Or you could flip that around and make testing so plentiful everyone is tested 3x (perhaps by independent labs), so the FP rate is more like...

1/10 * 1/10 * 1/10 (1/1000)

This would require that false positives are truly random and not influenced by sample-specific factors. Is this the case? In antibody testing, I would imagine that false positives result from cross-reactivity instead of random chance.

Sure, there are certainly protocols that could be made that would almost ensure FP is due to sample factors.... or even test related factors that would maximize 3x FP (same lab, same lab tech, same machine, same day, back to back, same reagent stock).

We shouldn't adopt those protocols.

No good reason for test errors to be uncorrelated, ie. quite possible same people who trigger false positives on the first test would be false positives on the second etc.

Sure, there will probably be some correlation. Depends heavily on how the 3x testing is designed.

T1. Drive up saliva swab by nurse, for serology test.

T2. Re-deploy census peeps to get samples (and census info) from entire neighborhoods.

T3. 23andMe-like home kit express mail to lab

The odds of people getting a FP on all three might not be 1 in 1000, but it probably isn't 1 in 10.

Anyway, as it is, leading 1 in 10 people to falsely believe they are immune, isn't much better than having no test at all.

Assuming false positives aren't correlated even with independently manufactured tests seems like a terrible idea to me, knowing little about medicine.

Sounds like you had a bad experience once, with 'double checking'.

This assumes that the cause for false positive is in the test execution and not a contamination in the sample itself.