# Hardy Weinberg principle

The principle is that sum total of all allelic frequencies is 1.

Individual frequencies for example can be named p,q.In a diploid cell , p and q represent the frequency of allele A and a respectively . The frequency of AA individuals in a population is p2 as the probability of an allele A with a frequency of p appear on both the chromosomes of a diploid individual is simply the product of the probabilities. Similarly for aa it will be q2.Can anybody explain that how it will be 2pq for Aa?

I am getting it as pq.

• This question is really more about probabilities than H-W. Basically, you can arrive at Aa in two ways; either getting A from the father and a from the mother (Aa), or the opposite (aA). The probability of each combination of Aa is pq and the total probablility of both (Aa and aA) is therefore 2pq. Jan 26, 2017 at 9:13
• @fileunderwater: oops, my explanation might be super misleading. Thanks for clarifying! Jan 26, 2017 at 9:16
• @AlexDeLarge The binomial expansion that you gave (if I remeber correctly) is also correct to apply. Jan 26, 2017 at 10:47

You almost said it all! Here are two ways to think of this problem

Finding the missing probability

Let's denote the fraction yz genotype in the population with $f(yz)$. The sum of of fraction of all genotypes must be equal to 1. In equation it means

$$f(aa) + f(aA) + f(AA) = 1$$

Here, of course I don't make a difference between $f(aA)$ and $f(Aa)$ (whether A is inherited from the mother or from the father). Knowing as you said

$$f(aa) = p^2$$

and

$$f(AA) = q^2$$

and feeding this value back to the first equation result into

$$p^2 + f(aA) + q^2 = 1$$

Solving for $f(aA)$ yields to

$$f(aA) = 2pq$$

Directly calculate the probability

Another to reach this result is to ask the following questions

What is the probability that the maternal allele is A?

The answer is $q$

What is the probability that the paternal allele is a?

The answer is $p$

What is the probability that the maternal allele is a?

The answer is $p$

What is the probability that the paternal allele is A?

The answer is $p$

Therefore, $$f(aA) = pq + pq = 2pq$$