Sicklecell anemia is a hereditary medical condition affectin

Sickle-cell anemia is a hereditary medical condition affecting red blood cells that is thought to protect against malaria, a debilitating parasitic infection of the liver and blood. That would explain why the sickle-cell trait is found in people who originally came from Africa, where malaria is widespread. A study in Africa tested 543 children for the sickle-cell trait and also for malaria infection. In all, 25% of the children had sickle-cell and 6.6% of the children had both sickle-cell and malaria. Overall, 34.6% of the children had malaria.

(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.
Please Explain all steps

Solution

(a) P(sickle-cell) = 0.25
P(malaria) = 0.346
P(sickle-cell and malaria) = 0.066

P(malaria or sickle cell anemia) =P(malaria) + P(sickle cell) - P(malaria and sickle cell anemia)
= 0.346 + 0.25 - 0.066
= 0.53 (Ans.)

b) What is the probability that a given child has either malaria or sickle-cell?
Answer: .53
Approach #1
Pr [nEITHER malaria or sickle cell ] = 1 Pr[either malaria nor sickle cell]
= 1 - .53
= .47

c) Answer:
Malaria and sickle cell are independent if:
Pr[ malaria | sickle cell ] = Pr[malaria]
Solution for Pr[malaria | sickle cell ] = .066/.25 = .264
Solution for Pr[malaria] = .346
Since the two are not equal, the two events are NOT independent.

d)These data suggest that the presence of sickle cell trait confers some protection against malaria, as
the probability of malaria is reduced in the presence of sickle cell trait.