They are dependent only when n=1 because, if E occurs then so does Fk, (k=1) and if E doesn't occur, then neither does Fk (k=0).
When n>1, E may or may not occur. If E occurs, Fk also occurs (1≤k≤n); but if E does not occur, Fk may occur (1≤k≤n-1). So Fk is independent of E. Also E is independent of Fk.