 That random mating will lead the genotypes to be in the above proportions (“Hardy-Weinberg proportions”) is a consequence of Mendel’s law of segregation. To see this, note that random mating is in effect equivalent to offspring being formed by randomly picking pairs of gametes from a large ‘gamete pool’ and fusing them into a zygote. The gamete pool contains all the successful gametes of the parent organisms. Since we are assuming the absence of selection, all parents contribute equal numbers of gametes to the pool. By the law of segregation, an \(A_1 A_2\) heterozygote produces gametes bearing the \(A_1\) and \(A_2\) alleles in equal proportion (on average). Therefore, the relative frequencies of the \(A\) and \(B\) alleles in the gamete pool will be the same as in the parental population, namely \(p\) and \(q\) respectively. Given that the gamete pool is very large, when we pick pairs of gametes from the pool at random, we will get the ordered genotypic pairs \(\{A_1 A_1\},\) \(\{A_1 A_2\},\) \(\{A_2 A_1\},\) \(\{A_2 A_2\}\) in the proportions \(p^2 :pq:qp:q^2\).  But order does not matter, so we can regard the \(\{A_1 A_2\}\) and \(\{A_2 A_1\}\) pairs as equivalent, giving the Hardy-Weinberg proportions for the unordered offspring genotypes.