Hardy's statement begins with a recurrence relation for the frequencies "p", "2q", and "r". These recurrence relations follow from fundamental concepts in probability, specifically independence, and conditional probability. For example, consider the probability of an offspring from the generation formula_30 being homozygous dominant. Alleles are inherited independently from each parent. A dominant allele can be inherited from a homozygous dominant parent with probability 1, or from a heterozygous parent with probability 0.5. To represent this reasoning in an equation, let formula_31 represent inheritance of a dominant allele from a parent. Furthermore, let formula_32 and formula_33 represent potential parental genotypes in the preceding generation.