Question

Dr. Micheal Osterholm, director of the Center for Infectious Disease Research and Policy at the University of Minnesota, in an interview with Chuck Todd on Sunday May 10, claimed that some COVID-19 tests have 50% false positives. Assuming that the same test has 2% false negatives, calculate

(a) Probability that a person who tested positive is really infected with COVID-19 virus.

(b) Probability that a person who tested negative is not sick.

As of today, there are 1.37 million confirmed COVID-19 cases. Population of the United States is about 330 million.

Answer 1

We are given here the false positive rate of 50%, that is:
P( + | no COVID) = 0.5,
P( - | no COVID) = 1 - 0.5 = 0.5

The false negative rate here is given to be 2% which means that:
P( - | COVID) = 0.02,
P( + | COVID) = 1 - 0.02 = 0.98

The COVID rate itself is computed here as:
= n(COVID) / n(Total population)

= 1.37 / 330 = 0.004152

a) Using law of total probability, we have here:
P(+) = P( + | no COVID)P(no COVID) + P(+ | COVID)P(COVID)

= 0.5*(1 - 0.004152) + 0.98* 0.004152

= 0.5020

Now using Bayes theorem, we have here:
P( COVID | +) = P(+ | COVID)P(COVID) / P(+)

= 0.98* 0.004152/ 0.5020

= 0.0081

Therefore 0.0081 is the required probability here.

b) Probability that a person who tested negative is not sick is again computed in the same way.

P(-) = 1 - P(+) = 1 - 0.5020 = 0.4980

Using Bayes theorem,
P(no COVID | -) = P(- | no COVID)P(no COVID) / P(-)

= 0.5*(1 - 0.004152) / 0.4980

= 0.9998

Therefore 0.9998 is the required probability here.