which ones are, have the heterozygous genotype? Well, these two, the one that are in the larger font, right? So the probability is two-thirds for having a heterozygous genotype just for R. That's also true for S and T, right? Since they follow exactly the same behavior, and they're inherited the same way because the parents have like one dominant and one recessive for those two genes as well. So for this question, all three genes must have a heterozygous genotype. Since the genes are independent, the probability of this happening is two-thirds times two-thirds times two-thirds, right? So the probability that you have a heterozygous genotype for each individual gene, right? So two-thirds times two-thirds times two-thirds is eight-twenty-seventh. Someone's asking why isn't it two-quarters, well, remember, we're only looking at these three cells. We're saying, given that the offspring is purple flowers, right? So if the offspring has the dominant phenotype, then it could not have this lowercase r, lowercase r. Okay, great. Now we're launching into the most complicated topic, but it's also our last one. So we have, I think, two more questions to go. So the Hardy-Weinberg equilibrium or Hardy-Weinberg principle, what does it mean? So it's a principle that predicts allele slash genotype frequencies in a population and describes how they will stay consistent after a generation. Now, there's a little, a lot of caveats here, right? A lot of like conditions, right? We're assuming there's no mutations. We're assuming there's no additions to the population. We're assuming that there's random mating. We're assuming that there's no catastrophic events, et cetera, right? So I feel like there's so many things I'm listing here, I feel like one of those ads for medicine. So please contact your doctor if you feel any side effects, right? So let's demonstrate this using a Punnett square, right? Assuming that there are two alleles of a gene in a population, capital H and lowercase h, we have four cells, right? Each cell does not necessarily have the same chance of occurring, right? So what we're just doing here so far is trying to figure out, okay, if a population has two alleles for a gene, right? Capital H and lowercase h, right? What are the possible combinations of genotypes that can be inherited, right? Now this doesn't mean that like every parent has capital H, lowercase h. It just means that there's a capital H and lowercase h somewhere in the population, right? This does not mean there's an equivalent probability of inheriting either one, okay? So now let's add probabilities into this, right? Assuming that the probability of inheriting allele H is p, because let's say H is represented as, like, it's p, you know, percentage or whatever of the entire alleles. And let's say the probability of H is q, right? Which means that, you know, q percent or something of the alleles are the recessive one, right? Then we can see that the probability of having HH is equivalent to the probability of having H and the probability of having another H, right? So that's p times p or p squared, right? The probability of having capital H, lowercase h, this cell gives us p times q, and this cell also gives us p times q. So together, right, the probability of having HH is in total 2 times p times q. And the probability of having this lowercase h, lowercase h is q squared, so q times q. All right, so we can expect the probabilities to look as follows. The probability of having capital H, capital H is p squared. The probability of having capital H, lowercase h is 2pq, and the probability of having HH is q squared. So, after one generation, the population will reach this equilibrium of genotypes and stay there. This is so well proven that if a population has two different of a proportion of genotypes after one generation, this suggests one of our many assumptions not hold or that we sequence the DNA incorrectly. So does everyone get what this is saying? Let's put this more concretely. Let's say, you know, 70% of your alleles in a population are capital H, 30% of your alleles in a population are lowercase h. That means 70% times 70%, that's 0.7 times 0.7 or 0.49, right? So after one generation, right, 0.49 of your alleles, or sorry, 0.49 of your organisms in the population will have the genotype of p squared, of capital H, capital H, right? Because that's the probability of p squared. And again, if we have maybe 0.7 for a capital H, right, 0.3 for a lowercase h, what's the probability of an organism having the genotype capital H lowercase h? Well, that's 0.7 times 0.3 times 2, right? So that's 0.42. Does this make sense so far? How it works and how we're predicting the genotypes? How we use these probabilities? Yes? Okay. I'm seeing some yeses. Okay. So here's a sanity check. Let's say if p plus q is equal to 1, which it must be, right, because the gene must be one of these alleles. So if