 So, let's go through one problem and look at it both with binomial expansion and look at it with probability so that we can see how both of those fit into polygenetic inheritance. Okay. So, our problem is, hypothetically, skin shade is determined by three alleles on different chromosomes. Okay. Dark skin is controlled by capital A, B, C. Light skin is lowercase A, B, C. Remember that these alleles have an additive effect, so an individual with a phenotype that contains any three dark or any three light skin alleles will be the same as any other. Okay. An example for you is an individual whose genotype contains three dark and three light alleles will have the same skin shade as someone that also contains three light and three dark alleles even though they're on different locations and they're different alleles. So the skin shade will be the same depending on how many capital letters you have which is dominant and how many lowercase letters you have which is going to be recessive. So our question that comes from this example is that we need to find the probability that the F2 generation will have one dark allele. So one capital letter while the rest will be lowercase letters and it will have a particular light shade. Okay. So what we're going to do is we're going to start off with the P generation. Remember that for the P generation we have a true cross which is a homozygous dominant and a homozygous recessive parent. So for the P generation you have one parent that is all capital letters and one parent that is going to be all lowercase letters. When you cross these together you get an F1 generation that is all heterozygous. Okay. Heterozygous meaning one capital A, one lowercase A. Very similar to what I've written up here. This is a heterozygous individual. One uppercase, one lowercase for each allele. So we now know the F1 generation and we need to figure out the F2 generation. The easiest way to do this is with binomial expansion. So binomial expansion starts off with a formula A plus B to the sixth power where A is going to represent the number of dominant alleles and B will represent the number of recessive alleles. So basically A is the number of capital letters and B is the number of lowercase letters all to the sixth power. So because you know you're going to raise it to the sixth power because skin shade is dictated by three alleles. Each individual has two copies of each allele therefore three times two is six. So that's how you know. If you are talking about, we have examples later on where it's two alleles so two times two would be four. So we do a binomial expansion to the fourth power. So each term in your expression is going to represent the number of offspring with that particular skin color shade. Remember that shade is something that you can see which is why it's referred to as the phenotype. So the first thing we do after we figure out which power we're going to raise this to, remember three times two gives you six so that's why we do the sixth power. We're going to write it all out. So A plus B times A plus B times A plus B six times. What you're going to do is you're going to start off with the first two sets of parentheses and multiply them together meaning that this A is multiplied to that A and then once again to that B. This B is multiplied to this A and then once again to this B which means A times A is A squared. A times B is AB. A times B once again is AB so you're going to add this up to make two AB and then B times B is B squared. So with the first two sets of parentheses you get A squared plus two AB plus B squared. Then the next thing you're going to do is you're going to take this expression and you're going to multiply it by the next set of parentheses. So once again A squared times A which is A to the third. A squared times B which is three AB blah blah blah so you just continue on. I actually have a slide for you on this that shows the math. So we have our binomial expansion. Remember we got the sixth power because we have three alleles, two copies of each allele. So it gives us the sixth power. We've written it all out. We've taken the first two parentheses, multiplied it out to get this. We add in the next parentheses, multiplied out to get this. Add in the next one. When you add it out we're going to multiply by each term. Now I'm going to work this out for you to show you how you're going to work out a larger multiplication and then you're going to add it to simplify before adding the next step. We're going to follow the same process no matter if it's two parentheses or if you're all the way down on the last one. So A cubed times A is A to the fourth. A cubed times B is A cubed B. Three A squared B times A, three A cubed B, three A squared B. You can see that when you multiply, let's say A cubed times A, you're just adding another exponential or another power. Same thing here. When you time it by A you're going to add another power. So that's all you're doing is you're adding a power every time you have a multiplication effect. So now we're moving on, three A squared B squared cubed. Here on the last one, A cubed plus B to the fourth. This is a really long problem. So when you look at this it can seem really overwhelming but what you're just going to do is add things together so you can simplify. So you've got A to the fourth. That's all by itself. There's nothing to add together. These have the same variables. You've got A cubed B, A cubed B. You always assume that there's a one in front of it if there's nothing else listed. So one plus three is four A cubed B. Those added together so you can take those away and simplify