Sickle-cell anemia is an inherited disease in which red blood cells are misshapen and sticky. Sickle cells tend to form clumps in blood vessels, inhibiting the flow of blood. Humans have two genes for sickle-cell anemia, either of which may be S for normal cells or s for sickle cells. A person with two copies of the s gene will have sickle-cell anemia. A person with one s gene and one S gene will not have the disease, but will be a carrier, which means that the s gene may be transmitted to the person’s offspring. If two carriers have a child, the probability is 0.25 that the child will have the disease and 0.5 that the child will be a carrier. Outcomes among children are independent.

Question

a. A mother and father who are both carriers have two children. What is the probability that neither child has the disease?
b. What is the probability that both children are carriers?
c. If neither child has the disease, what is the probability that both are carriers?
d. A woman who is the child of two carriers has a child by a man who is a carrier. What is the probability that this child has the disease?

Answer 1

a) Each child has probability 0.25 of having the disease. Since the children are independent, the probability that both are disease-free is = 0.5625.



(b) Each child has probability 0.5 of being a carrier. Since the children are independent, the probability that both are carriers is = 0.25.



(c) Let D denote the event that a child has the disease, and let C denote the event that the child is a carrier. Then P(D) = 0.25, P(C) = 0.5, and = 0.75. We first compute , the probability that a child who does not have the disease is a carrier. First, = P(C) = 0.5. Now







Since children are independent, the probability that both children are carriers given that neither has the disease is = 4?9.



(d) Let denote the event that the woman has the disease, Let denote the event that the woman is a carrier, and let WF denote the event that the woman does not have the disease and is not a carrier. Then = 0.25, = 0.5, and = 0.25, Let denote the event that the child has the disease.