# Simple question on probability computation for multiple comparisons

In class, we illustrated the multiple comparisons problem through a simple example (no, not the xkcd jelly beans comic). Essentially, with $$N$$ independent tests each at level $$\alpha$$, we said the probability of rejecting at least one true null is $$1-(1-\alpha)^N$$ which tends to 1 as $$N$$ gets large for fixed $$\alpha$$.

My question: doesn't this computation assume all $$N$$ nulls are correct? Aren't we interested in the probability of rejecting at least one true null, regardless of which ones are correct?

Update: On further thought, I think it was probably just an illustrative example under highly refined conditions e.g. independence and all true nulls. But I gather that methods to control familywise error rate such as Bonferroni and Holm do so under arbitrary dependence and under any number of true nulls. Any other thoughts are welcome!