 This is a two-part question to display both techniques. It says that, hypothetically, two alleles on different chromosomes control eye color. You've got B for brown eyes, capital G for green eyes, and then lowercase Bs for blue eyes. You can only have blue eyes in a homozygous recessive situation only. Realized result from at least one capital B and one capital G. So assuming the true cross of the P generation, remember true cross means that you're going to have homozygous dominant and homozygous recessive. All homozygous dominant will be big B, big B, big G, big G. All homozygous recessive would be little B, little B, little G, little G. When you look at this, this person will have hazel eyes. This person will have blue eyes. Remember they're blue because you contain this. Hazel because you have at least one capital B and one capital G. So this is your true cross for the P generation. Knowing this cross, we know that the F1 generation that will be produced is heterozygous. This means one capital B, one lowercase, one capital G, one lowercase. These also will have hazel eyes because you have one capital B and one capital G. Now that we know the F1 generation, we need to figure out the F2 generation. So we're going to conduct a binomial expansion. The first thing to figure out for binomial expansion is we have A plus B to some type of power. We need to figure out what that power is. So two alleles, we have two copies of each allele, two times two is four. So our binomial expansion will be A plus B to the fourth power. So now what we're going to do is we're going to set up the binomial expansion. So binomial expansion A plus B to the fourth. Remember you write that out four times. You're going to multiply the first two parentheses together. Remember A times A is A squared. A times B is AB. B times A is AB. B times B is B squared. We always simplify A squared plus, remember one plus one is two AB plus B squared. Then you have to add in another parentheses and perform the same calculation. Once you get that answer, you're going to perform another multiplication parentheses. When you get done, your end result is listed for you right here. We've got A to the fourth, four A cubed B, six A squared B squared, four AB cubed, and then B to the fourth. What this means when you go through is that this has four capital letters. This has three caps, two, one, and zero. If all four caps are present, then you know you're going to have a homozygous dominant genotype. If you have three caps, then you have at least one lower case letter. Two caps, you have several different combinations. It can be both capital B's, capital G's, or a mixture of the two. One capital, it's either going to be a B or G. And then zero caps means that you have all recessive traits. Remember that big B and big G means that you're going to have hazel eyes. So this will have hazel eyes, will have hazel eyes once again here, and once again here. We will not have hazel eyes in any of the other offspring. Our initial question was, assuming the true cross, what are the possible F2 generation phenotypes? We know that we can have one, five, eleven with hazel eyes. We could have a possible ten with brown eyes, and we could have ten with green eyes. We could only have one with blue eyes, and that blue eyes is very rare because you have to have all recessive traits for it to happen. So those are your phenotypes. You go through and list out the genotype, which will give you the phenotype. That's using binomial expansion. Second part of the question where we talked about eye color is we wanted to know what is the probability that an offspring from this cross will have hazel eyes. Remember, we're targeting the probability that we'll have a big B and a big G in the offspring. So first things first, we're going to look at the gametes. The big B here, gametes for brown eyes and blue eyes. When you combine these, you have a three-fourth chance of having your big B present. How we know this is that a lot of people aren't comfortable with probabilities. So they like to draw punent squares to start off initially, and then from there, you can start figuring probabilities out in your head. So let's do a punent square. If we mix big B, little B, little B, big B, what is our resulting offspring? When we bring them across, we find that three of the possible four combinations all contain a dominant allele. Remember that dominant allele is what we're after because it's going to lend us to that overall hazel genotype that we're looking for. So three-fourths of our offspring have the big B. That is the probability that we will pass that on. Now let's look at the G's. One parent, little G, little G, the other parent, big G, little G. Punent square once again. When we do this, we only find that one-half of our four has the big G that we're after. So one-half of the offspring. We then take those two probabilities, multiply them together to get three-sixteenth. There's a three-sixteenth chance that your offspring will have hazel eyes.