the probability of being capital H is 0.7, right, so if p is 0.7, then q must be 0.3, right? Because if an allele is not capital H, it must be lowercase h, because it's only one of these two alleles. So given that we know p plus q is equal to 1, we also know p plus q squared must equal to 1 squared. Or in other words, p squared plus 2pq plus q squared equals 1, right, if we expand it out with the binomial distribution. So if p squared plus 2pq plus p squared equals 1, we know this, right? That's actually exactly what we see in this Punnett square below, right? We have p squared of your population has capital H, capital H, 2pq of your population has capital H, lowercase h, and q squared of your population has the genotype lowercase h, lowercase h, right? So if you add all of these up, this should give you the genotypes for your entire population or 1, right? So p squared plus 2pq plus q squared should equal 1, right? And we already see that from squaring p plus q equals 1. Okay, so note that the probability of capital H and lowercase h stay the same. So the probability of capital H for the new generation is p squared plus half times 2pq equals to p squared plus pq equals to p squared plus p times 1 minus p equals to p. So what are we doing here? We're saying, okay, there's 100% chance, or like 100% of the alleles in the p squared of the population are H, right? And half of the alleles in the 2pq percentage of the population, right? Those are capital H. So, right, the probability of having a capital H is still p, right? And similarly, the probability of having a lowercase h is still q. So the frequencies of each allele stay the same, meaning that the frequencies of each genotype will as well. Okay, so solving Dr. Rue's quandary, right, if we know exactly what the alleles are for each of the organisms, then we can actually figure out, okay, generation 1, the probability of having capital H, the probability of having lowercase h, and generation 2, right, it should be the same. But in probability 2, sorry, in generation 2, the organism should reach Hardy-Weinstein or sorry, Hardy-Weinberg equilibrium. So we should be able to get, so there's a question about it, but we're running out of time. So let's skip it for now. And I'll just go over the answer. The Hardy-Weinberg equilibrium would be, if we have probability of seeing like gene A with 0.4 and probability of seeing lowercase a with 0.6, right, then we would expect to see 16% of guinea pigs having capital A, capital A, and 36% of guinea pigs having lowercase a. And that was the case in box 1, not in box 2. So box 1 must be from the second generation, it must belong to the kids. Box 2 is from the parents. So that brings us to the end of our presentation. If you'd like to learn more about STEAM and other subjects, my sister and I have a YouTube channel which we can share. Ms. Herpere, you wanted to collect some information and ask some questions at the end? Yes. So I have launched the name poll. So if you can quickly put the name, as soon as this Zoom webinar ends, we will tally the scores and we'll send out emails with the prizes. Mr. Bagheert, if you can go next slide. 50% have answered the name question, if you guys can just speeden it up, because I'm going to ask for email as well. Some courses, you know, some of you said you're interested in machine learning, in the fall, we teach AI 101 with Math Kangaroo, so make sure to check it out. And in the spring, we teach AI for social impact. So that's more like how to use AI tools. AI 101 is more like, mathematically, how does AI work? Okay, so name question, I see two people not submit, but I'm going to now end the poll because I assume, yeah. Okay, let me just put the email question. So we have that information as well. So we can be in touch, but other courses you can check out for the summer. If you like coding, we have competitive programming, it prepares you for USACO and other such competitions. So that's coming up. And then next slide, please. If you are interested in doing some projects with universities, this year, we are partnered up with UPenn, Montclair Saint, University of Birmingham, so many other universities. It's a six-week camp, AI machine learning research boot camp, definitely check it out. Next slide, please. And then we have the AI and visual arts live from August 4th to August 8th. So some of the prizes are going to be coupons for our intro to ethical hacking camp and AI and visual arts. 81% have written the email, can you please make sure to submit the emails? If anyone has any last questions, feel free to let us know. Otherwise, we really enjoyed presenting to you today. Yes. Thank you so much for coming and congratulations again.

Math Kangaroo 2025

genetics

Punnett squares

Hardy-Weinberg equilibrium

congenital diseases

DNA structure

interactive math

gene inheritance

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