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!