Question

Sickle cell anemia is a hereditary medical condition that affects red blood cells and is thought to protect against malaria, a debilitating parasitic infection of the liver and the blood. This protection would explain why the sickle-cell trait is found in people who originally came from Africa, where malaria is widespread. A landmark study in Africa tested children for the sickle-cell trait and for malaria infection. In all, 30% of the children had the sickle-cell trait, and 8.5% of the children had both the sickle-cell trait and malaria. Overall, 34.5% of the children had malaria.

Answer 1

The answer is below

Step-by-step explanation:

The complete question is:

(a) What is the probability that a child has either malaria or sickle-cell? Please use 3 decimal places.

(b) What is the probability that a child has neither malaria nor sickle-cell? Please use 3 decimal places.

(c) What is the probability that a child has malaria given that the child has the sickle-cell trait? Please use 3 decimal places.

(d) What is the probability that a child has malaria given that the child does not have the sickle-cell trait? Please use 3 decimal places.

(e) Are the events sickle-cell trait and malaria independent? What might that tell you about the relationship between sickle-cell and malaria

Answer:

Let S represent sickle cell anemia and M represent malaria. Hence, Given that:

P(S) = 30% = 0.3, P(S ∩ M) = 8.5% = 0.085, P(M) = 34.5% = 0.345

a) P(that a child has either malaria or sickle-cell) = P(S ∪ M) = P(M) + P(S) - P(S ∩ M)

P(S ∪ M) = P(M) + P(S) - P(S ∩ M) = 0.345 + 0.3 - 0.085 = 0.56 = 56%

b) P(that a child has neither malaria nor sickle-cell) = P(S ∪ M)' = 1 - P(S ∪ M) = 1 - 0.56 = 0.44 = 44%

c) P(child has malaria given that the child has the sickle-cell trait) = P(M / S)

From conditional probability, P(M / S) = P(M ∩ S) / P(S) = 0.085/0.3 = 0.283 = 28.3%

d) P(S') = 1 - P(S) = 1 - 0.3 = 0.7

P(M ∩ S') = P(M) - P(S ∩ M) = 0.345 - 0.085 = 0.26

P(child has malaria given that the child does not have sickle-cell trait) = P(M / S')

From conditional probability, P(M / S') = P(M ∩ S') / P(S') = 0.26/0.7 = 0.371 = 37.1%

e) Two events are independent if the occurrence of one event does not affect the other event. Malaria and sickle cell traits are not independent events because there is evidence of a biological link between the two conditions