saying a test has 85% accuracy tells you nothing about the results of that test. not being hyperbolic - literally nothing. simple proof:
i develop a test for disease X and give it to 1 billion walking around grocery stores in the whole world. it is 85% accurate. results are 200 million people are positive? how many of those people actually have disease X? do you have a guess?
disease X i was testing for was death. every single positive test was wrong. how close was your guess?
you must, at a minimum, know test accuracy _as well as_ disease prevalence to form a statistical guess. death is prevalent in 0% of alive people, so test accuracy is worthless.
As long as we are fairly confident about the specificity and sensitivity of the test, then it DOES tell us something, even if those numbers are as low as 85%.
Here's the flaw in your proof - If the test has 85% specificity, then the chance of 200 million positives is essentially 0% (you'd get ~150 million if no one really has the disease). Getting a result of 200 million positive means either you DO have a significant number of true positives (50 million), or your 85% number for specificity was wrong.
Thanks for showing why Bayesian reasoning is superior to frequentist thinking.
(Yes, adding the disease prevalence % is essentially bayesian, but testing everybody isn't)
Even better than knowing the disease prevalence is knowing possible proxies (again, Bayesian).
Your Covid-19 test might have a low sensitivity but if the person has a temperature and other symptoms you might want to redo the test later even if it comes out negative.
“Precision and recall” are positive predictive value and sensitivity, respectively.