My bathroom ceiling is plaster and the paint is peeling. I'm stripping off the old paint right now. There appears to be areas that were patched with plaster (?) on top of the origional paint/plaster. In some scattered areas the paint is peeling down to the origional plaster. Do I have to remove all of this down to the origional plaster or do I just need to remove where it's peeling? I've tried to remove down to the origional plaster but it's really difficult if not impossible.

Also, what kind of primer and paint should I use when this is all finished? I plan to install a ceiling fan before I paint the plaster. I'm sure that will help with peeling issues.