A genetic autoimmune disease is caused by a single recessive allele a that is located on the X-chromosome. The normal allele A is dominant hence, an individual with A does not develop the disease. However, an individual with alleles A and a is a carrier of the disease. Therefore, an individual with the genotype AA is healthy and not a carrier; an individual with the genotype Aa is only a carrier (i.e., the individual is healthy due to the dominant A, but can pass on the allele a to their offspring); and an individual with the genotype aa has the disease. The alleles A and a occur in the proportions p : q, with p + q = 1, in the population. It can be assumed that random mating has been the usual practice for many generations in this population. Assume that Alice and Bob with chromosome sets XX and XY, respectively, belong to this population. It is known that Bob has the autoimmune disease. However, it is not known whether Alice has the autoimmune disease or not. (a) Alice and Bob have a son, Luiz. Find the probability that Luiz does not have the autoimmune disease. (b) Alice and Bob have a daughter, Maria. Find the probability, in terms of p and q, that Maria (i) has two normal alleles AA, and (ii) is a carrier of the disease with alleles aA.

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