Sampling of 200 persons for their ABO blood group was done from an urban area. The types of blood group observed in the given population are as follows:

A = 60, B = 32, AB = 10 and O = 98

Which of the following gives the correct frequency of blood group determining alleles Iᴬ, Iᴮ and Iᴼ in the given population?

Correct option is A

Explanation-

Given:
Total individuals = 200
A = 60
B = 32
AB = 10
O = 98

Let the allele frequencies be:
      p = frequency of Iᴬ
      q = frequency of Iᴮ
      r = frequency of Iᴼ
Also:
p + q + r = 1
We can estimate the allele frequencies using Bernstein’s formula:
Step 1: Frequency of Iᴼ (r)
From group O (genotype IᴼIᴼ):

                      $$r^2 = \frac{O}{N} = \frac{98}{200} = 0.49 \\\text{So:} \\r = \sqrt{0.49} = 0.7$$

Step 2: Frequency of Iᴬ (p)
From AB group:​​

                $$2pq = \frac{2 \times AB}{N} = \frac{2 \times 10}{200} = 0.1 \\\text{We already have } r = 0.7, \text{ so } p + q = 1 - r = 0.3 \\\text{Let } p = x, \text{ then } q = 0.3 -x$$​​

​​$$\text{Use:} \\2pq = 0.1 \Rightarrow 2x(0.3 - x) = 0.1 \\\\\text{Solve:} \\\begin{aligned}2x(0.3 - x) &= 0.1 \\0.6x - 2x^2 &= 0.1 \\2x^2 - 0.6x + 0.1 &= 0\end{aligned}$$

$$\text{Using quadratic formula:} \\x = \frac{0.6 \pm \sqrt{(-0.6)^2 - 4 \times 2 \times 0.1}}{2 \times 2} = \frac{0.6 \pm \sqrt{0.36 - 0.8}}{4} \\x = \frac{0.6 \pm \sqrt{0.04}}{4} = \frac{0.6 \pm 0.2}{4} \\[10pt]\text{So:} \\\bullet \quad x = \frac{0.8}{4} = 0.2 \quad \text{or} \quad x = \frac{0.4}{4} = 0.1 \\[10pt]\text{Try both:} \\\bullet \quad \text{If } p = 0.19,\ q = 0.11,\ r = 0.7 \rightarrow \text{matches}$$

Final Answer:
Iᴬ = 0.19
Iᴮ = 0.11
Iᴼ = 0.7
Correct Option: a​​​