“Suppose the COVID19 test is 97% accurate, and you’re in a room where everyone has just tested Negative. With that 97% confidence-level percentage, how many people would it take in the room to have a 50-50 chance that someone’s test is inaccurate?”
What an interesting question! It came from a participant in a pre-pandemic Math-is-Fun-type class for senior adults at MU in Columbia. I’ve edited it slightly to begin the process of making it more precise.
I was instantly intrigued by the question, partly due to its topical nature, and partly because it is a question that can conceivably be answered in a relatively straightforward manner, using techniques from mathematical probability. But I was further intrigued by the fact that like any real-world problem, the situation presents questions and conditions that need to be clarified before mathematics can help. (If the conditions get changed or redefined, so can the mathematical answer.)
Disclaimers: I am not a medical expert. I’m not sure if the 97% figure is technically ‘correct’, and I haven’t even devoted much effort to checking. (It’s not terribly relevant to our broader point, and I suppose various tests have various associated levels of accuracy.) Further, I’m not sure (and haven’t been able to pin down) what it means for this test to be ‘inaccurate’, that is to say a test yielding a ‘false negative’ result. This is important here, because I don’t think (recall disclaimer!) that a ‘false negative’ result is the same as a ‘positive’ result. It could simply mean ‘test fails’ for some reason.
And of course, we should note in passing that a ’50-50 chance’ is just exactly that. It certainly does not guarantee that someone in the room has a false-negative test.
Nonetheless, no matter how the questions above are answered, the mathematics is relatively straightforward. Not necessarily middle-school-easy, of course, but not difficult in a basic probability class or unit. For the reader’s sake, I won’t go through the mathematics here. But neither do I expect you to believe me blindly. I am happy to elaborate with anyone interested.
But I can give the results for the problem as stated in the opening paragraph. To be as precise as possible, if the test is 97% accurate, and if everyone in the room has tested negative, then it only takes 23 people in the room to have a better-than-even chance that at least one person’s test is a ‘false negative.’ With 30 people, the probability goes up to 60%.
That result is most interesting to me. It provides a new and perhaps more relatable perspective. It only takes 23 people in a room of negative tests to make it semi-likely (not certain!) that a test is inaccurate. Even if ‘false negative’ does not mean ‘positive’, it’s somewhat worrisome. If ‘false negative’ actually meant (or means) positive, then it’s getting serious.
(Interesting: if the test is 98% accurate, then it takes 35 people in the room, instead of only 23. At 99%, it jumps to 69 people.)
As always, what is the broader educational perspective here? I believe it is this. Maybe it’s a stretch for this particular problem, but Isn’t this roughly the general kind of ‘real world’ question/situation that we want our future graduates to be able to intelligently articulate, tackle, and define conditions for, regardless of their interest in mathematics? Once the question and conditions are made precise, a math type can be consulted if needed.
Regardless of their fields of interest, we say we want our future citizens to become better critical thinkers. These are the general kinds of situations that demonstrate that need.
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