stages each with a probability of a false negative of 0.1 is indeed 0.01.
This is independent of what is done after the first test if it is positive, as
we are only considereing the results on subjects who do have the disease,
and test negative.
So we may think of it like this 100 diseased individuals come to the lab how
many (on avaerage) do we send home reassured? Well 10 of them as clear
on the first test, and of those 1 is clear on the secons and sent home
reassured, thus 1% or 0.01 probability of false negative on the two stage test.