The human A/B/O blood type system is controlled by one gene for which 3 variants ("alleles") are common in the human population - unsurprisingly called A, B, and O. As with most genes, everyone has 2 copies of this gene, one inherited from the mother and the other from the father, and everyone passes a randomly selected copy to each of their children (probability 1/2 for each copy, independently for each child). Focusing only on A and O, people with AA or AO gene pairs have groupA blood; those with OO have type O blood. (A is "dominant", O is "recessive".) Suppose Apple and both of her parents have groupA blood, but her sister Olive has type O. Give exact answers as simplified fractions.

a) What is the probability that Apple carries an O gene?

b) Apple marries a man with type O blood. What is the probability that their first child will have type O blood?

c) If their first child had groupA blood, what is the probability that Apple carries an O gene?

d) If their first child had groupA blood, what is the probability that their second child will as well?