As you may remember from basic biology, 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 type A blood; those with OO have type O blood. (A is "dominant", O is "recessive".) Suppose Apple and both of her parents have type A 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, given that she has blood type A? (b) Apple marries Oscar, who has type O blood. What is the probability that their first child will have type O blood? (c) If Apple and Oscar’s first child had type A blood, what is the probability that Apple carries an O gene? (d) If Apple and Oscar’s first child had type A blood, what is the probability that their second child will as well?

A is a dominant gene and O is a recessive gene, this means that for someone to be type A, the gene pairing could either be AA or AO, while a type O blood gene pairing would be OO

For Apple and her parents to have a type A blood, this means that they have either an AA or AO gene pair.

Since Apple's sister is a type O (which is an OO gene pair), this means that each of their parents must have an O gene in their pairs. Therefore, their parents must have the AO gene pair. This way, it is possible for Apple to have a type A blood and her sister to have a type O blood.

(a) Probability that Apple carries an O gene

Parents AO AO

Possible gene pairs of children AA AO AO OO

Since Apple is a type A, we have to consider all the possibilities where Apple carries an O gene while being a type A. These are the AO pairs

This is 2/4 =1/2

therefore, the probability is 1/2.

(b) Since Oscar is a type O, this means that his gene pair is OO

We have to consider two scenarios where Apple is either an AO or an AA

(i) Where Apple carries the AO gene pair

Parents AO OO

Possible gene pairs of children AO AO OO OO

Possibility of type O blood = 2/4 = 1/2

therefore, the probability is 1/2 if Apple carries the AO gene pair.

(ii) Where Apple carries the AA gene pair

Parents AA OO

Possible gene pairs of children AO AO AO AO

Possibility of type O blood = 0

therefore, the probability is 0 if Apple carries the AA gene pair.

Therefore, adding both probabilities = 1/2 + 0 = 1/2

(c) Considering both scenarios illustrated above, we have to look out for the possibilities where Oscar's first child would have a type A blood while Apple carries an O gene.

This can be found only in the first scenario where Apple carries the AO gene pair. Here, the probability that Apple carries an O gene 1/2.

In the second scenario where Apple carries an AA gene pair, the probability that Apple carries an O gene is 0.

Hence, the probability that Apple carries an O gene is 1/2 + 0 = 1/2

(d) The probability of gene pairing is the same for every pregnancy.

In the first scenario, the probability of having a type A child is 2/4 = 1/2

In the second scenario, the probability of having a type A child is 4/4 = 1

Therefore, the probability of having a second child who is type A is either 1/2 or 1