Question

6. Imagine that Zika virus has a 1% incidence in the population. A test for the virus has a 3% false positive rate and no false negative rate. If someone takes the test and gets a positive result, what is the chance that they are infected? 7. Imagine that human papillomavirus has a 40% incidence among people 21-30 years of age. A test for the virus has a 50% false negative rate but no false positive rate. If you get a negative test for the virus, what's the chance that you are infected?

Answer 1

6) We will use Bayes' theorem to solve this problem.

According to Bayes' theorem, P (Ai | B) = where, i = 1,2,...,n

P(BAPA) %= P(B|Ak)P(AR)

Given-

Probability of disease, P(Disease) = 0.01 (As 1% incidence rate)

False negative rate = 0. So, probability everyone who has zika virus will test positive, P (Positive| Disease) = 1

False positive rate = 3% = 0.03; so, among who tests positive 3% will not have the disease. Thus, probability who test positive has no disease, P (Positive| No Disease) = 0.03

So, probability of not having disease, P (No disease) = 1 - P(Disease) = 1 - 0.01 = 0.99

So, probability that someone is infected after getting positive result, P (Disease| Positive) = = (0.01 x 1) / (0.01 x 1 + 0.99 x 0.03) = 0.25189 (Up to 5 decimals)

P(Disease P(Positive Disease) P(Disease) P(Positive Disease) + P(NoDisease) P(Positive No Disease)