And the important thing I emphasized at the very end of the last video was,

with these sorts of assumptions that gametes just come together at random

based on their proportions, we end up with a stable equilibrium.

And, in essence, the allele frequencies don't change over time.

The genotype frequencies don't change over time.

That is, by definition, an equilibrium.

Now, this pattern was first described by these three gentlemen here.

This is Godfrey Hardy over here.

Wilhelm Weinberg over here.

So they're the ones who typically are named for this,

as the Hardy-Weinberg equilibrium.

There's also American, William Castle, who introduced a similar idea around 1903.

But, the important thing was the idea of self perpetuation.

This is what I showed you in the end of the last video that you had an allele,

a big A allele frequency of 0.6 in gametes.

They created the seven genotypes with a little allele, 0.36 with big A,

big A, 0.48 with big A, little A, 0.16 with little a, little a.

The following generation you would again have this .6 frequency of the big

A gametes and correspondingly the .4 for the little a.

Now, up until 1902, several people thought that it was possible that

the dominate alleles would intrinsically increase in the population.

They are after all called dominant.

And some people also assume that rare alleles would just tend to get lost and

have this inherent drive towards loss.

Now it was in 1908, and also probably from the 1903, contribution.

Hardy and Weinberg independently showed that both these assumptions are not true.

That if you can just use this purely probabilistic approach, allele and

genotype frequency will stay completely stable.

Now there are assumptions underlying this and

we're gonna come back to those assumptions in just a moment.

Before we come back to that, let's change this into a more mathematical notation.

So let's formalize the math.

Now let's say these are typical notations people use.

Let's say the frequency of big A is referred to as p.

Also the frequency of little a, is referred to as q.

Again we're assuming there's only two alleles in this population.

Because there are only two types, p plus q must necessarily equal one.

Plus if p is one, then q must necessarily be zero.

You can't have negative numbers because these are frequencies.

What would the frequency of big A, big A be?

Well, it would be the probability of a big A encountering another big A.

So necessarily, it would be p times p or p squared.

Right, we can push this in P squared for big A big A.

Q squared for little a,little a.

2pq for big A, little a.

Why is it 2pq?

Why isn't it just pq?

Doesn't it just mean a big A and a little a?

Again there are two different ways you can have it.

You can have a big A, sperm and a little a, egg or a little a, sperm and

a big A egg so there's two different ways you can get it.

Again, these genotype frequencies must necessarily add up to one.

So P squared plus two p q plus q squared equals one.

You note that this quantity squared will come out to that.

Right? P plus Q squared is 2pq plus q squared.

So, it all comes out very elegantly.

Let me show you how the frequencies would look if you were to plot

these things together with these assumptions.