Derivation of Hardy-Weinberg Ratios for dioecious organisms
Let individuals with genotypes AA, AB, & BB be distributed as p2 : 2pq : q2

Let ratios in females & males be equal.

I. Fraction of total offspring contributed by each type of mating
 p2 'AA' 2pq 'AB' q2 'BB' p2 'AA' p4 2p3q p2q2 2pq 'AB' 2p3q 4p2q2 2pq3 q2 'BB' p2q2 2pq3 q4

II. Expected genotype ratios from each type of mating
 'AA' 'AB' 'BB' 'AA' all AA 1/2 AA : 1/2 AB all AB 'AB' 1/2 AA : 1/2 AB 1/4 AA : 1/2 AB : 1/4 BB 1/2 AB : 1/2 BB 'BB' all AB 1/2 AB : 1/2 BB all BB

III. Expected proportions of genotypes produced by each type of mating
 p2 'AA' 2pq 'AB' q2 'BB' p2 'AA' p4 AA p3q AA + p3q AB p2q2AB 2pq 'AB' p3q AA + p3q AB p2q2AA + 2p2q2 AB + p2q2 BB pq3 AB + pq3BB q2 'BB' p2q2AB pq3 AB + pq3BB q4 BB

Then, summing over genotypes

f(AA) = p4 +p3q + p3q + p2q2 = (p2)(p2 + 2pq + q2) = p2

f(AB) = p3q + p2q2 + p3q + 2p2q2 +p2q2 + pq3
= 2p3q + 4p2q2 + 2pq = (2pq)(p2 + 2pq + q2) = 2pq

f(BB) = p2q2 +pq+ pq3 +q4 = (q2)(p2 + 2pq + q2) = q2

Conclusion: random matings between dioecious organisms produce the same genotype ratios as random union of gametes in monoecious organisms

All text material © 2010 by Steven M. Carr