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 = 0.992 = 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 = 0.993 = 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 0.9969 = 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 - .991000) = 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."