things. Next same variable, A squared B squared. Three plus three is six. Six A squared B squared. So far so good. Assume the one. Same variable. Three plus one is four. A, B cubed as cancel plus B, four. So what we've done is we've taken something that looks completely overwhelming and simplified it on down. We still have two more parentheses to add. So you're going to repeat the process that we just did with the second parentheses, A plus B. And you're going to start all over again. This times this, this times this, and so on and so forth all the way down the line. So now that you continue on with the binomial expansion, you end up with an overall result of A to the sixth, six A fifth B, fifteen A fourth B two, twenty A cubed B cubed, fifteen A squared B four, six AB five and B six. This is your overall result from that binomial expansion. If you did not get that result from your overall binomial expansion, you need to go back. Sometimes it's a simple addition or multiplication over sites that can mess this whole thing up. But once you get into a habit of one times this and one times this, and you repeat that process, you're going to see that it's very easy to get your overall response, okay? So once you finish that and you get all the way down with your end result, remember the capital letters, okay, or these numbers up here tell you how many offspring would have that genotype. So this says 20 offspring will have three capital letters and three lowercase letters. It doesn't matter which ones are capital or which ones are lowercase because if there's three capital, three lowercase, they will have the same shade because it's additive effect, okay? Our original question said, what was the probability that the F2 generation will have one dark allele, which means one capital letter? You can see this has all capital letters. Remember A represents capitals. It says five, four, three, two, one. This is what we want, okay? There are six of the progeny that will have one capital letter, okay? Six out of a possible 64 combinations, meaning that the overall probability would be 9.38% or six out of 64. So now that we've solved that problem with binomial expansion, we want to solve the same problem with probability. So looking with probability, remember that the P generation is a true cross, okay? We're still looking for that one dark allele. After we look at the P generation and the F1 generation, the F1 generation are going to be heterozygates with one capital, one lowercase per allele, okay? In order to find the F2 generation, we first need to find the gametes, okay? Using probabilities. You have to go through the step of finding gametes when you use probabilities. Now the easiest way to do this is to start off with ABC, all in capitals, okay? Moving from right to left on your screen, you're going to make it lowercase while the rest are capital, okay? Now B is lowercase, the rest are capital, and then A is lowercase all the way across. Once you get all the way across, now you're going to repeat the process in the reverse direction with a capital letter. So you've got uppercase A, everything else is lower, B and C. The last one is going to be all recessive, alleles, okay? So the easiest way to do that to make sure you don't miss any is start with dominates, move left when it comes to recessive alleles, and then move right for dominant, and with homozygous recessive. If you follow that same back and forth pattern, you always make sure that you've listed all the gametes. After you're done listing the gametes, you'll find that there are eight possible gametes for the F1 generation. Eight possible gametes that will combine to make the F2. Okay? So we still want to know what is the probability that our F2 will have one dark allele? Well, looking through, let's talk about the probabilities when these combine, okay? So the probabilities of inheriting this particular gamete from one parent is one eighth. One out of eight chances that you will have this one gamete. Remember that each gamete is individual and it is just a random occurrence that this gamete will reach fertilization versus this or even this one. So one eighth chance that you'll get this gamete from one parent. One eighth a chance that you'll get this gamete from another parent. So the probability that the offspring is going to be all homozygous dominates is one eighth times one eighth, also one sixty fourth, okay? So that is the probability that you'll have this particular combination. The combination we want is one dark allele, one capital letter. So the genotypes that give you the shade with only one dark letter are as follows, okay? You can see how I've just taken that capital letter and moved it all the way down the line. So when you add all of these probabilities together, remember you get one eighth from one parent, one eighth from the other makes one sixty fourth. When you add it all up, you get six sixty fourths. Which is also that nine point three eight percent chance, same answer as before, okay? So when I do these problems you can see that you could use either technique, binomial expansion or probability in order to get the same answer. The easiest way to do this is to figure out which works best for you and to use a combination of some of their parts to get the answers, okay? I will kind of go through a couple more of these problems with you. Feel free to carry along on scratch paper as we go to make sure that you can keep up. There's a lot of extra problems if you don't feel like you're getting enough in this one tutorial. There's a lot of problems online with answer codes and you can go through those as well.