Answer: The probability for A is p², for B is q² and for O is r².
The probability for two different genes is, in this case, 2pq+2qr+2pr.
The maximum percentage is 2/3.
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 <u><em>p</em></u> 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.
