Hardy-Weinberg in dioecious organisms

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

p^{2 '}AA' 2pq 'AB' q²'BB'

p^{2 '}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

p^{2 '}AA' 2pq 'AB' q²'BB'

p^{2 '}AA' p⁴ AA p³q AA + p³q AB p²q²AB

2pq 'AB' p³q AA + p³q AB p²q²AA + 2p²q² AB + p²q²BB 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+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}

	p^{2 '}AA'	2pq 'AB'	q²'BB'
p^{2 '}AA'	p⁴	2p³q	p²q²
2pq 'AB'	2p³q	4p²q²	2pq³
q²'BB'	p²q²	2pq³	q⁴

	'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