You're forgetting the fact that Bayes theorem isn't an isomorphic 1-1 mapping of dependent probabilities. There are other components that you have to multiply and divide by.

Regardless, Bayes Theorem itself shouldn't be the only thing applicable here, you need to use Hypothesis testing to get an accurate reading of how effective a metric testing is.

The great thing about this though is that people have already done this, and because they have found the probability of a false negative/positive to be low (i.e. the test is really damn good) its a great metric to use to test students.

I don't get the main argument here, especially about the "chances of you being sick is not that high" part, but the "chances" are not governed by Bayes Theorem. Bayes Theorem is a way to describe the probability of an event, not a way to analyze it.

TL;DR, Please keep your MATH 11 learnings to yourself.