Algebraic derivation of Mendelian ratios
as binomial traits
Mendelian ratios may also be obtained from the calculation of the binomial
distribution of two independent traits. For two alleles at one
locus, the genotype ratios are
1A + 1a
1A + 1a
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1Aa + 1aa
1AA + 1 Aa
_________________
1AA + 2Aa + 1aa gives a 1 : 2 : 1 ratio
and the phenotype ratios are therefore (1AA + 2Aa) + 1 aa =
(1+2) A + 1 a = 3:1
For two alleles at two loci, the genotypic and phenotypic
ratios above give:
1AA + 2Aa + 1aa
1BB + 2Bb +
1bb
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+ 1AAbb + 2Aabb +
1aabb
+
2AABb + 4AaBb + 2aaBb
1AABB + 2AaBB + 1aaBB
= 1 : 2 : 1 : 2 : 4 : 2
: 1 : 2 : 1
for which the phenotypes are
1AB +
2AB + 1aB + 2AB
+ 4AB + 2aB + 1Ab
+ 2Ab + 1ab
rearranging
(1 + 2 + 2 + 4)AB + (1 + 2) aB + (1 + 2) Ab + 1 ab gives a 9
: 3 : 3 : 1 ratio
Alternatively,
3A + 1a
3B + 1b
____________________
3Ab + 1ab
9AB + 3aB
____________________
9AB + 3aB + 3Ab + 1ab