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

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

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

f(BB) = p²q² +pq³  + 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}