Human blood is generally classified in the "ABO" system, with four blood types: A, B, O, and AB. These four types reflect six gene pairs (genotypes), with blood type A corresponding to gene pairs AA and AO, blood type B corresponding to gene pairs BB and BO, blood type O corresponding to gene pair 00, and blood type AB corresponding to gene pair AB. Let p be the proportion of gene A in the population, q be the proportion of gene B in the population, and r be the proportion of gene O in the population. Note that p + q + r = 1. The Hardy-Weinberg principle states that p, q, and r are fixed from generation to generation, as are the frequencies of the different genotypes. Under this assumption, what is the probability that an individual has genotype AA? BB? OO? What is the probability of an individual having two different genes? Find the maximum percentage of the population that can have two different genes under the Hardy-Weinberg principle in two different ways, by directly maximizing a function of only two variables and by using the method of Lagrange multipliers. Can you say what the Lagrange multiplier represents in the above example?

Step-by-step explanation: According to the Hardy-weinberg principle states, in a population where there's no external factors changing the proportion of genes, the probability of frequencies would be calculate as (p+q+r)*(p+q+r)=1.

Calculating :

(p+q+r)²= p²+2pq+2pr+2qr+q²+r²

As p represents the frequency of genotype AA, the probability is P(A)=p²; as if for genotype BB is P(B)=q² and genotype OO is P(O)=r².

For the probability of two different genes, the answer is P(A,B,O)=2pq+2qr+2pr, since these represents the "mixture" of the genes.

The maximum percentage calculated by using Lagrange multipliers is demonstrated below:

F(p,q,r)= 2pq+2pr+2qr-λ(p,q,r)

Note: λ(p,q,r) are the Lagrange multipliers and F(p,q,r) is the function which relates the three frequencies.

We use partial derivatives:

Fp= 2q + 2r - λ

Fq=2p + 2r - λ

Fr=2p + 2q -λ

p + q + r = 1

Solving by λ and dividing by 2, we have

q + r = p + r = p + q

Using p+q+r=1 and substituting p by

p= 1 - q - r

we have q + r = 1 - q = 1 - r

From the first equation, we have r= 1 - 2q

Substituting into the 2nd equation, we have

1 - q = 1 - (1-q) =2q so q = 1/3

Resolving for the others, we have p=1/3 and r=1/3.

Calculating probability:

P(A,B,O)=2(1/3)(1/3)+2(1/3)(1/3)+2(1/3)(1/3)

P(A,B,O)=2/3

The Lagrange multipliers represents the external factors, for example mutations or migrations, capable of changing the frequency of the genes.