Question: A certain hereditary disease can be passed from a mother to her children. Given that the mother has the disease, her children independently will have it with probability 1/2. Given that she doesn't have the disease, her children won't have it either. A certain mother, who has probability 1/3 of having the disease, has two children.
(a) Find the probability that neither child has the disease.
(b) Is whether the elder child has the disease independent of whether the younger child has the disease? Explain.
(c) The elder child is found not to have the disease. A week later, the younger child is also found not to have the disease. Given this information, find the probability that the mother has the disease.