# Basic question regarding probability

I quote this question from "All of Statistics":

Suppose a gene can be type A or type a. There are three types of people (called genotypes): AA, Aa, and aa. Let (p, q, r) denote the fraction of people of each genotype. We assume that everyone contributes one of their two copies of the gene at random to their children. We also assume that mates are selected at random. The latter is not realistic however, it is often reasonable to assume that you do not choose your mate based on whether they are AA, Aa, or aa. (This would be false if the gene was for eye color and if people chose mates based on eye color.) Imagine if we pooled everyone’s genes together. The proportion of A genes is P = p + (q/2) and the proportion of a genes is Q = r + (q/2). A child is AA with probability P^2, aA with probability 2PQ, and aa with probability Q^2. Thus, the fraction of A genes in this generation is P^2 + PQ

This is probably a basic question. I'm asking: why is A = P^2 + PQ and not P^2 + 2PQ?

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Each person contributes two alleles to the gene population. AA individuals contribute 2 instances of A each, Aa individuals contribute 1, and aa individuals contribute none. So the proportion of A present must be $(2\cdot P^2 + 1\cdot 2PQ)/2$. –  cardinal Jun 17 '11 at 18:49
If $A$ is dominant and $a$ is recessive, then the proportion of people showing $A$ is $P^2+2PQ$, but not all of the genes they carry are $A$. –  Henry Jun 17 '11 at 19:22

For now, I'm going to assume that you understand everything up until the last sentence of your block quote, i.e. you understand why the probability of Aa is 2PQ. If I'm wrong, reply with a comment and I'll try to explain the earlier stuff.

• So we have a proportion (2PQ) of Aa individuals in the population. Each of those individuals has an A allele and an a allele. So half (one out of two) of their alleles is A. So the proportion of alleles that are both A and in an Aa individual is PQ (2PQ times one half).

• Meanwhile, a different proportion (P^2) of the individuals are AA. 100% (2/2) of their alleles are A. So the proportion of alleles that are in an AA individual and are A is P^2 (P^2 times one).

• No A alleles are found in the aa individuals (Q^2 times zero).

We can find the total proportion of A alleles in the population by adding those three, which gives us PQ from the mixed individuals, P^2 from the AA individuals, and 0 from the aa individuals.

Hope this helps.

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And "The proportion of A genes is P = p + (q/2)" has the same reason –  Theta30 Jun 17 '11 at 18:49