A new medical drug to cure HIV/AIDS has just been developed. The probability that it will cure HIV/AIDS is 0.9 if the patient has HIV/AIDS. The probability that it will cure HIV/AIDS falls to 0.4 if the patient does not have the disease. There is a probability of 0.3 that a person chosen at random from the community has HIV/AIDS. Find the following: i. Probability that a patient is cured of HIV/AIDS [12 Marks) Probability that a person is cured of HIV/AIDS actually had HIV/AIDS. [8 Marks)
To find the probabilities, we can use the following information:
Let:
A = Event that a patient has HIV/AIDS
C = Event that a patient is cured of HIV/AIDS
Given:
P(A) = 0.3 (Probability that a person chosen at random has HIV/AIDS)
P(C|A) = 0.9 (Probability that a patient is cured of HIV/AIDS if they have the disease)
P(C|A') = 0.4 (Probability that a patient is cured of HIV/AIDS if they do not have the disease)
i. Probability that a patient is cured of HIV/AIDS:
To find this probability, we can use the law of total probability.
P(C) = P(C|A) * P(A) + P(C|A') * P(A')
Now, we can substitute the given values:
P(C) = 0.9 * 0.3 + 0.4 * (1 - 0.3)
= 0.27 + 0.4 * 0.7
= 0.27 + 0.28
= 0.55
Therefore, the probability that a patient is cured of HIV/AIDS is 0.55.
ii. Probability that a person who is cured of HIV/AIDS actually had HIV/AIDS:
To find this probability, we can use Bayes' theorem.
P(A|C) = (P(C|A) * P(A)) / P(C)
Now, we have already calculated P(C) as 0.55. Substituting the given values into Bayes' theorem:
P(A|C) = (0.9 * 0.3) / 0.55
= 0.27 / 0.55
≈ 0.491
Therefore, the probability that a person who is cured of HIV/AIDS actually had HIV/AIDS is approximately 0.491.