Current estimates are that somewhere between 1% and 5% of theU.S. population hasbeen infected with Covid-19. A serological testfor Covid-19 is supposed to determine if a personhas antibodies forthat virus, which would indicate a previous infection. Some claimthat if a persontests positive, then the person should be able toreturn to normal activity since they would haveimmunity (thoughwhether an infected person develops immunity seems to be currentlyunknown).One particular test has a “specificity†of 95.6%, whichmeans that the probability of a false positiveis 4.4%. The test hasa “sensitivity†of 93.8%, which means that the probability of afalse negativeis 6.2%. These numbers look impressive, and thenumbers for this test are much better than most.However, noticethat a false positive rate of around 4.4% is substantial comparedto the proportionof people infected (1% to 5%).(1) Suppose that 5%of the people have been infected. Given that the test comes backpositive fora randomly selected person, what is the probabilitythat that person has been infected withCovid-19?(2) Suppose that 1%of the people have been infected. Given that the test comes backpositive fora randomly selected person, what is the probabilitythat that person has been infected withCovid-19?(3) Even if weassume that a prior infection bestows immunity, is it reasonablefor a person receiv-ing a positive serological test to assume thatthey have immunity?
