Probability problem involving multiple coronavirus tests in the same household

Mark Tuttle writes:

Here is a potential homework problem for your students.

The following is a true story.

Mid-December, we have a household with five people. My wife and myself, and three who arrived from elsewhere.

Subsequently, various diverse symptoms ensue – nothing too serious, but everyone is concerned, obviously. Video conference for all five of us with an Urgent Care doc who interviews each of regarding identity, onset, severity, etc. of symptoms. This leads to him scheduling virus tests for all of us. Doc says he’s seen too many cases where patients with symptoms postpone tests because they think it’s not serious, then it both gets serious and others are infected. No surprise there.

We all agree to self-quarantine if anyone gets a positive test. (Maybe we should have anyway, but that’s another story.)

The symptoms resolve themselves, and all five tests are negative.

In email follow-up from the doc (good for him) he observes that since all five tests are negative they probably do not include a false negative.

I recount this story – and the doc’s observation re reduced likelihood of false negative – to various smart folks, but no statistics/probability nerds, with fascinating results. It shows how hard it is for normal folks to deal with these concepts. One friend who heard the story consulted his quantitative expert who confirmed the doc’s observation.

I think it’s a wonderful example because we probably all had the same “bug”, but, apparently, not THE virus. Thus, if we all had THE virus, it does make it more difficult to get five (false) negatives. Again, the implication from the doc was that he had had to deal with false negatives, generally – apparently, they were frequent enough to be a concern (ignoring the technical challenge of confirming a false negative in real practice).

The test was some kind of PCR thing; I could get the exact name if that ever mattered.

So, the homework problem is to assign probabilities to the various inputs in a way that justifies the doc’s observation that false negatives were unlikely (or not), and unlikely vs. the false negative rate for a single test on a single random person with symptoms.