What is the probability that an individual has these symptoms and tests positively actually has COVID-19

Question

The probability that a person with certain symptoms has COVD-19 is 0.81. One test being used to confirm this diagnosis gives positive results for 93% of people with the virus and 3% of those without the virus. What is the probability that an individual has these symptoms and tests positively actually has COVID-19?

Answer 1

This is a classic Bayes' Law problem (and conditional probability). Let C represent COVID-19 and + represent a positive test. From the problem statement,

P(C) = 0.81

P(+ | C) = 0.93

P(+ | not C) = 0.03

P(C | +) = ?

Using Bayes' Law, P(C | +) = (P(+ | C) * P(C)) / P(+)

You can find P(+) using the law of total probability: P(+) = P(+ | C) * P(C) + P(+ | not C) * P(not C)

To find P(not C), you can use the fact that either people have COVID-19 or not. So, P(C) + P(not C) = 1.