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What if you consider external factors too? You can have a simple survey asking whether they have been traveled recently, whether they have symptoms, whether they are essential workers, if they know somebody that was infected, where they are living, with how many people, etc... You can easily use theses answers to remove the ones most likely to test postive with theses answers.

If you include that your goal is to test everyone too, I have no doubt that you can easily reach 96% negative test quite easily.

I remember in Quebec while we were only testing people with symptoms and that had traveled (or have been in contact with someone that has been tested positive), we were still way higher than 90% of negative tests (even right now we are a 90.1%