Inbreeding theory and generation rates:

• Rate of inbreeding under regular full-sib mating:
  The recurrence relation is $F_t = \frac{1}{4} + \frac{1}{2} F_{t-1} + \frac{1}{4} F_{t-2}$.
  Generation 1: $F = 0.25$
  Generation 5: $F \approx 0.672$
  Generation 10: $F \approx 0.886$ (NOT $0.986$; $F$ exceeds $0.98$ only after $20\text{ generations}$). Therefore, Statement (I) is incorrect.
• Inbreeding coefficient ($F$): Formally introduced and defined by American population geneticist Sewall Wright in $1921\text{--}1922$ as the correlation between uniting gametes. Therefore, Statement (II) is correct.
  
  Option analysis:
  Statement (I) is incorrect but Statement (II) is correct, matching Option 4 (Key 4).

Hardy-Weinberg equilibrium genotype calculation:

• Let allele $A_1$ have frequency $p = 0.30$.
• For a biallelic locus, $p + q = 1$, where $q$ is the frequency of allele $A_2$.
• Therefore, $q = 1 - p = 1 - 0.30 = 0.70$.
• Under Hardy-Weinberg equilibrium, the expected frequency of homozygous $A_2A_2$ individuals in the progeny is:
  $f(A_2A_2) = q^2 = (0.70)^2 = 0.49$.
  
  Option analysis:
  Option 4 ($0.49$) is the exact homozygous genotype frequency $q^2$ (Key 4).

Carrier frequency among unaffected/normal individuals in population genetics:

• Given recessive allele frequency $q = 0.50$, dominant allele frequency $p = 1 - q = 0.50$.
• In genetic counseling and population genetics, when referring to a recessive genetic disorder, 'frequency of carriers' is evaluated among the phenotypically normal (non-affected) segment of the population, because affected individuals ($aa$) display the disease phenotype and are not merely healthy 'carriers':
  $\text{Carrier frequency among phenotypically normal} = \frac{2pq}{p^2 + 2pq} = \frac{2q}{1 + q}$.
• Calculation:
  $\frac{2 \times 0.50}{1 + 0.50} = \frac{1.00}{1.50} = \frac{2}{3} \approx 0.667 \approx 0.67$.
  
  Option analysis:
  Option 1 ($0.67$) is the exact proportion of carriers among phenotypically normal individuals (Key 1).