Assuming you have a test for disease X. The false positive rate of the test is 2%, the true positive rate is 100%. You know that it's a low prevalence disease, only 1% of the population has it. The country has a population of 10 million. The government implements a mass testing program in order to find out everyone who has it. To save resources, samples from 10 citizens are mixed together to be tested. If it's a positive result, these 10 people are tested again individually to figure out who actually has it. Given this testing system, how many true negative people will be falsely reported as positive cases?
My incomplete reasoning is as follows:
Let $-$ denote true negative, $+$ denote true positive, $*$ denote the test is positive. For the first 10 mixed test, the probability of person A being negative given the test is positive is: $$ \begin{align} p(- | *) &= \frac{p(-)p(* | -)}{p(*)} \\ &= \frac{p(-)p(* | -)}{p(*|-)p(-) + p(*|+)p(+)} \end{align} $$ We know $p(-) = 0.99, p(+) = 0.01$, $p(*|+) = 1$
I'm a bit stuck on $p(*|-)$ and how to proceed from here. I know $p(*|-)$ has to combine the false positive rate 0.02 and the probability of other 9 people being positive, but not sure how?