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
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    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
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                       3Ab + 1ab
9AB + 3aB
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9AB + 3aB + 3Ab + 1ab




All figure & text material ©2016 by Steven M. Carr