# Hardy-WeinbergÂ equation

## Video transcript

Voiceover: Now that we're
familiar with the idea of allele frequency, let's build on that to develop the Hardy,
do this in a new color, and actually, let me
do it right over here, the Hardy Weinberg
principle, which is a really useful principle for thinking through what allele frequencies might
be, or what probability you would have if you found someone, what percentage of the population might be homozygous recessive,
or homozygous dominant, or might be a heterozygote. And it really builds on the work we've already seen with
allele frequencies. Now before we go into
that, we're gonna make some assumptions, and these
are all just assumptions that get us a stable allele
frequency in the population from generation to generation. We're going to assume
that there's no selection, no natural selection is, or
even unnatural selection, is going on that would change
the allele frequencies. So it's not like people with
one of the alleles or another are going to be more or less likely to reproduce and have viable offspring. We're also going to assume no mutation, so we're going to assume that
one of these alleles can't, isn't from generation to
generation turning into another one, or turning
into maybe a different, a new type of trait,
whether it's green eyes, or whatever else it might be. And we're also going to
assume large populations. So that would definitely
throw out the example that we looked at in the last
video, which I did just to understand the notion of allele
frequencies, where we said, hey, look, one out of
the four of the genes in this population, or
one-fourth of the alleles in this population,
are the dominant brown, while three-fourths, or 75
percent, were the recessive blue. We're gonna assume large
populations, so many, many, many, and so that's so that if you
have very small populations, you can imagine that, depending
on how these reproduce, it's very easy to get to
changes in frequencies, but at larger populations, that
helps us make the assumption that we have stable allele frequencies. So once again, this is also that we have stable allele frequencies. Now based on that,
we've already seen if we take the frequency of the dominant trait, which we can denote with
p, and to that we add the frequency of the recessive trait, of the recessive, I
should say, allele, let me be very careful here, the frequency of the dominant allele, and to
that we add the frequency of the recessive allele,
what's that going to be? Well, you see in this case,
it adds up to 100 percent, or one, and it's always going to
add up to 100 percent, or one. Because we're assuming that there's only one of two alleles in the population, so you have 100 percent chance
of getting one of these two, that whatever percentage is going to be, whatever the frequency here is, 100 percent minus that is
going to be whatever q is. So these two things are going to be equal to 100 percent, or equal to one. And now we can start to do a little bit of interesting mathematics. It'll allow us to start thinking about things like homozygotes and heterozygotes, and so to do that,
let's square both sides. So let's square both sides of this, a little bit of algebra in biology class. And so when you square the left-hand side, this is just squaring a
binomial, you might want to review it if this looks like
Latin to you, there's many algebra videos on Khan
academy that go into this, this is going to be p
squared, plus two times pq, plus q squared, and of course one squared is still going to be equal to one. Now what are each of these terms here? What are each of these terms? Well, let's just think about something. p squared is the same thing as p times p. Well, p is the frequency
of your dominant allele. So this is the frequency
of your dominant allele, the percentage of the
allele population, I guess you could say, the allele
frequency, that is dominant, and you're multiplying
that times it again. Well, another way to
think about p, is this is the probability, if you
were randomly to pick one of these four genes,
and here I'm using my over-simplified population, of course, the truths that we're about
to surface to be true, you're going to have to
assume a large population, but in this one right over
here, one way to view p is what's the probability if I
were to pick a gene at random, what's the probability
that it is the variant, or it is the dominant allele,
what is the probability that it represents the brown variant? So that's one way to view
p, so, the probability of getting a, let's
just write it that way, a capital B, a dominant brown allele. So what's p times p? That's the probability of
getting two dominant alleles. Or another way of thinking about
it, this is the probability for someone in the population
to be homozygous dominant, so it's the probability of someone being capital B and capital B. And so by the same
logic, what is q squared? Well, q squared, that's just q times q, q is the probability of
getting one recessive allele, so this is the probability of
getting two recessive alleles. One from your mother,
and one from your father, so this is the probability of, if you were kind of randomly
born into this population, of getting two recessive alleles. Now what is this middle
term, right over here? Well, p times q, so pq, so
one way to think about it, if you said, what's the
probability that from your mother, you get, randomly, you
know nothing about them, or if you pick a random
mother and a random father, what's the probability, I
have to be careful here, so if you're just
randomly getting alleles, what's the probability
that from on one side you're gonna get the dominant, and from the second side you're
gonna get the recessive? So that would be pq,
that would be p times q, so that's getting it from,
say, from one parent, and then that's from the second parent, but what about the other way around? From the first parent,
you have a q probability of getting the recessive one,
and from the second parent, you have a p probability of
getting the dominant one. So there's two ways of
becoming a heterozygote. And so if you add these two
probabilities, what do you get? These are both pq, I'm just changing the order of multiplication. You sum these two, you get two pq. So this is the probability
of being a heterozygote. So this is a pretty neat result. Just by making a few assumptions, and reasoning through this
notion of allele frequency, we're able to come up with this expression that actually is fairly
powerful in thinking about allele frequency in
a population, and actually, the different genotype
frequencies in a population. You see it all makes sense,
these all add up to one. The probability of someone
being homozygous dominant, plus the probability of
someone being a heterozygote, plus the probability of someone being a homozygous recessive, they're
gonna add up to 100 percent, because someone's going to have to be one of these three things. Now I'm gonna leave you
there in this video, and the next video,
we're actually gonna use this Hardy Weinberg equation
to actually come up with some very interesting
results about a population.