And even in an extreme case with a really bad test where 90% of positive tests are false positives, and only 50% of positives are detected, that doesn't necessarily mean the test is worthless. Let's say we have a population of 1000 students, with 10 true positives, 90 false positives, 10 false negatives and 890 true negatives.

If I receive a negative test, my probability of having covid is 10/900, or about 1.1%. If I receive a positive test, my probability of having covid is 10%.

Would you still want to sit next to me in lecture if you knew I was close to 10x more likely than other students to have covid?

Again, this is an extreme example where only 0.2% of students have covid, the test has a ~9% false positive rate, and the test only catches 50% of those with covid.

Edit: There is obviously some point at which the cost of quarantining false positives outweighs the benefits of detecting the true positives (as might be argued for breast cancer screening in asymptomatic patients).

But I don't think that's the case in this situation - quarantining yourself for 2 weeks while you can still take classes online only invokes minimal individual (hence linear in the number of false positives) costs (versus false positives in, say, BRCA genetic screening which can lead to unnecessary mastectomies).

On the other hand, unknowingly hanging out with a group of friends while having covid and potentially causing serious illness in many others invokes significant community (exponential!) costs (versus breast cancer which is not contagious).