Type B = 11.52%
Type AB = 3.84%
Since blood type is determined by the presence or absence of the A and B genes and neither gene is dominant, we have a 3x3 grid where the rows and columns are both labeled A,B,O and the resulting 9 intersections are labeled A, B, AB, and O. Since the only homozygous entry we have numbers for is the O box, let's determine the absolute percentage of O alleles.
The absolute percentage of the population that has an type O allele is the square root of the population exhibiting that allele. To illustrate this, imaging the classic 50/50 population of a dominant and recessive allele. In that situation, only 25% of the population exhibits the recessive trait. And the square root of 0.25 is 0.5. The same principle applies for any ratio of recessive vs dominate traits. Since we see 46.24% showing that, the number with a type O allele will be sqrt(0.4624) = 0.68 = 68%.
Now we need to figure out the overall percentage of the population that are homozygous A. The population of those exhibiting the type A trait are a mixture of homozygous and heterozygous. There are 3 ways to exhibit the A trait. Having a genotype of AA, AO, or OA. So we can create the following equation.
PA = A^2 + 2OA
where
PA = Percent showing the trait
A = absolute percentage having the A allele
O = absolute percentage having the O allele
Substitute known values and solve.
PA = A^2 + 2OA
0.384 = A^2 + 2*0.68*A
0.384 = A^2 + 1.36A
0 = A^2 + 1.36A - 0.384
We now have a quadratic equation with A=1, B=1.36, and C=-0.384. Use the quadratic formula to get the roots of -1.6 and 0.24. The root of -1.6 doesn't make sense for this problem, so the absolute percentage of A alleles is 0.24
Since the total of all the alleles has to add to 1, that leaves us with B being 1 - 0.24 - 0.68 = 0.08
So for the total population we have A=0.24, B=0.08, and O=0.68. With those numbers we can calculate the percentage of the population that exhibits any specified blood type.
Type B = B*B + B*O + O*B
= 0.08*0.08 + 0.08*0.68 + 0.68*0.08
= 0.0064 + 0.0544 + 0.0544
= 0.1152
= 11.52%
Type AB = A*B + B*A
= 0.24*0.08 + 0.08*0.24
= 0.0192 + 0.0192
= 0.0384
= 3.84%
And since they all have to add to 100%, let's check that.
38.40 + 46.24 + 11.52 + 3.84 = 100
