I was listening to an interview the other day about COVID-19 testing and the likelihood of a "false negative" test result. If I heard the interview correctly, the numbers are rather startling.

Suppose that the likelihood for any one test is that detection of the virus has a 99% or 0.99 probability of being correct for a genuinely infected patient. That would mean a 1% chance of not detecting the virus in a genuinely infected patient, a 1% chance of a false negative test result for that genuinely infected patient.

Yes, I repeated that phrase over and over again for emphasis.

Now suppose you have __two__ genuinely infected patients whom you test. The probability of correctly detecting __both__ infections would be 0.99 x 0.99 = 0.99^{2} = 0.9801 which is a 98.01% probability of correctly detecting __both__ infections and which therefore means a 0.0199 probability, or 1.99% probability of at least one false negative test result.

Now suppose you have __three__ genuinely infected patients whom you test in the same way. The probability of correctly detecting __all three__ infections would be 0.99 x 0.99 x 0.99 = 0.99^{3} = 0.9703 which is a 97.03% probability of correctly detecting all three infections and which therefore means a 0.0297 probability, or 2.97% probability of at least one false negative test result.

Carry this calculation on to larger numbers of tests and you come to sixty-nine patients for whom 0.99^{69} = 0.4998 which means a false negative probability from someone in the group of 50.01%. As the tested group is made larger, the probability of getting at least one false negative test result gets closer and closer to absolute certainty of 100%.

For example, in testing one thousand infected people: (1 - .99^{1000}) = 0.99996 which is 99.996% probability of there being at least one false negative test result. If that one false negative person goes out into the community, that one person becomes a new vector of virus transmission to lots of other people.

If only one percent of a total population is infected, then testing one __hundred__ thousand people would yield the same probability of a false negative person going out there and doing harm. The present population of the Borough of Queens in New York City is approximately 2.2 million. Think about the implications of __that__ number.

The lesson we must learn is that a negative test result does not mean the test subject is free to go out there and not wear a mask and not maintain social distancing because that negative test result might be false. If your test result is a false negative and you violate the rules, you can do terrible harm.

To someone who says: "You don't have the right to make me wear a mask and take away my freedom.", I say the following:

"Your freedoms do __not__ give you the right to drive your car with an expired inspection and your freedoms do __not__ give you the right to not wear a mask and thereby risk taking away someone else's life."