In humans (and many other organisms), genes come in pairs. A certain gene comes in two types (alleles): type a and type A. The genotype of a person for that gene is the types of the two genes in the pair: AA, Aa, or aa (aA is equivalent to Aa). Assume that the Hardy-Weinberg law applies here, which means that the frequencies of AA, Aa, aa in the population are ,2p(1 p),respectively, for some p with 0 < p < 1. When a woman and a man have a child, the child's gene pair consists of one gene contributed by each parent. Suppose that the mother is equally likely to contribute either of the two genes in her gene pair, and likewise for the father, independently. Also suppose that the genotypes of the parents are independent of each other (with probabilities given by the Hardy-Weinberg law).

(a) Find the probabilities of each possible genotype (AA,Aa,aa) for a child of two random parents. Explain what this says about stability of the Hardy-Weinberg law from one generation to the next. Hint: Condition on the genotypes of the parents.

Would you please help me find the answer and explain? Thanks.

(b) A person of type AA or aa is called homozygous (for the gene under consideration), and a person of type Aa is called heterozygous (for that gene). Find the probability that a child is homozygous, given that both parents are homozygous. Also, nd the probability that a child is heterozygous, given that both parents are heterozygous.